From 37ffea5ce7eef872906a87e3aa2fa16ebcc6c540 Mon Sep 17 00:00:00 2001 From: Aaron Caldwell Date: Thu, 19 Mar 2026 14:33:17 -0500 Subject: [PATCH 01/40] Updated --- R/htest_helpers.R | 2 +- papers/Avocado_Update/Avocado_Update.Rmd | 247 ++++++++++++++++---- papers/Avocado_Update/interactcadsample.bib | 120 ++++++++++ vignettes/references.bib | 6 + 4 files changed, 324 insertions(+), 51 deletions(-) diff --git a/R/htest_helpers.R b/R/htest_helpers.R index 239206e0..46847b86 100644 --- a/R/htest_helpers.R +++ b/R/htest_helpers.R @@ -443,7 +443,7 @@ printable_pval = function(pval, #' @title Plot Estimate from 'htest' Object #' #' @description -#' `r lifecycle::badge('stable')` +#' `r lifecycle::badge('maturing')` #' #' Creates a simple point estimate plot with confidence interval from any 'htest' object #' that contains an estimate and confidence interval. This provides a visual representation diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index e611c8fd..6303e7f1 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -4,8 +4,10 @@ title: | shorttitle: "Updated TOSTER R Package" author: - name: Aaron R. Caldwell - affil: a - email: arcaldwell49@gmail.com + affil: Department of Biostatistics, Fay W. Boozman College of Public Health, University of Arkansas for Medical Sciences Northwest + city: Springdale + state: AR + email: caldwellaaron@uams.edu type: PREPRINT output: papaja::apa6_docx: default @@ -66,7 +68,7 @@ library(patchwork) Researchers often erroneously declare that no statistical effect exists based on a single "non-significant" p-value [@blandaltman95]. In many of these cases, the data may corroborate the researcher's claim, but the interpretation of a null hypothesis significance test (NHST), wherein the lack of significance is considered evidence of "no effect", is nonetheless incorrect. In order to statistically test for whether there is practically no effect, researchers could use equivalence testing. -Equivalence testing is used when the goal of a statistical test is to demonstrate that the difference between two conditions is too small to be meaningful. For example, if a researcher wanted to test whether a new drug was no worse than a standard drug, the null hypothesis would be that the new drug is worse than the standard drug by more than a meaningful amount, and the alternative hypothesis would be that the difference between the two drugs is small enough to be meaningless. A very simple equivalence testing approach is the use of “two one-sided tests” (TOST) [@schuirmann1987]. +Equivalence testing is used when the goal of a statistical test is to demonstrate that the difference between two conditions is too small to be meaningful. For example, if a researcher wanted to test whether a new drug was no worse than a standard drug, the null hypothesis would be that the new drug is worse than the standard drug by more than a meaningful amount, and the alternative hypothesis would be that the difference between the two drugs is small enough to be meaningless. A very simple equivalence testing approach is the use of "two one-sided tests" (TOST) [@schuirmann1987]. The TOST procedure is a statistical test of whether a parameter (e.g., mean difference) is within a specified interval. The TOST procedure can be used to test the equivalence of two means, two proportions, two regression coefficients, and even two variances. An upper ($\Delta_U$) and lower ($\Delta_L$) equivalence bound is specified based on the smallest effect size of interest (SESOI). @@ -94,7 +96,7 @@ In this manuscript, I will detail the updates to the TOSTER package, and give so # TOST with t-tests -In an effort to make TOSTER more informative and easier to use, a new function `t_TOST` was created. This function operates very similarly to base R's `t.test` function, but performs 3 t-tests (one two-tailed and two one-tailed tests). In addition, this function has a generic method where two vectors can be supplied or a formula can be given (e.g.,`y ~ group`). This function also makes it easier to switch between types of t-tests. All three types (two sample, one sample, and paired samples) can be performed/calculated from the same function. Moreover, the output from this function is verbose, and should make the decisions derived from the function more informative and user-friendly. +In an effort to make TOSTER more informative and easier to use, the `t_TOST` function was created. This function operates very similarly to base R's `t.test` function, but performs 3 t-tests (one two-tailed and two one-tailed tests). In addition, this function has a generic method where two vectors can be supplied or a formula can be given (e.g.,`y ~ group`). This function also makes it easier to switch between types of t-tests. All three types (two sample, one sample, and paired samples) can be performed/calculated from the same function. Moreover, the output from this function is verbose, and should make the decisions derived from the function more informative and user-friendly. Also, `t_TOST` is not limited to equivalence tests. Minimal effects testing (MET) is possible. MET is useful for situations where the hypothesis is about a minimal effect and the *null hypothesis is equivalence* (see Figure 1) [@mazzolari2022myths]. @@ -379,35 +381,159 @@ res_tsum = tsum_TOST( res_tsum ``` +# Simplified Hypothesis Testing + +The `t_TOST` function, and related `*_TOST` functions, are designed to be comprehensive — they run three tests (one two-tailed and two one-sided), compute standardized effect sizes, and produce verbose output. For researchers who want the full picture, this is ideal. However, for users who *only* want to test a single hypothesis of equivalence (or minimal effects), the output can feel unnecessarily heavy. The three-test structure may be confusing when the researcher only cares about one of those tests. + +To address this, I created the `simple_htest` function as a lighter interface. It returns a standard R `htest` object — the same class returned by `t.test()` and `wilcox.test()` — but with expanded `alternative` options that include `"equivalence"` and `"minimal.effect"`. This means researchers can perform equivalence testing with the same familiar interface they already use for standard hypothesis tests. + +Alongside `simple_htest`, I also developed a set of helper functions for working with any `htest` object. These helpers make it easy to convert results to data frames (`df_htest`), generate (almost) publication-ready text descriptions (`describe_htest`), create plots (`plot_htest_est`), and bridge between the verbose TOST output and the simpler `htest` ecosystem (`as_htest`). Importantly, these tools work not only with TOSTER functions but with any base R hypothesis test. + +## simple_htest + +The `simple_htest` function supports `t.test` and `wilcox.test` as the underlying test (via the `test` argument) and all five alternative hypotheses: `"two.sided"`, `"less"`, `"greater"`, `"equivalence"`, and `"minimal.effect"`. The `mu` parameter works like `t.test` for standard alternatives (a single null value), but for equivalence and minimal effect testing a single value creates symmetric bounds (i.e., `mu = 0.5` becomes bounds at -0.5 and 0.5), or two values can be given for asymmetric bounds. + +For instance, we can perform an equivalence test on the `sleep` data with symmetric bounds of $\pm$ 0.5: + +```{r} +res_sh1 <- simple_htest(extra ~ group, data = sleep, + alternative = "equivalence", mu = 0.5) +res_sh1 +``` + +Notice how the output is a single clean `htest` object — much more concise than the verbose `t_TOST` output shown previously. + +Similarly, a minimal effect test on paired data from the `bugs` dataset can be performed: + +```{r} +res_sh2 <- simple_htest(x = bugs$LDHF, y = bugs$LDLF, + paired = TRUE, + alternative = "minimal.effect", mu = 1) +res_sh2 +``` + +The same interface works for non-parametric tests simply by changing the `test` argument: + +```{r} +res_sh3 <- simple_htest(extra ~ group, data = sleep, + test = "wilcox.test", + alternative = "equivalence", mu = 0.5) +res_sh3 +``` + +## Converting TOST Results with as_htest + +Users who have already run `t_TOST` or `wilcox_TOST` and want to use the `htest` helper functions can convert their results with `as_htest`. This extracts the TOST-relevant test and packages it as a standard `htest` object. + +```{r} +res1_htest <- as_htest(res1) +res1_htest +``` + +## Describing Results with describe_htest + +For writing up results, `describe_htest` generates a formatted text string following typical scientific reporting conventions. It includes the significance decision, test statistic, p-value, estimate, confidence interval, and the hypothesis being tested. This works with any `htest` object — from base R or from TOSTER. + +```{r} +describe_htest(simple_htest(extra ~ group, data = sleep)) +``` + +This also works with standard R tests. Note the difference from the difference between the output below from `t.test` and `simple_htest` above. The differences, despite both descriptions coming from the same underlying statistical tests, are caused by differences in what output is produced by each function (i.e., `simple_htest` has slight differences in output from `t.test`). + +```{r} +describe_htest(t.test(extra ~ group, data = sleep)) +``` + +## Tabulating Results with df_htest + +When combining results from multiple tests (e.g., running equivalence tests across several outcomes), it is useful to have results in a data frame. The `df_htest` function converts any `htest` object into a single-row data frame with standardized column names. Multiple results can then be combined with `rbind`. + +```{r} +rbind( + df_htest(res_sh1), + df_htest(res_sh2), + df_htest(res_sh3) +) +``` + +## Plotting Results with plot_htest_est + +A visual summary of the estimate and confidence interval can be produced with `plot_htest_est`. For equivalence and minimal effect tests, the equivalence bounds are shown as dashed vertical lines, making it easy to see whether the confidence interval falls within (or outside) the bounds. + +```{r shplot1, fig.width=6, fig.height=3.5, fig.cap = "Estimate plot for an equivalence test via simple_htest."} +plot_htest_est(res_sh1) +``` + +For standard tests, only a single null reference line (at zero) appears: + +```{r shplot2, fig.width=6, fig.height=3.5, fig.cap = "Estimate plot for a standard two-sided t-test."} +plot_htest_est(t.test(extra ~ group, data = sleep)) +``` + \newpage # Robust Methods for Equivalence Testing In some cases, the use of t-test may be less than ideal. Any serious violation to the assumptions of a t-test (e.g., normality or homoscedasticity) could greatly inflate the type 1 error rate of TOST. -Therefore, it may be useful to explore alternatives to the t-test for TOST that either do not have those assumptions or are robust to violating those assumptions. +Therefore, it may be useful to explore alternatives to the t-test for TOST that either do not have those assumptions or are robust to violations of those assumptions. + +The TOSTER package provides robust alternatives to the t-test organized into three categories. +First, there are *rank-based tests* — the Brunner-Munzel test (`brunner_munzel`) for stochastic equality and the Wilcoxon-Mann-Whitney TOST (`wilcox_TOST`) for location shift — which target entirely different estimands than means or medians. +Second, there are *resampling methods* — the bootstrap t-test (`boot_t_test`), permutation t-test (`perm_t_test`), and bootstrap TOST (`boot_t_TOST`) — which relax distributional assumptions while still testing hypotheses about means [@efron93]. +Third, there are *log-transformed methods* — `log_TOST` and `boot_log_TOST` — which are parametric and bootstrap approaches for ratio-scale data commonly used in pharmaceutical bioequivalence studies [@he2022]. +Where possible, the package now favors functions that return standard `htest` objects, which enforce a single hypothesis per test and integrate cleanly with the existing R ecosystem. + +## Rank-Based Tests -The TOSTER package currently provides 4 robust alternatives to the t-test for TOST. -First, there is the `wilcox_TOST` function which uses the Wilcoxon-Mann-Whitney (WMW) type tests (i.e., `wilcox.test`) to perform TOST as a test of symmetry. -Second, there is the `boot_t_TOST` function which uses the bootstrap method outlined by @efron93. -Third, there is the `log_TOST` function which performs log-transformed t-tests, which is a parametric approach commonly used in pharmaceutical bioequivalence studies on ratio data [@he2022]. -Fourth, there is the `boot_log_TOST` function which uses the same bootstrap method outlined by @efron93 but on the log-transformed data, which is more robust than parametric log t-test [@he2022]. +Rank-based tests avoid distributional assumptions by operating on the ranks of the data rather than the raw values. +An important consideration is that these tests are *not* tests of means or medians — they target completely different estimands. +The choice between rank-based methods should therefore be driven by the estimand of interest: stochastic equality (i.e., the probability that a random observation from one group exceeds a random observation from another) for the Brunner-Munzel test, or pseudo-median of pairwise differences (under a location-shift assumption) for WMW tests. -In the following sections, I will briefly outline the available robust TOST functions within the TOSTER package. +### Brunner-Munzel Test -## Tests of Symmetry (rank based tests) +The Brunner-Munzel test [@brunner2000] provides a non-parametric test of stochastic equality. +Rather than testing whether two groups differ in location, it directly estimates the relative effect $\hat{p} = P(X > Y) + 0.5 \cdot P(X = Y)$, which is the probability that a randomly chosen observation from one group exceeds a randomly chosen observation from the other. +This estimand is interpretable without additional distributional assumptions, unlike the WMW test which requires a location-shift assumption to say anything about location differences. +@karch2021 argues that the Brunner-Munzel test should be preferred over the Mann-Whitney U test as the default nonparametric procedure because it is valid under heteroscedasticity and provides a more interpretable effect size. -The WMW group of tests (e.g., Mann-Whitney U-test) provide a non-parametric test of differences between groups, or within samples, based on *ranks*. This provides a test of location shift, which is a fancy way of saying differences in the center of the distribution (i.e., in parametric tests the location is mean). Within the TOST framework, there are two separate tests of directional location shift to determine if the location shift is within (equivalence) or outside (minimal effect) the equivalence bounds. Many researchers mistakenly think these are tests of medians, but this is not the case (See @median_test for details). Using a WMW-based TOST is useful for testing whether the differences between groups/conditions is symmetric around the equivalence bounds^[Care should be taken when considering paired samples; a test on the rank transformed data [@kornbrot1990rank] or another robust test may be more prudent.]. For equivalence testing, the TOST would be testing whether there is asymmetry towards no effect with a null hypothesis of symmetry at the equivalence bound. +The `brunner_munzel` function in TOSTER supports equivalence and minimal effect testing via the `alternative` argument, with bounds specified on the probability scale through the `mu` parameter. +Results can be reported on different scales via the `scale` argument ("probability", "difference", "logodds", or "odds"), and three test methods are available via `test_method`: "t" (default, t-approximation), "logit" (range-preserving CIs), and "perm" (studentized permutation, recommended for small samples; @neubert2007). -In the TOSTER package, we accomplish this "test of symmetry" with the `wilcox_TOST` function. -This function operates in an extremely similar implementation to the `t_TOST` function. -The exact calculations utilized in this function can be explored via the documentation of the `wilcox.test` function. -A standardized mean difference (SMD) is *not* calculated in this function since this would be an inappropriate measure of effect size alongside the non-parametric test statistics. -Instead, a standardized effect size (SES) is calculated for *all* types of comparisons (e.g., two sample, one sample, and paired samples). -The function can produce a rank-biserial correlation [@Kerby_2014], a WMW Odds [@wmwodds], or a "common language effect size" [@Kerby_2014] (Also known as the non-parametric probability of superiority, or concordance probability).^[There is no plotting capability at this time for the output of this function.] +```{r} +bm_test = brunner_munzel(formula = extra ~ group, + data = sleep, + alternative = "equivalence", + mu = c(0.3, 0.7)) +print(bm_test) +``` \newpage +### Wilcoxon-Mann-Whitney TOST + +The WMW group of tests (e.g., Mann-Whitney U-test) provide a non-parametric test of differences between groups, or within samples, based on *ranks*. +Within the TOST framework, there are two separate tests of directional location shift to determine if the location shift is within (equivalence) or outside (minimal effect) the equivalence bounds^[Care should be taken when considering paired samples; a test on the rank transformed data [@kornbrot1990rank] or another robust test may be more prudent.]. + +However, there are important caveats to the interpretation of WMW tests. +Many researchers mistakenly believe these are tests of medians, but this is not the case. +Without additional assumptions, the WMW test is a test of stochastic equality — identical to what the Brunner-Munzel test estimates. +Only under a location-shift assumption (the two distributions have identical shapes, differing only in location) does the WMW test become a test of the Hodges-Lehmann pseudo-median of pairwise differences, and only with additional symmetry can it be interpreted as a test of median or mean differences. +@median_test demonstrate several striking counterexamples: groups can have equal medians yet produce a significant WMW test (because distributions differ in shape), very different medians yet a non-significant WMW test (because neither group stochastically dominates the other), or even significance in the opposite direction from the median comparison. + +This interpretive difficulty is especially pronounced for equivalence testing. +When performing TOST with Wilcoxon tests, the equivalence bounds are applied to the pseudo-median of pairwise differences (i.e., the Hodges-Lehmann estimator), which corresponds to the difference in means or medians only under the location shift assumption [@median_test]. +When distributions differ in shape, bounds on the pseudo-median do not guarantee that means or medians are within those bounds. +For testing hypotheses about mean differences, `perm_t_test` or `boot_t_test` are more appropriate. +For stochastic superiority, `brunner_munzel` provides a clearer framework. +For testing effect sizes directly, `ses_calc` supports hypothesis testing on the standardized effect size such as the rank-biserial correlation. +The `wilcox_TOST` function remains useful for situations where rank-based tests are specifically desired, such as with ordinal data or when consistency with prior analyses is needed. + +In the TOSTER package, `wilcox_TOST` operates similarly to `t_TOST`. +A standardized mean difference (SMD) is *not* calculated since this would be an inappropriate measure of effect size alongside the non-parametric test statistics. +Instead, a standardized effect size (SES) is calculated for *all* types of comparisons (e.g., two sample, one sample, and paired samples). +The function can produce a rank-biserial correlation [@Kerby_2014], a WMW Odds [@wmwodds], or a "common language effect size" [@Kerby_2014] (also known as the non-parametric probability of superiority, or concordance probability)^[While `wilcox_TOST` does not have a dedicated plotting method, the result can be converted to an `htest` object with `as_htest`, or users can obtain an `htest` object directly via `simple_htest`. The `ses_calc` function also returns an `htest` object. Any of these `htest` objects can be plotted with `plot_htest_est`.]. + As an example, we can use the sleep data to make a non-parametric comparison of equivalence. ```{r} @@ -418,41 +544,63 @@ test1 = wilcox_TOST(formula = extra ~ group, print(test1) ``` -Based on these results, we would have conclude there is no significant difference but not equivalent differences either (i.e., inconclusive result). +Based on these results, we would conclude there is no significant difference but not equivalent differences either (i.e., inconclusive result). \newpage -## Bootstrap TOST +## Resampling Methods -The bootstrap refers to resampling with replacement and can be used for statistical estimation and inference. Bootstrapping techniques are very useful because they are considered somewhat robust to the violations of assumptions for a simple t-test and provide better estimations of SMDs [@Kirby2013]. Therefore, I added a bootstrapping function, `boot_t_TOST`, to the package to provide another robust alternative to the `t_TOST` function. +When the goal is to test hypotheses about means or mean differences while relaxing distributional assumptions, resampling methods provide a powerful set of tools. +TOSTER provides two primary resampling approaches — bootstrap and permutation — each with distinct inferential logic. -In this function we provide a percentile bootstrap solution outlined by @efron93 (see chapter 16, page 220). The bootstrapped p-values are derived from the "studentized" version of a test of mean differences [@efron93]. Overall, the results should be similar to the results of `t_TOST`. **However**, for paired samples, the Cohen's d(rm) effect size *cannot* be calculated by this function. +The *bootstrap* [@efron93] resamples with replacement from the observed data to approximate the sampling distribution of the test statistic. +Bootstrap confidence intervals can be constructed using several methods (studentized, basic, percentile, or bias-corrected and accelerated), and the p-value computation is tied to the CI method via the CI inversion principle. With the way the package now works for bootstraps methods, a p-value is significant at level $\alpha$ if and only if the corresponding $1 - 2\alpha$ confidence interval excludes the null value. +This ensures consistency between reported CIs and p-values [@Thulin_2024]. -### Two Sample Algorithm +The *permutation* approach shuffles group labels (for two-sample tests) or flips signs (for one-sample/paired tests) to construct the null distribution directly. +When the number of possible permutations is small enough, exact p-values can be computed; otherwise, a randomized approximation is used with the "+1" correction to avoid zero p-values [@phipson2010]. +Studentized permutation tests [@janssen1997; @chung2013], which recompute the standard error for each permutation, provide valid inference even under heteroscedasticity *and* allow for accurate equivalence tests [@Arboretti_Pesarin_Salmaso_2020]. -The steps by which the bootstrapping occurs are fairly simple. +Both `boot_t_test` and `perm_t_test` return standard `htest` objects and support `alternative = "equivalence"` and `alternative = "minimal.effect"` for equivalence and minimal effect testing respectively. +Both also support Yuen's trimmed means via the `tr` parameter, providing additional robustness to outliers by trimming a specified fraction from each tail of the distribution before computing means and variances. -1. Form B bootstrap data sets from x* and y* wherein x* is sampled with replacement from $\tilde x_1,\tilde x_2, ... \tilde x_n$ and y* is sampled with replacement from $\tilde y_1,\tilde y_2, ... \tilde y_n$ +### Bootstrap t-test -2. t is then evaluated on each sample, but the mean of each sample (y or x) and the overall average (z) are subtracted from each (i.e., null distribution is formed) +The `boot_t_test` function provides a bootstrap t-test that returns a standard `htest` object. +The `boot_ci` argument controls the CI method ("stud", "basic", "perc", or "bca"), and as noted above, the p-value method follows the CI method to maintain consistency. +Equivalence bounds are specified via the `mu` argument (two values for equivalence/minimal effect testing). -$$ -t(z^{*b}) = \frac {(\bar x^*-\bar x - \bar z) - (\bar y^*-\bar y - \bar z)}{\sqrt {sd_y^*/n_y + sd_x^*/n_x}} -$$ +```{r} +set.seed(4522) +boot_t_test(formula = extra ~ group, + data = sleep, + alternative = "equivalence", + mu = c(-0.5, 0.5), + R = 999) +``` -3. An approximate p-value can then be calculated as the number of bootstrapped results greater than the observed t-statistic from the sample. +\newpage -$$ -p_{boot} = \frac {\#t(z^{*b}) \ge t_{sample}}{B} -$$ +### Permutation t-test -The same process is completed for the one sample case but with the one sample solution for the equation outlined by $t(z^{*b})$. The paired sample case in this bootstrap procedure is equivalent to the one sample solution because the test is based on the difference scores. +The `perm_t_test` function provides a permutation-based t-test. +When `R = NULL`, the function attempts to enumerate all possible permutations for an exact test; otherwise, it draws `R` random permutations. +By default, the test is studentized (`perm_se = TRUE`), meaning the standard error is recomputed for each permuted sample, which provides valid inference under heteroscedasticity [@janssen1997; @chung2013]. +For randomized permutations, the "+1" correction is applied by default to ensure proper type I error control [@phipson2010]. -\newpage +```{r} +set.seed(8812) +perm_t_test(formula = extra ~ group, + data = sleep, + alternative = "equivalence", + mu = c(-0.5, 0.5), + R = 999) +``` -### Example of Bootsrapping +### Bootstrap TOST -We can use the sleep data to see an example of the bootstrapped results. If you plot the bootstrap samples, it will show how the resampling via bootstrapping indicates the instability of Hedges' d(z). Just looking at the printed results you will notice some differences between confidence intervals from the bootstrapped result and the t-test. +For situations where both a two-tailed nil hypothesis test and an equivalence test are desired simultaneously, the `boot_t_TOST` function provides both in a single call, returning a `TOSTt` object. +This function uses the same bootstrap methods as `boot_t_test` and additionally provides bootstrapped SMD estimates [@Kirby2013]. ```{r} set.seed(891111) @@ -461,14 +609,12 @@ test1 = boot_t_TOST(formula = extra ~ group, paired = TRUE, eqb = .5, R = 999) - - print(test1) ``` \newpage -## Log TOST +## Log-Transformed Methods The natural logarithmic (log) transformation is often utilized to stabilize the variance of a measure, and it often provides the best approximation of the normal distribution [@logtest]. However, another, less often reported, advantage of the log transformation is that the back transformation of the differences of the log-transformed data is a *ratio* [@logtest]. For example, if we had a two samples (x & y) with an geometric mean^[The mean of log-transformed data is the *geometric* not *arithmetic* mean. I highly recommend reading @logtest and @caldwell2019basic for more details] or 7 and 10.5, x and y respectively in the code below, we could represent the differences as ratio of y:x where y is 1.5 times greater than x. @@ -486,11 +632,13 @@ In pharmaceutical reserach, bioequivalence testing involves determining whether In my personal experience as a physiologist, it is not uncommon that biological/physiological phenomenon present have longer right-tailed distributions, and are often adequately normalized with a natural log transformation. The additional advantage is the how equivalence bounds can, almost, be universally applied when making comparisons on the log scale. The FDA considers to drugs to be bioequivalent when the maximal concentration and AUC differences between drugs are less than 1.25. To put it another way, ratio between two means must be between 1.25 and 0.8 (i.e., 1/1.25) [@fda]. +An alternative to using the dedicated `log_TOST` function is to simply log-transform the data and analyze it with `t_TOST` (or any of the resampling methods described above). The differences on the log scale can then be interpreted as "sympercents" — symmetric percentage differences on the $100 \cdot \ln$ scale [@cole2000]. Sympercents have the attractive property that a difference of +10 sympercent and -10 sympercent are symmetric and of equal magnitude, unlike ordinary percentages where +10% and -10% are not. This approach provides more flexibility (e.g., one-sample tests become possible) at the cost of requiring the user to specify equivalence bounds on the log scale rather than as a ratio. + Therefore, I have implemented two functions to allow for the comparison of data that is believed to be left skewed (long right tail), and is on a ratio scale^[Ratio scale means the outcome is measured on a numerical scale that has equal distances between adjacent values and true zero. ]. The first function is a parametric t-test on the log transformed scale while the second function is a bootstrapping test which is more robust than parametric version [@he2022]. ### Example of Log TOST -The `log_TOST` function is almost exactly the same as the `t_TOST` function. First, the primary differences is that it only accepts paired and two sample comparisons. One sample tests are not support (i.e., there is no ratio to calculate). Second, standardized mean differences are not calculated, but a ratio of means is instead reported [@lajeunesse2015bias]^[Also, referred to as a "response ratio" in ecology. Like an SMD, the response ratio can be utilized in meta-analysis.]. Third, the default equivalence bounds are by default set to the FDA standards (i.e., `eqb = 1.25`), but can be changed by the user^[Only one value needs to be supplied to eqb; the reciprocal value of eqb is taken as the other equivalence bound. For example, if `eqb = 0.85` then the upper equivalence bound is 1/0.85 (~1.333)]. +The `log_TOST` function is almost exactly the same as the `t_TOST` function. First, the primary differences is that it only accepts paired and two sample comparisons. One sample tests are not support (i.e., there is no ratio to calculate). Second, standardized mean differences are not calculated, but a ratio of means is instead reported [@lajeunesse2015bias]^[Also, referred to as a "response ratio" in ecology. Like an SMD, the response ratio is often utilized in meta-analysis.]. Third, the default equivalence bounds are by default set to the FDA standards (i.e., `eqb = 1.25`), but can be changed by the user^[Only one value needs to be supplied to eqb; the reciprocal value of eqb is taken as the other equivalence bound. For example, if `eqb = 0.85` then the upper equivalence bound is 1/0.85 (~1.333)]. As an example we can use the `mtcars` data to compare the type of transmission (`am`) effects on the gas mileage (`mpg`). We can see from the data below there are significant, non-equivalent, differences in mpg between transmission types. @@ -502,7 +650,7 @@ log_TOST(mpg ~ am, data = mtcars) ### Example of Bootstrap Log TOST -The bootstrap version of `log_TOST`, `boot_log_TOST`, uses the same bootstrapping method detailed above (`boot_t_TOST`), but it uses the log-transformed values and produces the ratio of means as the effect size. +The bootstrap version of `log_TOST`, `boot_log_TOST`, uses the same bootstrap methods as `boot_t_test`, but on the log-transformed values, producing the ratio of means as the effect size. ```{r, error=FALSE} boot_log_TOST(mpg ~ am, data = mtcars, R=999) @@ -648,7 +796,7 @@ The above results are only based on an approximating the differences between the SMDs. If the raw data is available, then the optimal solution is the bootstrap. This can be accomplished with the `boot_compare_smd` function. The only drawback to this function is that TOST is -currently not avaiable, and users would instead have to run 2 one-sided tests +currently not available, and users would instead have to run 2 one-sided tests manually using the `null` and `alternative` arguments. For this example, we will simulate some data. As an alternative approach to TOST, @@ -666,15 +814,14 @@ boot_test # Conclusions -In this manuscript I have demonstrated most of the new functions and features within the TOSTER R package. -This constitutes a major update to the package over the past 2 years. +In this manuscript I have demonstrated most of the newest functions and features within the TOSTER R package. I hope that updates to the package builds upon the original impact of the TOSTER package^[In my opinion, the impact of the @lakens_ori cannot be overstated considering it is cited by over 1000 other papers!], and has been made TOST more accessible to the average researcher. In addition, I have added a number of other functions that offer robust alternatives to the t-test for performing TOST analyses. I would strongly recommend users of TOSTER to explore these functions, and, at the very least, compare the robust results to the t-test results to ensure that the conclusions do not change -due to the chosen analysis^[If they do change, then it would be prudent to explore what features in the data might explain this discrepancy.]. +due to the chosen analysis. If your interpretation of the results substantially changes depending on the test, then it would be prudent to explore what features in the data, and the underlying data generating process, might explain this discrepancy. Lastly, to my knowledge, this is the first package to offer equivalence testing options for ANOVAs or for comparing SMDs between studies. Overall, this package and its functions offer an easily accessible option for researchers to explore equivalence testing, and hopefully improve their statistical analyses. @@ -682,11 +829,11 @@ Overall, this package and its functions offer an easily accessible option for re # Additional Information -All analyses/code in this manuscript are from TOSTER v0.6.0: +All analyses/code in this manuscript are from TOSTER v0.9.0: ``` # Install the exact release with this code -devtools::install_github("Lakens/TOSTER@v0.6.0") +devtools::install_github("Lakens/TOSTER@v0.9.0") ``` ## Acknowledgement(s) {-} @@ -706,7 +853,7 @@ No funding was provided for this work. Daniel Lakens provided a review of many of the materials that have been incorporated into the update of TOSTER, and was the original author of this package. -Without his help and encouragment, the TOSTER package and this update would not exist. +Without his help and encouragement, the TOSTER package and this update would not exist. ## Nomenclature/Notation {-} diff --git a/papers/Avocado_Update/interactcadsample.bib b/papers/Avocado_Update/interactcadsample.bib index aa8293bc..122b3f90 100644 --- a/papers/Avocado_Update/interactcadsample.bib +++ b/papers/Avocado_Update/interactcadsample.bib @@ -243,6 +243,10 @@ @article{repSES doi = {10.1007/s40279-022-01749-1} } +@misc{Thulin_2024, title={Modern Statistics with R}, url={http://dx.doi.org/10.1201/9781003401339}, +DOI={10.1201/9781003401339}, publisher={Chapman and Hall/CRC}, +author={Thulin, Måns}, year={2024}, month=june, language={en} } + @article{median_test, title={The Wilcoxon--Mann--Whitney procedure fails as a test of medians}, author={Divine, George W and Norton, H James and Bar{\'o}n, Anna E and Juarez-Colunga, Elizabeth}, @@ -267,6 +271,12 @@ @article{ doi = {10.1177/09622802221093721} } +@article{Arboretti_Pesarin_Salmaso_2020, title={A unified approach to permutation testing for equivalence}, volume={30}, +url={http://dx.doi.org/10.1007/s10260-020-00548-0}, DOI={10.1007/s10260-020-00548-0}, +abstractNote={AbstractThe notion of testing for equivalence of two treatments is widely used in clinical trials, pharmaceutical experiments, bioequivalence and quality control. It is traditionally operated within the intersection–union principle (IU). According to this principle the null hypothesis is stated as the set of effects the differences $$delta$$ δ of which lie outside a suitable equivalence interval and the alternative as the set of $$delta$$ δ that lie inside it. In the literature related solutions are essentially based on likelihood techniques, which in turn are rather difficult to deal with. A recently published paper goes beyond most of likelihood limitations by using the IU approach within the permutation theory. One more paper, based on Roy’s union–intersection principle (UI) within the permutation theory, goes beyond some limitations of traditional two-sided tests. Such UI approach, effectively a mirror image of IU, assumes a null hypothesis where $$delta$$ δ lies inside the equivalence interval and an alternative where it lies outside. Since testing for equivalence can rationally be analyzed by both principles but, as the two differ in terms of the mirror-like roles assigned to the hypotheses under study, they are not strictly comparable. The present paper’s main goal is to look into these problems and provide a sort of comparative analysis of both by highlighting the related requirements, properties, limitations, difficulties, and pitfalls so as to get practitioners properly acquainted with their correct use in practical contexts.}, number={3}, journal={Statistical Methods & Applications}, publisher={Springer Science and Business Media LLC}, +author={Arboretti, Rosa and Pesarin, Fortunato and Salmaso, Luigi}, +year={2020}, month=nov, pages={1033–1052}, language={en} } + @article{wmwodds, title={Exploiting the link between the Wilcoxon-Mann-Whitney test and a simple odds statistic}, author={O’Brien, Ralph G and Castelloe, John}, @@ -277,6 +287,13 @@ @article{wmwodds organization={Citeseer} } +@article{Bauer_1972, title={Constructing Confidence Sets Using Rank Statistics}, volume={67}, +url={http://dx.doi.org/10.1080/01621459.1972.10481279}, +DOI={10.1080/01621459.1972.10481279}, number={339}, +journal={Journal of the American Statistical Association}, +publisher={Informa UK Limited}, author={Bauer, David F.}, year={1972}, +month=sept, pages={687-690}, language={en} } + @article{logtest, title={Statistics notes: the use of transformation when comparing two means}, author={Bland, J Martin and Altman, Douglas G}, @@ -390,3 +407,106 @@ @article{rafi2020 publisher={Springer} } +@article{brunner2000, + title={The nonparametric {Behrens-Fisher} problem: Asymptotic theory and a small-sample approximation}, + author={Brunner, Edgar and Munzel, Ullrich}, + journal={Biometrical Journal}, + volume={42}, + number={1}, + pages={17--25}, + year={2000}, + publisher={Wiley}, + doi={10.1002/(SICI)1521-4036(200001)42:1<17::AID-BIMJ17>3.0.CO;2-U}, + url={https://doi.org/10.1002/(SICI)1521-4036(200001)42:1<17::AID-BIMJ17>3.0.CO;2-U} +} + +@article{neubert2007, + title={A studentized permutation test for the non-parametric {Behrens-Fisher} problem}, + author={Neubert, Karin and Brunner, Edgar}, + journal={Computational Statistics \& Data Analysis}, + volume={51}, + number={10}, + pages={5192--5204}, + year={2007}, + publisher={Elsevier}, + doi={10.1016/j.csda.2006.05.024}, + url={https://doi.org/10.1016/j.csda.2006.05.024} +} + +@article{karch2021, + title={Psychologists should use {Brunner-Munzel's} instead of {Mann-Whitney's} {U} test as the default nonparametric procedure}, + author={Karch, Julian D}, + journal={Advances in Methods and Practices in Psychological Science}, + volume={4}, + number={2}, + pages={2515245921999602}, + year={2021}, + publisher={SAGE Publications}, + doi={10.1177/2515245921999602}, + url={https://doi.org/10.1177/2515245921999602} +} + +@article{munzel2003, + title={A nonparametric test for proving noninferiority in clinical trials with ordered categorical data}, + author={Munzel, Ullrich and Hauschke, Dieter}, + journal={Pharmaceutical Statistics}, + volume={2}, + number={1}, + pages={31--37}, + year={2003}, + publisher={Wiley}, + doi={10.1002/pst.59}, + url={https://doi.org/10.1002/pst.59} +} + +@article{phipson2010, + title={Permutation p-values should never be zero: Calculating exact p-values when permutations are randomly drawn}, + author={Phipson, Belinda and Smyth, Gordon K}, + journal={Statistical Applications in Genetics and Molecular Biology}, + volume={9}, + number={1}, + year={2010}, + publisher={De Gruyter}, + doi={10.2202/1544-6115.1585}, + url={https://doi.org/10.2202/1544-6115.1585} +} + +@article{chung2013, + title={Exact and asymptotically robust permutation tests}, + author={Chung, EunYi and Romano, Joseph P}, + journal={The Annals of Statistics}, + volume={41}, + number={2}, + pages={484--507}, + year={2013}, + publisher={Institute of Mathematical Statistics}, + doi={10.1214/13-AOS1090}, + url={https://doi.org/10.1214/13-AOS1090} +} + +@article{janssen1997, + title={Studentized permutation tests for non-i.i.d.\ hypotheses and the generalized {Behrens-Fisher} problem}, + author={Janssen, Arnold}, + journal={Statistics \& Probability Letters}, + volume={36}, + number={1}, + pages={9--21}, + year={1997}, + publisher={Elsevier}, + doi={10.1016/S0167-7152(97)00043-6}, + url={https://doi.org/10.1016/S0167-7152(97)00043-6} +} + +@article{cole2000, + title={Sympercents: Symmetric percentage differences on the 100 log\_e scale simplify the presentation of log transformed data}, + author={Cole, TJ}, + journal={Statistics in Medicine}, + volume={19}, + number={22}, + pages={3109--3125}, + year={2000}, + publisher={Wiley}, + doi={10.1002/1097-0258(20001130)19:22<3109::AID-SIM558>3.0.CO;2-F}, + url={https://doi.org/10.1002/1097-0258(20001130)19:22<3109::AID-SIM558>3.0.CO;2-F} +} + diff --git a/vignettes/references.bib b/vignettes/references.bib index 8ae8637e..fb39d36a 100644 --- a/vignettes/references.bib +++ b/vignettes/references.bib @@ -450,3 +450,9 @@ @article{arboretti2021 url={https://doi.org/10.1002/sim.9015} } +@article{Arboretti_Pesarin_Salmaso_2020, title={A unified approach to permutation testing for equivalence}, volume={30}, +url={http://dx.doi.org/10.1007/s10260-020-00548-0}, DOI={10.1007/s10260-020-00548-0}, +abstractNote={AbstractThe notion of testing for equivalence of two treatments is widely used in clinical trials, pharmaceutical experiments, bioequivalence and quality control. It is traditionally operated within the intersection–union principle (IU). According to this principle the null hypothesis is stated as the set of effects the differences $$delta$$ δ of which lie outside a suitable equivalence interval and the alternative as the set of $$delta$$ δ that lie inside it. In the literature related solutions are essentially based on likelihood techniques, which in turn are rather difficult to deal with. A recently published paper goes beyond most of likelihood limitations by using the IU approach within the permutation theory. One more paper, based on Roy’s union–intersection principle (UI) within the permutation theory, goes beyond some limitations of traditional two-sided tests. Such UI approach, effectively a mirror image of IU, assumes a null hypothesis where $$delta$$ δ lies inside the equivalence interval and an alternative where it lies outside. Since testing for equivalence can rationally be analyzed by both principles but, as the two differ in terms of the mirror-like roles assigned to the hypotheses under study, they are not strictly comparable. The present paper’s main goal is to look into these problems and provide a sort of comparative analysis of both by highlighting the related requirements, properties, limitations, difficulties, and pitfalls so as to get practitioners properly acquainted with their correct use in practical contexts.}, number={3}, journal={Statistical Methods & Applications}, publisher={Springer Science and Business Media LLC}, +author={Arboretti, Rosa and Pesarin, Fortunato and Salmaso, Luigi}, +year={2020}, month=nov, pages={1033–1052}, language={en} } + From 67646a9e07ca60572741e33f1160e0ecfd8c5c0c Mon Sep 17 00:00:00 2001 From: Aaron Caldwell Date: Thu, 19 Mar 2026 15:25:48 -0500 Subject: [PATCH 02/40] Update Avocado_Update.Rmd --- papers/Avocado_Update/Avocado_Update.Rmd | 10 ++++------ 1 file changed, 4 insertions(+), 6 deletions(-) diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index 6303e7f1..ebeae3b7 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -4,9 +4,7 @@ title: | shorttitle: "Updated TOSTER R Package" author: - name: Aaron R. Caldwell - affil: Department of Biostatistics, Fay W. Boozman College of Public Health, University of Arkansas for Medical Sciences Northwest - city: Springdale - state: AR + affil: a email: caldwellaaron@uams.edu type: PREPRINT output: @@ -19,12 +17,12 @@ abstract: | Equivalence testing is arguably under utilized by experimental researchers. Due to limited software support for such analyses, and little education on the topic in graduate programs, the utilization of - equivalence testings still appares to be low. + equivalence testings still appears to be low. One option for equivalence testing is the use of two one-sided tests (TOST). The TOSTER R package and jamovi module, originally developed by Daniel Lakens in 2017, was created to make TOST more accessible to the average researcher. - In the past two years, I have made significant changes to the TOSTER package + In recent years, I have made significant changes to the TOSTER package in order to increase its accessibility and provide more robust analysis options for researchers. In this paper, I will detail the changes to the package and highlight new analysis options that will make TOST easier for the @@ -45,7 +43,7 @@ header-includes: | affiliation: - num: a address: | - Natick, MA, + Department of Biostatistics, Fay W. Boozman College of Public Health, University of Arkansas for Medical Sciences Northwest, Springdale, AR --- ```{r setup, echo=FALSE} From b42be01175efe26e36b8e819b3315474ff1920a5 Mon Sep 17 00:00:00 2001 From: Aaron Caldwell Date: Thu, 26 Mar 2026 15:07:25 -0500 Subject: [PATCH 03/40] Main fixes made Now need to put meat on the bones --- .claude/settings.local.json | 7 +- papers/Avocado_Update/Avocado_Update.R | 142 +++-- papers/Avocado_Update/Avocado_Update.Rmd | 580 ++++++------------ papers/Avocado_Update/Avocado_Update.docx | Bin 494825 -> 100616 bytes .../figure-docx/cdplot-1.pdf | Bin 19010 -> 18730 bytes .../figure-docx/cdplot-1.png | Bin 135677 -> 27237 bytes .../figure-docx/hypplot-1.pdf | Bin 4770 -> 4622 bytes .../figure-docx/hypplot-1.png | Bin 80968 -> 20487 bytes .../figure-docx/shplot1-1.pdf | Bin 0 -> 4660 bytes .../figure-docx/shplot1-1.png | Bin 0 -> 20562 bytes papers/Avocado_Update/interactcadsample.bib | 56 ++ 11 files changed, 319 insertions(+), 466 deletions(-) create mode 100644 papers/Avocado_Update/Avocado_Update_files/figure-docx/shplot1-1.pdf create mode 100644 papers/Avocado_Update/Avocado_Update_files/figure-docx/shplot1-1.png diff --git a/.claude/settings.local.json b/.claude/settings.local.json index c7c82dfb..a86f3bc0 100644 --- a/.claude/settings.local.json +++ b/.claude/settings.local.json @@ -128,7 +128,12 @@ "Bash(ls:*)", "Bash(\"C:/Program Files/R/R-4.5.1/bin/Rscript.exe\" -e \"git diff HEAD -- tests/testthat/test-smd_calc_htest.R\")", "Bash(\"C:/Program Files/R/R-4.5.1/bin/Rscript.exe\" junk/diag_bca.R)", - "Bash(git -C /Users/aaroncaldwell/Documents/GitHub/TOSTER check-ignore vignettes/robustTOST.html)" + "Bash(git -C /Users/aaroncaldwell/Documents/GitHub/TOSTER check-ignore vignettes/robustTOST.html)", + "Read(//c/Program Files/R//**)", + "Read(//c/Program Files/R/**)", + "Read(//c/Users/caldwellaaron/AppData/Local/Programs/R//**)", + "Bash(\"C:/Program Files/R/R-4.5.1/bin/Rscript.exe\" -e \":*)", + "Bash(\"C:/Program Files/R/R-4.4.1/bin/Rscript.exe\" -e \":*)" ] } } diff --git a/papers/Avocado_Update/Avocado_Update.R b/papers/Avocado_Update/Avocado_Update.R index bde8f50d..c1b0c2d8 100644 --- a/papers/Avocado_Update/Avocado_Update.R +++ b/papers/Avocado_Update/Avocado_Update.R @@ -142,52 +142,15 @@ library(jmv) data('bugs') ## ----------------------------------------------------------------------------- -head(sleep,2) - -## ----------------------------------------------------------------------------- -# Formula Interface -res1 = t_TOST(formula = extra ~ group, data = sleep, +res1 = t_TOST(formula = extra ~ group, data = sleep, eqb = .5, smd_ci = "t") -# x & y Interface -res1a = t_TOST(x = subset(sleep,group==1)$extra, - y = subset(sleep,group==2)$extra, eqb =.5) ## ----------------------------------------------------------------------------- print(res1) -## ----cdplot,fig.width=6, fig.height=5,fig.cap="Example of consonance density plot."---- +## ----cdplot,fig.width=6, fig.height=5,fig.cap="Consonance density plot for the independent-samples equivalence test."---- plot(res1, type = "cd") -## ----shadeplot,fig.width=6, fig.height=5, fig.cap = "Demonstrating the shading in plot method."---- -plot(res1, type = "cd", - ci_shades = c(.9,.95)) - -## ----conplot,fig.width=6, fig.height=5, fig.cap = "Example of consonance plot."---- -plot(res1, type = "c", - ci_lines = c(.9,.95)) - -## ----------------------------------------------------------------------------- -res2 = t_TOST(formula = extra ~ group, - data = sleep, - paired = TRUE, - eqb = .5) -res2 - -## ----------------------------------------------------------------------------- -res3 = t_TOST(x = bugs$LDHF, - y = bugs$LDLF, - paired = TRUE, - eqb = 1) -res3 - -## ----------------------------------------------------------------------------- -res3a = t_TOST(x = bugs$LDHF, - y = bugs$LDLF, - paired = TRUE, - hypothesis = "MET", - eqb = 1) -res3a - ## ----------------------------------------------------------------------------- res4 = t_TOST(x = bugs$LDHF, hypothesis = "EQU", @@ -204,6 +167,55 @@ res_tsum = tsum_TOST( res_tsum +## ----------------------------------------------------------------------------- +res_sh1 <- simple_htest(extra ~ group, data = sleep, + alternative = "equivalence", mu = 0.5) +res_sh1 + +## ----------------------------------------------------------------------------- +describe_htest(simple_htest(extra ~ group, data = sleep)) + +## ----shplot1, fig.width=6, fig.height=3.5, fig.cap = "Estimate plot for an equivalence test via simple_htest."---- +plot_htest_est(res_sh1) + +## ----------------------------------------------------------------------------- +smd_calc(formula = extra ~ group, + data = sleep, + paired = FALSE, + bias_correction = TRUE, + alternative = "equivalence", + null.value = c(-0.5,0.5), + output = "htest") + +## ----------------------------------------------------------------------------- +X = rnorm(30) +Y = rnorm(30) + +boot_cor_test( + x = X, + y = Y, + method = "pearson", + alternative = "equivalence", + null = 0.4) + +## ----------------------------------------------------------------------------- +compare_smd(smd1 = 0.95, + n1 = 25, + smd2 = 0.23, + n2 = 50, + paired = TRUE) + +## ----------------------------------------------------------------------------- +compare_smd(smd1 = 0.95, n1 = 25, smd2 = 0.23, n2 = 50, + paired = TRUE, TOST = TRUE, null = .25) + +## ----------------------------------------------------------------------------- +bm_test = brunner_munzel(formula = extra ~ group, + data = sleep, + alternative = "equivalence", + mu = c(0.3, 0.7)) +print(bm_test) + ## ----------------------------------------------------------------------------- test1 = wilcox_TOST(formula = extra ~ group, data = sleep, @@ -211,6 +223,22 @@ test1 = wilcox_TOST(formula = extra ~ group, eqb = .5) print(test1) +## ----------------------------------------------------------------------------- +set.seed(4522) +boot_t_test(formula = extra ~ group, + data = sleep, + alternative = "equivalence", + mu = c(-0.5, 0.5), + R = 999) + +## ----------------------------------------------------------------------------- +set.seed(8812) +perm_t_test(formula = extra ~ group, + data = sleep, + alternative = "equivalence", + mu = c(-0.5, 0.5), + R = 999) + ## ----------------------------------------------------------------------------- set.seed(891111) test1 = boot_t_TOST(formula = extra ~ group, @@ -218,21 +246,12 @@ test1 = boot_t_TOST(formula = extra ~ group, paired = TRUE, eqb = .5, R = 999) - - print(test1) -## ----------------------------------------------------------------------------- -x = 7; y = 10.5 -log(y) - log(x) -log(y/x) -exp(log(y) - log(x)) -y/x - -## ---- error=FALSE------------------------------------------------------------- +## ----error=FALSE-------------------------------------------------------------- log_TOST(mpg ~ am, data = mtcars) -## ---- error=FALSE------------------------------------------------------------- +## ----error=FALSE-------------------------------------------------------------- boot_log_TOST(mpg ~ am, data = mtcars, R=999) ## ----warning=FALSE, message=FALSE--------------------------------------------- @@ -245,29 +264,22 @@ anova(aovtest) equ_ftest(Fstat = 34.70228, df1 = 5, df2 = 66, eqb = 0.35) ## ----------------------------------------------------------------------------- -# Example using a purely within-subjects design +# Example using a purely within-subjects design # (Maxwell & Delaney, 2004, Chapter 12, Table 12.5, p. 578): library(afex) data(md_12.1) -aovtest2 = aov_ez("id", "rt", md_12.1, within = c("angle", "noise"), +aovtest2 = aov_ez("id", "rt", md_12.1, within = c("angle", "noise"), anova_table=list(correction = "none", es = "none")) equ_anova(aovtest2, eqb = 0.35) ## ----------------------------------------------------------------------------- -compare_smd(smd1 = 0.95, - n1 = 25, - smd2 = 0.23, - n2 = 50, - paired = TRUE) - -## ----------------------------------------------------------------------------- -compare_smd(smd1 = 0.95, n1 = 25, smd2 = 0.23,n2 = 50, - paired = TRUE, TOST = TRUE, null = .25) +power_t_TOST(delta = 0, sd = 1, eqb = 0.5, + alpha = 0.05, power = 0.8, + type = "two.sample") ## ----------------------------------------------------------------------------- -set.seed(4522) -boot_test = boot_compare_smd(x1 = rnorm(25,.95), x2 = rnorm(50), - paired = TRUE, alpha = .1) -boot_test +power_z_cor(rho = 0, power = 0.8, + null = 0.3, alpha = 0.05, + alternative = "equivalence") diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index ebeae3b7..a3d978d6 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -9,26 +9,28 @@ author: type: PREPRINT output: papaja::apa6_docx: default - rticles::tf_article: + rticles::tf_article: extra_dependencies: ["float"] bibliography: interactcadsample.bib #appendix: appendix.tex abstract: | - Equivalence testing is arguably under utilized by experimental - researchers. Due to limited software support for such analyses, and - little education on the topic in graduate programs, the utilization of - equivalence testings still appears to be low. - One option for equivalence testing is the use of two one-sided tests (TOST). - The TOSTER R package and jamovi module, originally developed by - Daniel Lakens in 2017, was created to make TOST - more accessible to the average researcher. - In recent years, I have made significant changes to the TOSTER package - in order to increase its accessibility and provide more robust analysis - options for researchers. In this paper, I will detail the changes to the - package and highlight new analysis options that will make TOST easier for the - average quantitative researcher. + Despite decades of methodological guidance, researchers continue to + misinterpret non-significant results as evidence for the absence of an effect. + Equivalence testing via two one-sided tests (TOST) provides a principled + alternative, but adoption remains limited by software accessibility and + lack of training. The TOSTER R package, originally developed by Lakens (2017), + made TOST accessible to experimental psychologists. This paper describes + a major update to the package that expands its scope with (1) a unified + t-test interface supporting both equivalence and minimal effect testing, + (2) a simplified hypothesis testing interface returning standard R objects, + (3) tools for computing and testing standardized effect sizes and correlations, + (4) robust and resampling-based alternatives including bootstrap and + permutation tests, (5) equivalence testing for ANOVAs, and (6) power + analysis and sample size planning. Each capability is demonstrated with + worked examples. TOSTER is also available as a jamovi module for + researchers who prefer a graphical interface. keywords: | - statistics, bootstrap, minimal effects test, NHST, TOST + statistics, bootstrap, minimal effects test, NHST, TOST, R package, robust statistics header-includes: | \usepackage{hyperref} \usepackage[utf8]{inputenc} @@ -37,7 +39,7 @@ header-includes: | colorlinks=true, citecolor = cyan, linkcolor=blue, - filecolor=magenta, + filecolor=magenta, urlcolor=blue } affiliation: @@ -60,43 +62,35 @@ library(ggdist) library(patchwork) ``` - # Introduction -Researchers often erroneously declare that no statistical effect exists based on a single "non-significant" p-value [@blandaltman95]. -In many of these cases, the data may corroborate the researcher's claim, but the interpretation of a null hypothesis significance test (NHST), wherein the lack of significance is considered evidence of "no effect", is nonetheless incorrect. -In order to statistically test for whether there is practically no effect, researchers could use equivalence testing. -Equivalence testing is used when the goal of a statistical test is to demonstrate that the difference between two conditions is too small to be meaningful. For example, if a researcher wanted to test whether a new drug was no worse than a standard drug, the null hypothesis would be that the new drug is worse than the standard drug by more than a meaningful amount, and the alternative hypothesis would be that the difference between the two drugs is small enough to be meaningless. A very simple equivalence testing approach is the use of "two one-sided tests" (TOST) [@schuirmann1987]. - -The TOST procedure is a statistical test of whether a parameter (e.g., mean difference) is within a specified interval. The TOST procedure can be used to test the equivalence of two means, two proportions, two regression coefficients, and even two variances. -An upper ($\Delta_U$) and lower ($\Delta_L$) equivalence bound is specified based on the smallest effect size of interest (SESOI). -If the TOST is below a pre-specified alpha level, then the effect can be considered close enough to zero to be practically equivalent [@lakens_ori]. - -Both the complaints about erroneous conclusions regarding equivalence [@blandaltman95] and proposed statistical solutions [@schuirmann1987] have existed for decades now. Yet, the problem appears to persist in many applied disciplines. -I believe the continued dissonance is due to a general lack of education on equivalence testing and a struggle for many applied researchers to implement equivalence testing. -In my experience, most researchers have received some degree of statistical training in their doctoral or master's studies, but it is rare that any have idea of how to use TOST. -It may also be difficult to implement equivalence testing for many researchers. -This may be caused by most statistical software defaulting to a null hypothesis of zero, -or even completely lacking an ability to change the null hypothesis. -Therefore, I feel the continued development of educational content on TOST, and software to help with such analyses, would be beneficial to many quantitative researchers. - -The TOSTER R package^[All updates to the package can be found on the package's website https://aaroncaldwell.us/TOSTERpkg ] was originally developed in by @lakens_ori to introduce -experimental psychologists to the concept of equivalence testing and provide an easy-to-use implementation in R. -In the years since that publication, I have made a significant update to the package in order to improve the user interface and expand the tools available within the package. -An experienced R programmer may have no problem performing equivalence testing within R, -but beginners may struggle with both writing the code and interpreting the output. -If you fall into that category, I would suggest using jamovi, -an open-source statistical software, -that has a TOSTER module to perform equivalence/TOST analyses. -Not all the features listed in this manuscript are available in the jamovi module, but it is a good starting point for most researchers without statistical programming experience. - -In this manuscript, I will detail the updates to the TOSTER package, and give some basic usage examples of some of the new functions. This is meant to just be an introduction to *how* to perform such analyses, and provide a little bit of context for when such analyses are appropriate. For a greater introduction to equivalence testing, I would suggest reading other methodological tutorials [@lakens_ori;@lakens2018equivalence;@lakens2020improving;@mazzolari2022myths]. +Researchers routinely interpret non-significant p-values as evidence that no effect exists --- a logical error that has been recognized for decades [@blandaltman95]. A non-significant result means only that the data are compatible with the null hypothesis; it does not mean the null hypothesis is true. + +Equivalence testing provides a principled framework for evaluating whether an effect is small enough to be considered negligible. The two one-sided tests (TOST) procedure is the most widely used approach [@schuirmann1987]. Researchers specify an upper ($\Delta_U$) and lower ($\Delta_L$) equivalence bound based on the smallest effect size of interest (SESOI). Two one-sided tests then evaluate whether the observed effect falls within those bounds. If both tests reject at a pre-specified alpha level, the effect can be considered practically equivalent to zero [@lakens_ori]. + +The TOSTER R package^[All updates to the package can be found on the package's website ] was originally developed by @lakens_ori to introduce experimental psychologists to the concept of equivalence testing and provide an accessible implementation in R. The package provided dedicated functions for independent, paired, and one-sample t-tests, as well as correlations and meta-analytic equivalence tests. It has been widely adopted, with over 1000 citations to date. + +However, the original package had a number of limitations. It offered a function-per-design interface (separate functions for each t-test type), lacked robust alternatives to parametric tests, provided no support for ANOVA equivalence testing, and had no tools for comparing effect sizes across studies. The package also did not integrate with standard R workflows and usually just printed results to the console rather than returning objects that could be manipulated or plotted. + +This paper provides a tutorial-style introduction to the updated TOSTER package, organized around its major capabilities: (1) a unified t-test interface (`t_TOST` and `simple_htest`) that pairs a standard significance test with an equivalence test and supports minimal effect testing, (2) a simplified hypothesis testing interface (`simple_htest`) that returns standard `htest` objects, (3) tools for computing and testing standardized effect sizes --- including standardized mean differences, rank-based effect sizes, and correlations, (4) robust and resampling-based alternatives including the Brunner-Munzel test, bootstrap and permutation t-tests, and log-transformed methods, (5) equivalence testing for ANOVAs, and (6) power analysis and sample size planning. Each section motivates when and why a researcher would use the tool before demonstrating how. + +TOSTER occupies a specific niche in the R ecosystem in that it is a frequentist toolkit for equivalence testing of standard designs. Researchers interested in Bayesian approaches to evidence for the null have excellent options (e.g., bayestestR). For equivalence-style contrasts in more complex models --- mixed models, marginal means --- the emmeans and marginaleffects packages can accommodate non-zero null hypotheses. TOSTER fills the space of accessible, self-contained equivalence testing for common, albeit simple, research designs. For a broader introduction to the methodology, see @lakens_ori, @lakens2018equivalence, @lakens2020improving, and @mazzolari2022myths. TOSTER is also available as a jamovi module for researchers who prefer a graphical interface. + +# Setting Equivalence Bounds + +Before applying any equivalence test, the researcher must specify the equivalence bounds --- the range of effect sizes considered too small to be meaningful. There is no single correct method for choosing these bounds; the appropriate approach depends entirely on the research question and domain. + +If a researcher has an equivalence hypothesis, they should be able to articulate what magnitude of effect would be too small to matter in context. The SESOI flows from the hypothesis, not from a formula. Several general strategies exist. Anchor-based methods tie the bound to an externally meaningful criterion, such as a clinically meaningful difference or a just-noticeable difference [@sesoi]. Distribution-based methods define the bound as a fraction of the standard deviation, though this approach has limitations (discussed below in the Standardized Effect Sizes section). Cost-benefit reasoning sets the bound at the point where the cost of treating a difference as real outweighs the benefit --- for example, when additional measurement or intervention resources exceed what a small effect would justify. Researchers can draw on existing literature or normative data to identify effect magnitudes that prior work suggests are negligible. When theory is sufficiently precise, however, the bound may require no external benchmark at all, and the hypothesis itself implies what magnitude of effect would be too small to matter. + +Using "default" bounds based on Cohen's benchmarks (e.g., SMD = 0.2, 0.5, 0.8 to imply small, medium, or large effects, respectively) without domain-specific justification is strongly discouraged. Cohen himself stated that these conventions were "recommended for use only when no better basis for estimating the index is available" (p. 25, as cited in @panzarella2021) and reportedly came to regret having proposed them at all [@funder2019]. The benchmarks were never intended to serve as universal thresholds, and their uncritical application can lead to equivalence bounds that are either far too liberal or far too conservative for the question at hand. + +If a researcher cannot specify meaningful bounds, this itself is informative --- it suggests the equivalence hypothesis is not well enough developed to test. In such cases, it may be more productive to focus on effect estimation with confidence intervals rather than forcing a dichotomous equivalence decision. Throughout the examples in this paper, equivalence bounds are chosen for illustration; readers should not treat them as direct recommendations for their own research. # TOST with t-tests -In an effort to make TOSTER more informative and easier to use, the `t_TOST` function was created. This function operates very similarly to base R's `t.test` function, but performs 3 t-tests (one two-tailed and two one-tailed tests). In addition, this function has a generic method where two vectors can be supplied or a formula can be given (e.g.,`y ~ group`). This function also makes it easier to switch between types of t-tests. All three types (two sample, one sample, and paired samples) can be performed/calculated from the same function. Moreover, the output from this function is verbose, and should make the decisions derived from the function more informative and user-friendly. +The `t_TOST` function provides a unified interface for equivalence and minimal effect testing with t-tests. It runs a standard nil significance test ($H_0: $) alongside the equivalence (or minimal effect) test, because in practice researchers may want both: "Is there a significant effect?" and "Can I rule out a meaningful effect?" The function handles independent, paired, and one-sample designs through the `paired` argument and accepts data via either a formula interface or raw vectors, making it a direct analog to base R's `t.test`. For researchers who only want to test a single hypothesis at a time, the `simple_htest` function (described in the next section) provides a lighter alternative. -Also, `t_TOST` is not limited to equivalence tests. Minimal effects testing (MET) is possible. MET is useful for situations where the hypothesis is about a minimal effect and the *null hypothesis is equivalence* (see Figure 1) [@mazzolari2022myths]. +Minimal effect testing (MET) is also supported. MET is useful when the hypothesis is that an effect *exceeds* a meaningful threshold --- the null hypothesis is equivalence, and rejection implies a practically meaningful effect (see Figure 1) [@mazzolari2022myths]. ```{r hypplot, fig.width=6, fig.height=2.75, echo=FALSE, message = FALSE, warning = FALSE, fig.show='hold', fig.cap = "Type of Hypothesis"} @@ -226,7 +220,9 @@ p1 + p2 + plot_annotation(tag_levels = 'A') \newpage -In these examples of `t_TOST`, we will use the `bugs` data from the `jmv` R package and the `sleep` data. +## Independent Groups Example + +In these examples we will use the `sleep` data and the `bugs` data from the `jmv` R package. ```{r} data('sleep') @@ -234,32 +230,14 @@ library(jmv) data('bugs') ``` - -## Independent Groups - -For this example, we will use the sleep data. In this data, there is a `group` variable and an outcome `extra`. +For the first example, we use the `sleep` data with equivalence bounds of $\pm$ 0.5 units. The formula interface works identically to `t.test`: ```{r} -head(sleep,2) -``` - -We will assume the data are independent (in reality this is paired data), and that we have equivalence bounds of +/- 0.5 units of `extra`. All we need to do is provide the `formula`, `data`, and `eqb` arguments for the function to run appropriately. In addition, we can set the `var.equal` argument (to assume equal variance), and the `paired` argument (sets if the data is paired or not). Both are logical indicators that can be set to TRUE or FALSE. The `alpha` is automatically set to 0.05 but this can also be adjusted by the user depending on the desired alpha-level^[I strongly recommend users "justify their alpha" [@jya1;@jya2], and the justification process can be aided by my other R package [Superpower](https://aaroncaldwell.us/Superpower)]. - -Standardize mean differences (SMDs) are provided in the output for any t-test based TOST analysis (e.g., Cohen's d). The Hedges's corrected SMD [@hedges_bias] is automatically calculated, but this can be overridden with the `bias_correction` argument^[Glass's delta can also be produced in the output by using the `glass` argument]. In previous versions of this package, the equivalence bounds could be set by the SMD (e.g., equivalence bound of 0.5 SD), but this is an erroneous approach since the bound would be dependent upon the *sample* variance. However, users can opt for such an analysis by setting `eqbound_type` to SMD, which will produce a noticeable warning to the R console. - -The `hypothesis` argument is automatically set to "EQU" for equivalence, but if a minimal effect is of interest then "MET" can be supplied. - -```{r} -# Formula Interface -res1 = t_TOST(formula = extra ~ group, data = sleep, +res1 = t_TOST(formula = extra ~ group, data = sleep, eqb = .5, smd_ci = "t") -# x & y Interface -res1a = t_TOST(x = subset(sleep,group==1)$extra, - y = subset(sleep,group==2)$extra, eqb =.5) ``` -Once the function has run, we can print the results with the `print` method. -This provides a verbose summary of the results. +The `print` method provides a verbose summary including the two-tailed test, both one-sided equivalence tests, and the standardized mean difference (Hedges' $g$ by default): ```{r} print(res1) @@ -267,81 +245,19 @@ print(res1) \newpage -Another nice feature is the generic `plot` method that can provide a visual summary of the results. Most of the plots in this package were inspired by the [concurve](https://cran.r-project.org/package=concurve) R package [@rafi2020]. -There are two types of plots that can be produced. The first, and default, is the consonance density plot (`type = "cd"`). +The generic `plot` method produces a consonance density plot, inspired by the concurve R package [@rafi2020], that shows the estimate, confidence intervals, and equivalence bounds simultaneously: -```{r cdplot,fig.width=6, fig.height=5,fig.cap="Example of consonance density plot."} +```{r cdplot,fig.width=6, fig.height=5,fig.cap="Consonance density plot for the independent-samples equivalence test."} plot(res1, type = "cd") ``` -\newpage - -The shading pattern can be modified with the `ci_shades`. - -```{r shadeplot,fig.width=6, fig.height=5, fig.cap = "Demonstrating the shading in plot method."} -plot(res1, type = "cd", - ci_shades = c(.9,.95)) -``` +Other plot types are available: `type = "c"` produces a consonance plot showing multiple confidence interval levels, and shading can be customized with the `ci_shades` argument. \newpage -Consonance plots, where all confidence intervals can be simultaneous plotted, can also be produced. The advantage here is multiple confidence interval lines can plotted at once. - -```{r conplot,fig.width=6, fig.height=5, fig.cap = "Example of consonance plot."} -plot(res1, type = "c", - ci_lines = c(.9,.95)) -``` - -\newpage - -## Paired Sample - -To perform TOST on paired samples, the process does not change much. We could process the test the same way by providing a formula. All we would need to then is change `paired` to TRUE. - -```{r} -res2 = t_TOST(formula = extra ~ group, - data = sleep, - paired = TRUE, - eqb = .5) -res2 -``` - -\newpage +## One-Sample t-test with Asymmetric Bounds -However, we may have two vectors of data that are paired. So instead we may want to just provide those separately rather than using a data set and setting the formula. This can be demonstrated with the "bugs" data. - -```{r} -res3 = t_TOST(x = bugs$LDHF, - y = bugs$LDLF, - paired = TRUE, - eqb = 1) -res3 -``` - -\newpage - -Additionally, a MET, instead of equivalence testing, can be performed with the `hypothesis` argument set to "MET". With this setting, the hypothesis being tested is whether the effect is *greater* than the equivalence bound. - -```{r} -res3a = t_TOST(x = bugs$LDHF, - y = bugs$LDLF, - paired = TRUE, - hypothesis = "MET", - eqb = 1) -res3a -``` - -The data would indicate that we should accept the MET hypothesis. - -\newpage - -## One Sample t-test - -In other cases we may have a one sample test. -If that is the case, only `x` argument for the data is needed. -This is useful in situations where you may have hypotheses to test about a single samples mean. -In order for the two-sample test to be correct, we also need to supply the `mu` argument. -In the example below, we hypothesize that the mean of `LDHF` is not more than 1.5 points greater or less than 7. With the way the `mu` and `eqb` arguments are set, we are testing whether the mean of `LDHF` is significantly different from 7.5 (two-tailed tests) and ($\pm$) than 1.5 points 7.5 as well (equivalence bounds at 5.5 and 8.5). +When the equivalence bounds are not symmetric around the null, two values can be supplied to `eqb`. This is common when the research question implies asymmetric consequences. Here we test whether the mean of `LDHF` falls within the bounds of 5.5 and 8.5 around a hypothesized value of 7.5: ```{r} res4 = t_TOST(x = bugs$LDHF, @@ -351,23 +267,13 @@ res4 = t_TOST(x = bugs$LDHF, res4 ``` -We would conclude that `LDHF` is practically equivalent to the hypothesized mean (7.5). +The same interface handles paired designs by setting `paired = TRUE`. Standardized mean differences are reported automatically; Hedges' correction is applied by default but can be toggled with `bias_correction`, and Glass's delta can be requested via the `glass` argument. Setting equivalence bounds in SMD units (via `eqbound_type = "SMD"`) is possible but discouraged because the bound becomes dependent on sample variance (see the Standardized Effect Sizes section below). \newpage - ## Using Summary Statistics -In some cases you may only have access to the summary statistics (e.g., when reviewing an article or attempting to perform a meta-analysis). -Therefore, I created a function, `tsum_TOST`, to perform the same tests just based on the summary statistics. -This involves providing the function with a number of different arguments. - -* `n1 & n2` the sample sizes (only n1 needs to be provided for one sample case) -* `m1 & m2` the sample means -* `sd1 & sd2` the sample standard deviation -* `r12` the correlation between each if paired is set to TRUE^[The `extract_r_paired` function can be used if the correlation between paired observations is not readily available.] - -The results from the `bugs` example can be replicated with the `tsum_TOST`: +When only summary statistics are available (e.g., from a published article), the `tsum_TOST` function performs the same tests using sample means, standard deviations, sample sizes, and (for paired designs) the correlation between measures^[The `extract_r_paired` function can be used if the correlation between paired observations is not readily available.]: ```{r} res_tsum = tsum_TOST( @@ -381,17 +287,13 @@ res_tsum # Simplified Hypothesis Testing -The `t_TOST` function, and related `*_TOST` functions, are designed to be comprehensive — they run three tests (one two-tailed and two one-sided), compute standardized effect sizes, and produce verbose output. For researchers who want the full picture, this is ideal. However, for users who *only* want to test a single hypothesis of equivalence (or minimal effects), the output can feel unnecessarily heavy. The three-test structure may be confusing when the researcher only cares about one of those tests. +The `t_TOST` function is designed to be comprehensive --- it runs three tests, computes standardized effect sizes, and produces verbose output. This is useful for exploration or situations where researchers want to pair a traditional nil significance test with an equivalence test. However, sometimes a researcher has a single specific hypothesis to test. The `simple_htest` function provides a lighter interface for this purpose. It returns a standard R `htest` object --- the same class returned by `t.test()` and `wilcox.test()` --- but with expanded `alternative` options that include `"equivalence"` and `"minimal.effect"`. This means researchers can perform equivalence testing with the same familiar interface they already use for standard hypothesis tests. -To address this, I created the `simple_htest` function as a lighter interface. It returns a standard R `htest` object — the same class returned by `t.test()` and `wilcox.test()` — but with expanded `alternative` options that include `"equivalence"` and `"minimal.effect"`. This means researchers can perform equivalence testing with the same familiar interface they already use for standard hypothesis tests. - -Alongside `simple_htest`, I also developed a set of helper functions for working with any `htest` object. These helpers make it easy to convert results to data frames (`df_htest`), generate (almost) publication-ready text descriptions (`describe_htest`), create plots (`plot_htest_est`), and bridge between the verbose TOST output and the simpler `htest` ecosystem (`as_htest`). Importantly, these tools work not only with TOSTER functions but with any base R hypothesis test. +Alongside `simple_htest`, a set of helper functions makes it easy to work with any `htest` object: `as_htest` converts verbose TOST output to the simpler format, `describe_htest` generates formatted text for reporting, `df_htest` converts results to data frames for tabulation, and `plot_htest_est` produces estimate plots with equivalence bounds. ## simple_htest -The `simple_htest` function supports `t.test` and `wilcox.test` as the underlying test (via the `test` argument) and all five alternative hypotheses: `"two.sided"`, `"less"`, `"greater"`, `"equivalence"`, and `"minimal.effect"`. The `mu` parameter works like `t.test` for standard alternatives (a single null value), but for equivalence and minimal effect testing a single value creates symmetric bounds (i.e., `mu = 0.5` becomes bounds at -0.5 and 0.5), or two values can be given for asymmetric bounds. - -For instance, we can perform an equivalence test on the `sleep` data with symmetric bounds of $\pm$ 0.5: +The `simple_htest` function supports `t.test` and `wilcox.test` as the underlying test (via the `test` argument) and all five alternative hypotheses: `"two.sided"`, `"less"`, `"greater"`, `"equivalence"`, and `"minimal.effect"`. The `mu` parameter specifies equivalence bounds --- a single value creates symmetric bounds, or two values can be given for asymmetric bounds. ```{r} res_sh1 <- simple_htest(extra ~ group, data = sleep, @@ -399,104 +301,117 @@ res_sh1 <- simple_htest(extra ~ group, data = sleep, res_sh1 ``` -Notice how the output is a single clean `htest` object — much more concise than the verbose `t_TOST` output shown previously. +Notice how the output is a single clean `htest` object --- much more concise than the verbose `t_TOST` output shown previously. + +## Describing and Tabulating Results -Similarly, a minimal effect test on paired data from the `bugs` dataset can be performed: +For writing up results, `describe_htest` generates a formatted text string following typical scientific reporting conventions: ```{r} -res_sh2 <- simple_htest(x = bugs$LDHF, y = bugs$LDLF, - paired = TRUE, - alternative = "minimal.effect", mu = 1) -res_sh2 +describe_htest(simple_htest(extra ~ group, data = sleep)) ``` -The same interface works for non-parametric tests simply by changing the `test` argument: -```{r} -res_sh3 <- simple_htest(extra ~ group, data = sleep, - test = "wilcox.test", - alternative = "equivalence", mu = 0.5) -res_sh3 +## Plotting Results + +A visual summary of the estimate and confidence interval can be produced with `plot_htest_est`. For equivalence and minimal effect tests, the equivalence bounds are shown as dashed vertical lines: + +```{r shplot1, fig.width=6, fig.height=3.5, fig.cap = "Estimate plot for an equivalence test via simple_htest."} +plot_htest_est(res_sh1) ``` -## Converting TOST Results with as_htest +\newpage -Users who have already run `t_TOST` or `wilcox_TOST` and want to use the `htest` helper functions can convert their results with `as_htest`. This extracts the TOST-relevant test and packages it as a standard `htest` object. +# Standardized Effect Sizes -```{r} -res1_htest <- as_htest(res1) -res1_htest -``` +This section provides a unified treatment of the standardized effect size tools in TOSTER --- standardized mean differences (SMDs), rank-based effect sizes, correlation equivalence tests, and tools for comparing SMDs between studies. Before demonstrating these tools, it is worth discussing the limitations of standardized effect sizes, because these limitations directly affect how equivalence bounds should be set. -## Describing Results with describe_htest +## Problems with Standardized Effect Sizes -For writing up results, `describe_htest` generates a formatted text string following typical scientific reporting conventions. It includes the significance decision, test statistic, p-value, estimate, confidence interval, and the hypothesis being tested. This works with any `htest` object — from base R or from TOSTER. +Standardized effect sizes --- Cohen's $d$, correlation coefficients, standardized regression coefficients --- are appealing because they allow comparison across studies and scales, are required by many journals, and serve as the primary currency of meta-analysis. However, they have well-documented problems that are especially relevant to equivalence testing. -```{r} -describe_htest(simple_htest(extra ~ group, data = sleep)) -``` +Standardized mean differences are sensitive to the sample variance: the "same" real-world effect produces different $d$ values across populations and study designs. Cohen's $d$ conflates the magnitude of the mean difference with the precision of measurement, which means that a narrow equivalence bound on $d$ may correspond to a very different raw-scale bound depending on the variability of the sample [@Caldwell2020]. This makes it unclear what, substantively, is being declared "equivalent" --- are we claiming the means are close, or that the means are close *relative to the variance*? [@baguley2009]. The choice of denominator (pooled SD, control SD, within-subject SD) further complicates interpretation. + +Correlations have analogous issues. @tukey1969 noted that correlations are influenced by the range of both variables, so the "same" bivariate relationship produces different coefficients depending on sample selection. A correlation of 0.3 in a restricted-range sample may correspond to a much larger association in the full population. Standardized regression coefficients share these vulnerabilities: they depend on the variances of both the predictor and outcome, which makes them unstable across samples and difficult to interpret substantively [@greenland1991]. -This also works with standard R tests. Note the difference from the difference between the output below from `t.test` and `simple_htest` above. The differences, despite both descriptions coming from the same underlying statistical tests, are caused by differences in what output is produced by each function (i.e., `simple_htest` has slight differences in output from `t.test`). +The implication for equivalence testing is straightforward. When the measurement scale is meaningful, researchers should generally prefer *unstandardized* (raw-scale) equivalence bounds. When standardized bounds are used, the choice should be justified with reference to the specific population and measurement context, not borrowed from generic benchmarks. + +With these caveats in mind, the following subsections describe the tools TOSTER provides for computing and testing standardized effects when they are appropriate. + +## SMD Functions + +The `smd_calc` function computes standardized mean differences --- Cohen's $d$, Hedges' $g$ (bias-corrected), or Glass's $\Delta$ --- with confidence intervals and optional hypothesis testing. It supports one-sample, paired, and two-sample designs, and can return either an `htest` object or a data frame. The SMD is already reported in `t_TOST` output, but `smd_calc` allows standalone computation and supports equivalence testing directly on the effect size. ```{r} -describe_htest(t.test(extra ~ group, data = sleep)) +smd_calc(formula = extra ~ group, + data = sleep, + paired = FALSE, + bias_correction = TRUE, + alternative = "equivalence", + null.value = c(-0.5,0.5), + output = "htest") ``` -## Tabulating Results with df_htest -When combining results from multiple tests (e.g., running equivalence tests across several outcomes), it is useful to have results in a data frame. The `df_htest` function converts any `htest` object into a single-row data frame with standardized column names. Multiple results can then be combined with `rbind`. +## Equivalence Tests for Correlations + +Researchers sometimes need to test that a correlation is practically zero --- for example, when evaluating discriminant validity or confirming that a potential confounder is unrelated to an outcome. The `boot_cor_test` function supports provides robust support for testing equivalence on correlation coefficients. Bounds are specified via the `null` argument: ```{r} -rbind( - df_htest(res_sh1), - df_htest(res_sh2), - df_htest(res_sh3) -) +X = rnorm(30) +Y = rnorm(30) + +boot_cor_test( + x = X, + y = Y, + method = "pearson", + alternative = "equivalence", + null = 0.4) ``` -## Plotting Results with plot_htest_est +This extends the original TOSTER package's correlation coverage by supporting all three major correlation methods and returning standard `htest` objects that integrate with the helper functions described above. If summary statistics are only available, an approximate method is provided through `corsum_test`, which uses the Fisher z-transformation to compute confidence intervals and p-values. -A visual summary of the estimate and confidence interval can be produced with `plot_htest_est`. For equivalence and minimal effect tests, the equivalence bounds are shown as dashed vertical lines, making it easy to see whether the confidence interval falls within (or outside) the bounds. +## Comparing SMDs Between Studies -```{r shplot1, fig.width=6, fig.height=3.5, fig.cap = "Estimate plot for an equivalence test via simple_htest."} -plot_htest_est(res_sh1) +When evaluating whether a replication study produced results consistent with the original, researchers need a way to formally compare effect sizes. The `compare_smd` function tests whether two SMD estimates --- from an original study and a replication, for example --- differ more than would be expected by sampling variability alone. + +Consider an original study reporting Cohen's $d_z$ = 0.95 with 25 participants and a replication with $d$ = 0.23 and 50 participants. Are these estimates compatible? + +```{r} +compare_smd(smd1 = 0.95, + n1 = 25, + smd2 = 0.23, + n2 = 50, + paired = TRUE) ``` -For standard tests, only a single null reference line (at zero) appears: +Setting `TOST = TRUE` with an equivalence bound via the `null` argument tests whether the two SMDs are practically equivalent: -```{r shplot2, fig.width=6, fig.height=3.5, fig.cap = "Estimate plot for a standard two-sided t-test."} -plot_htest_est(t.test(extra ~ group, data = sleep)) +```{r} +compare_smd(smd1 = 0.95, n1 = 25, smd2 = 0.23, n2 = 50, + paired = TRUE, TOST = TRUE, null = .25) ``` +Based on these results, we would reject the null significance hypothesis (the SMDs differ significantly) but fail to reject the equivalence null hypothesis (we cannot conclude the SMDs are practically equivalent). When raw data are available, `boot_compare_smd` provides a bootstrap alternative; see the package vignettes for details. + \newpage -# Robust Methods for Equivalence Testing +**A note on argument names for equivalence bounds.** The equivalence bound argument is named differently across TOSTER functions for historical and compatibility reasons: `eqb` in `t_TOST`, `wilcox_TOST`, `equ_ftest`, and `equ_anova`; `mu` in `simple_htest`, `brunner_munzel`, `boot_t_test`, and `perm_t_test` (for compatibility with base R's `t.test`); and `null` in `compare_smd` and `z_cor_test`. Despite the different names, the interpretation is the same: the bound(s) defining the region of practical equivalence. -In some cases, the use of t-test may be less than ideal. -Any serious violation to the assumptions of a t-test (e.g., normality or homoscedasticity) could greatly inflate the type 1 error rate of TOST. -Therefore, it may be useful to explore alternatives to the t-test for TOST that either do not have those assumptions or are robust to violations of those assumptions. +# Robust Methods for Equivalence Testing -The TOSTER package provides robust alternatives to the t-test organized into three categories. -First, there are *rank-based tests* — the Brunner-Munzel test (`brunner_munzel`) for stochastic equality and the Wilcoxon-Mann-Whitney TOST (`wilcox_TOST`) for location shift — which target entirely different estimands than means or medians. -Second, there are *resampling methods* — the bootstrap t-test (`boot_t_test`), permutation t-test (`perm_t_test`), and bootstrap TOST (`boot_t_TOST`) — which relax distributional assumptions while still testing hypotheses about means [@efron93]. -Third, there are *log-transformed methods* — `log_TOST` and `boot_log_TOST` — which are parametric and bootstrap approaches for ratio-scale data commonly used in pharmaceutical bioequivalence studies [@he2022]. -Where possible, the package now favors functions that return standard `htest` objects, which enforce a single hypothesis per test and integrate cleanly with the existing R ecosystem. +In some cases, serious violations of t-test assumptions (normality, homoscedasticity) could inflate the type I error rate of TOST. TOSTER provides robust alternatives organized into three categories: rank-based tests, resampling methods, and log-transformed methods. ## Rank-Based Tests Rank-based tests avoid distributional assumptions by operating on the ranks of the data rather than the raw values. -An important consideration is that these tests are *not* tests of means or medians — they target completely different estimands. -The choice between rank-based methods should therefore be driven by the estimand of interest: stochastic equality (i.e., the probability that a random observation from one group exceeds a random observation from another) for the Brunner-Munzel test, or pseudo-median of pairwise differences (under a location-shift assumption) for WMW tests. +These tests are *not* tests of means or medians --- they target different estimands. The choice between them should be driven by the estimand of interest. ### Brunner-Munzel Test -The Brunner-Munzel test [@brunner2000] provides a non-parametric test of stochastic equality. -Rather than testing whether two groups differ in location, it directly estimates the relative effect $\hat{p} = P(X > Y) + 0.5 \cdot P(X = Y)$, which is the probability that a randomly chosen observation from one group exceeds a randomly chosen observation from the other. -This estimand is interpretable without additional distributional assumptions, unlike the WMW test which requires a location-shift assumption to say anything about location differences. -@karch2021 argues that the Brunner-Munzel test should be preferred over the Mann-Whitney U test as the default nonparametric procedure because it is valid under heteroscedasticity and provides a more interpretable effect size. +The Brunner-Munzel test [@brunner2000] estimates the relative effect $\hat{p} = P(X > Y) + 0.5 \cdot P(X = Y)$ --- the probability that a randomly chosen observation from one group exceeds a randomly chosen observation from the other. This estimand is interpretable without distributional assumptions, unlike the WMW test which requires a location-shift assumption to say anything about location differences. @karch2021 argues it should be preferred as the default nonparametric procedure because it is valid under heteroscedasticity and provides a more interpretable effect size. -The `brunner_munzel` function in TOSTER supports equivalence and minimal effect testing via the `alternative` argument, with bounds specified on the probability scale through the `mu` parameter. -Results can be reported on different scales via the `scale` argument ("probability", "difference", "logodds", or "odds"), and three test methods are available via `test_method`: "t" (default, t-approximation), "logit" (range-preserving CIs), and "perm" (studentized permutation, recommended for small samples; @neubert2007). +The `brunner_munzel` function supports equivalence and minimal effect testing via `alternative`, with bounds on the probability scale via `mu`. Results can be reported on different scales (`"probability"`, `"difference"`, `"logodds"`, or `"odds"`), and three test methods are available: `"t"` (default), `"logit"` (range-preserving CIs), and `"perm"` (studentized permutation; @neubert2007). ```{r} bm_test = brunner_munzel(formula = extra ~ group, @@ -510,29 +425,11 @@ print(bm_test) ### Wilcoxon-Mann-Whitney TOST -The WMW group of tests (e.g., Mann-Whitney U-test) provide a non-parametric test of differences between groups, or within samples, based on *ranks*. -Within the TOST framework, there are two separate tests of directional location shift to determine if the location shift is within (equivalence) or outside (minimal effect) the equivalence bounds^[Care should be taken when considering paired samples; a test on the rank transformed data [@kornbrot1990rank] or another robust test may be more prudent.]. - -However, there are important caveats to the interpretation of WMW tests. -Many researchers mistakenly believe these are tests of medians, but this is not the case. -Without additional assumptions, the WMW test is a test of stochastic equality — identical to what the Brunner-Munzel test estimates. -Only under a location-shift assumption (the two distributions have identical shapes, differing only in location) does the WMW test become a test of the Hodges-Lehmann pseudo-median of pairwise differences, and only with additional symmetry can it be interpreted as a test of median or mean differences. -@median_test demonstrate several striking counterexamples: groups can have equal medians yet produce a significant WMW test (because distributions differ in shape), very different medians yet a non-significant WMW test (because neither group stochastically dominates the other), or even significance in the opposite direction from the median comparison. +The WMW group of tests provide non-parametric tests based on ranks. Within the TOST framework, two directional location-shift tests determine if the shift is within (equivalence) or outside (minimal effect) the equivalence bounds. -This interpretive difficulty is especially pronounced for equivalence testing. -When performing TOST with Wilcoxon tests, the equivalence bounds are applied to the pseudo-median of pairwise differences (i.e., the Hodges-Lehmann estimator), which corresponds to the difference in means or medians only under the location shift assumption [@median_test]. -When distributions differ in shape, bounds on the pseudo-median do not guarantee that means or medians are within those bounds. -For testing hypotheses about mean differences, `perm_t_test` or `boot_t_test` are more appropriate. -For stochastic superiority, `brunner_munzel` provides a clearer framework. -For testing effect sizes directly, `ses_calc` supports hypothesis testing on the standardized effect size such as the rank-biserial correlation. -The `wilcox_TOST` function remains useful for situations where rank-based tests are specifically desired, such as with ordinal data or when consistency with prior analyses is needed. +However, there are important caveats. Many researchers mistakenly believe these are tests of medians, but without a location-shift assumption, the WMW test is a test of stochastic equality. Only under a location-shift assumption does it become a test of the Hodges-Lehmann pseudo-median of pairwise differences. @median_test demonstrate striking counterexamples: groups can have equal medians yet produce a significant WMW test, very different medians yet a non-significant WMW test, or even significance in the opposite direction from the median comparison. -In the TOSTER package, `wilcox_TOST` operates similarly to `t_TOST`. -A standardized mean difference (SMD) is *not* calculated since this would be an inappropriate measure of effect size alongside the non-parametric test statistics. -Instead, a standardized effect size (SES) is calculated for *all* types of comparisons (e.g., two sample, one sample, and paired samples). -The function can produce a rank-biserial correlation [@Kerby_2014], a WMW Odds [@wmwodds], or a "common language effect size" [@Kerby_2014] (also known as the non-parametric probability of superiority, or concordance probability)^[While `wilcox_TOST` does not have a dedicated plotting method, the result can be converted to an `htest` object with `as_htest`, or users can obtain an `htest` object directly via `simple_htest`. The `ses_calc` function also returns an `htest` object. Any of these `htest` objects can be plotted with `plot_htest_est`.]. - -As an example, we can use the sleep data to make a non-parametric comparison of equivalence. +For equivalence testing, bounds are applied to the pseudo-median, which corresponds to the mean or median difference only under the location-shift assumption. For testing hypotheses about mean differences, `perm_t_test` or `boot_t_test` are more appropriate. For stochastic superiority, `brunner_munzel` provides a clearer framework. ```{r} test1 = wilcox_TOST(formula = extra ~ group, @@ -542,32 +439,20 @@ test1 = wilcox_TOST(formula = extra ~ group, print(test1) ``` -Based on these results, we would conclude there is no significant difference but not equivalent differences either (i.e., inconclusive result). +Based on these results, we would conclude there is no significant difference but the results are also not equivalent (i.e., an inconclusive result). But again, the estimate we are testing is the pseudo-median, not the mean or median difference, so this conclusion may not be relevant to the research question. \newpage ## Resampling Methods -When the goal is to test hypotheses about means or mean differences while relaxing distributional assumptions, resampling methods provide a powerful set of tools. -TOSTER provides two primary resampling approaches — bootstrap and permutation — each with distinct inferential logic. - -The *bootstrap* [@efron93] resamples with replacement from the observed data to approximate the sampling distribution of the test statistic. -Bootstrap confidence intervals can be constructed using several methods (studentized, basic, percentile, or bias-corrected and accelerated), and the p-value computation is tied to the CI method via the CI inversion principle. With the way the package now works for bootstraps methods, a p-value is significant at level $\alpha$ if and only if the corresponding $1 - 2\alpha$ confidence interval excludes the null value. -This ensures consistency between reported CIs and p-values [@Thulin_2024]. +When the goal is to test hypotheses about means while relaxing distributional assumptions, resampling methods provide powerful tools. TOSTER provides bootstrap and permutation approaches, each with distinct inferential logic. -The *permutation* approach shuffles group labels (for two-sample tests) or flips signs (for one-sample/paired tests) to construct the null distribution directly. -When the number of possible permutations is small enough, exact p-values can be computed; otherwise, a randomized approximation is used with the "+1" correction to avoid zero p-values [@phipson2010]. -Studentized permutation tests [@janssen1997; @chung2013], which recompute the standard error for each permutation, provide valid inference even under heteroscedasticity *and* allow for accurate equivalence tests [@Arboretti_Pesarin_Salmaso_2020]. +The bootstrap [@efron93] resamples with replacement to approximate the sampling distribution. P-values are tied to the CI method via CI inversion, ensuring consistency between reported CIs and p-values [@Thulin_2024]. The permutation approach shuffles group labels (two-sample) or flips signs (one-sample/paired) to construct the null distribution directly. Studentized permutation tests [@janssen1997; @chung2013] provide valid inference under heteroscedasticity and allow accurate equivalence tests [@Arboretti_Pesarin_Salmaso_2020]. -Both `boot_t_test` and `perm_t_test` return standard `htest` objects and support `alternative = "equivalence"` and `alternative = "minimal.effect"` for equivalence and minimal effect testing respectively. -Both also support Yuen's trimmed means via the `tr` parameter, providing additional robustness to outliers by trimming a specified fraction from each tail of the distribution before computing means and variances. +Both `boot_t_test` and `perm_t_test` return standard `htest` objects and support `alternative = "equivalence"` and `alternative = "minimal.effect"`. Both also support trimmed means via the `tr` parameter for additional robustness to outliers. ### Bootstrap t-test -The `boot_t_test` function provides a bootstrap t-test that returns a standard `htest` object. -The `boot_ci` argument controls the CI method ("stud", "basic", "perc", or "bca"), and as noted above, the p-value method follows the CI method to maintain consistency. -Equivalence bounds are specified via the `mu` argument (two values for equivalence/minimal effect testing). - ```{r} set.seed(4522) boot_t_test(formula = extra ~ group, @@ -577,14 +462,9 @@ boot_t_test(formula = extra ~ group, R = 999) ``` -\newpage - ### Permutation t-test -The `perm_t_test` function provides a permutation-based t-test. -When `R = NULL`, the function attempts to enumerate all possible permutations for an exact test; otherwise, it draws `R` random permutations. -By default, the test is studentized (`perm_se = TRUE`), meaning the standard error is recomputed for each permuted sample, which provides valid inference under heteroscedasticity [@janssen1997; @chung2013]. -For randomized permutations, the "+1" correction is applied by default to ensure proper type I error control [@phipson2010]. +When `R = NULL`, the function enumerates all possible permutations for an exact test; otherwise, it draws `R` random permutations with the "+1" correction [@phipson2010]. By default the test is studentized (`perm_se = TRUE`). ```{r} set.seed(8812) @@ -597,8 +477,7 @@ perm_t_test(formula = extra ~ group, ### Bootstrap TOST -For situations where both a two-tailed nil hypothesis test and an equivalence test are desired simultaneously, the `boot_t_TOST` function provides both in a single call, returning a `TOSTt` object. -This function uses the same bootstrap methods as `boot_t_test` and additionally provides bootstrapped SMD estimates [@Kirby2013]. +For situations where both a nil hypothesis test and an equivalence test are desired simultaneously with bootstrapping, `boot_t_TOST` returns a `TOSTt` object with bootstrapped SMD estimates [@Kirby2013]: ```{r} set.seed(891111) @@ -614,31 +493,13 @@ print(test1) ## Log-Transformed Methods -The natural logarithmic (log) transformation is often utilized to stabilize the variance of a measure, and it often provides the best approximation of the normal distribution [@logtest]. However, another, less often reported, advantage of the log transformation is that the back transformation of the differences of the log-transformed data is a *ratio* [@logtest]. For example, if we had a two samples (x & y) with an geometric mean^[The mean of log-transformed data is the *geometric* not *arithmetic* mean. I highly recommend reading @logtest and @caldwell2019basic for more details] or 7 and 10.5, x and y respectively in the code below, we could represent the differences as ratio of y:x where y is 1.5 times greater than x. - -```{r} -x = 7; y = 10.5 -log(y) - log(x) -log(y/x) -exp(log(y) - log(x)) -y/x -``` - -The log transformation thereby acts as a useful tool help tame data into conforming to the normality assumption, and makes the interpretation fairly simple. -In addition, some regulatory agencies, such as the United States Food and Drug Administration (FDA) [@fda], specifically require bioequivalence studies to report the geometric means and make statistical comparisons on the log transformed data [@he2022]. -In pharmaceutical reserach, bioequivalence testing involves determining whether two drugs, a test drug and a reference drug, have the same rate and extent of absorption in the body. This is typically accomplished by testing whether the blood concentrations of the drug after administration of the test drug are sufficiently close to the blood concentrations after administration of the reference drug. If the two drugs are bioequivalent, they can be used interchangeably. The area under the curve (AUC) is the measure of the extent of absorption, and the peak concentration is the measure of the rate of absorption. In order to determine bioequivalence, the AUC and peak concentration of the test drug must be within a certain percentage of the AUC and peak concentration of the reference drug. - -In my personal experience as a physiologist, it is not uncommon that biological/physiological phenomenon present have longer right-tailed distributions, and are often adequately normalized with a natural log transformation. The additional advantage is the how equivalence bounds can, almost, be universally applied when making comparisons on the log scale. The FDA considers to drugs to be bioequivalent when the maximal concentration and AUC differences between drugs are less than 1.25. To put it another way, ratio between two means must be between 1.25 and 0.8 (i.e., 1/1.25) [@fda]. - -An alternative to using the dedicated `log_TOST` function is to simply log-transform the data and analyze it with `t_TOST` (or any of the resampling methods described above). The differences on the log scale can then be interpreted as "sympercents" — symmetric percentage differences on the $100 \cdot \ln$ scale [@cole2000]. Sympercents have the attractive property that a difference of +10 sympercent and -10 sympercent are symmetric and of equal magnitude, unlike ordinary percentages where +10% and -10% are not. This approach provides more flexibility (e.g., one-sample tests become possible) at the cost of requiring the user to specify equivalence bounds on the log scale rather than as a ratio. - -Therefore, I have implemented two functions to allow for the comparison of data that is believed to be left skewed (long right tail), and is on a ratio scale^[Ratio scale means the outcome is measured on a numerical scale that has equal distances between adjacent values and true zero. ]. The first function is a parametric t-test on the log transformed scale while the second function is a bootstrapping test which is more robust than parametric version [@he2022]. +The natural log transformation stabilizes variance and approximates normality for right-skewed data [@logtest]. An additional advantage is that differences on the log scale back-transform to *ratios*, making interpretation straightforward: if two samples have geometric means of 7 and 10.5, the ratio is 1.5 (i.e., 10.5/7). This property is central to pharmaceutical bioequivalence, where the FDA requires comparisons on the log-transformed scale with equivalence bounds of 0.80 to 1.25 for the ratio of geometric means [@fda; @he2022]. -### Example of Log TOST +An alternative to `log_TOST` is to log-transform data manually and analyze with `t_TOST` or the resampling methods. Differences on the log scale can then be interpreted as "sympercents" --- symmetric percentage differences on the $100 \cdot \ln$ scale [@cole2000] --- which have the attractive property that +10 and -10 sympercents are of equal magnitude, unlike ordinary percentages. -The `log_TOST` function is almost exactly the same as the `t_TOST` function. First, the primary differences is that it only accepts paired and two sample comparisons. One sample tests are not support (i.e., there is no ratio to calculate). Second, standardized mean differences are not calculated, but a ratio of means is instead reported [@lajeunesse2015bias]^[Also, referred to as a "response ratio" in ecology. Like an SMD, the response ratio is often utilized in meta-analysis.]. Third, the default equivalence bounds are by default set to the FDA standards (i.e., `eqb = 1.25`), but can be changed by the user^[Only one value needs to be supplied to eqb; the reciprocal value of eqb is taken as the other equivalence bound. For example, if `eqb = 0.85` then the upper equivalence bound is 1/0.85 (~1.333)]. +### Log TOST -As an example we can use the `mtcars` data to compare the type of transmission (`am`) effects on the gas mileage (`mpg`). We can see from the data below there are significant, non-equivalent, differences in mpg between transmission types. +The `log_TOST` function accepts paired and two-sample comparisons (one-sample tests are not supported since there is no ratio to calculate). Instead of an SMD, a ratio of means is reported [@lajeunesse2015bias]. The default equivalence bounds are set to the FDA standard (`eqb = 1.25`; the reciprocal is taken as the other bound). ```{r, error=FALSE} log_TOST(mpg ~ am, data = mtcars) @@ -646,9 +507,9 @@ log_TOST(mpg ~ am, data = mtcars) \newpage -### Example of Bootstrap Log TOST +### Bootstrap Log TOST -The bootstrap version of `log_TOST`, `boot_log_TOST`, uses the same bootstrap methods as `boot_t_test`, but on the log-transformed values, producing the ratio of means as the effect size. +The bootstrap version uses the same bootstrap methods as `boot_t_test` on the log-transformed values: ```{r, error=FALSE} boot_log_TOST(mpg ~ am, data = mtcars, R=999) @@ -660,53 +521,13 @@ From this analysis, we would conclude there is a significant effect that is not # Equivalence Testing with ANOVAs -Many researchers utilize ANOVA as an omnibus test for the absence/presence of effects before inspecting multiple pairwise comparisons. -This is very useful when implementing factorial designs wherein multiple experimental factors are tested and/or manipulated. -As @Campbell_2021 suggest, the lack of a significant result at the ANOVA-level does not necessarily indicate that a factor or interaction of factors have no effect. -However, @Campbell_2021 only suggest an equivalence test for one-way ANOVAs and therefore exclude multi-factor or factorial ANOVAs. -Therefore, I have extended the work of @Campbell_2021 to include functions that allow for equivalence testing of the partial $\eta^2$ (eta-squared) effect size from ANOVAs. +Many researchers use ANOVA as an omnibus test before inspecting pairwise comparisons, particularly in factorial designs. As @Campbell_2021 note, the lack of a significant ANOVA result does not indicate that a factor has no effect. Equivalence testing for ANOVAs addresses this directly by asking: can we reject the hypothesis that the partial $\eta^2$ exceeds a specified bound? -## F-test Calculations +The equivalence test uses the non-central *F* distribution. The non-centrality parameter is computed from the equivalence bound and the degrees of freedom, and the *p*-value is obtained from the cumulative distribution function of the non-central *F*. I have extended the approach of @Campbell_2021 --- who proposed equivalence testing for one-way ANOVAs --- to accommodate multi-factor and factorial designs by generalizing the non-centrality parameter calculation. The full derivation is described in the package documentation; here we demonstrate usage. -Statistical equivalence testing^[Also called "omnibus non-inferiority testing" by @Campbell_2021] for *F*-tests are special use case of the cumulative distribution function of the non-central *F* distribution. -As @Campbell_2021 states, this type of statistical test answers the question: "Can we reject the hypothesis that the total proportion of variance in outcome Y attributable to X is greater than or equal to the equivalence bound $\Delta$?" +## Example with Summary Statistics -### Hypothesis Tests - -$$ -H_0 = 1 > \eta^2_p \geq \Delta -$$ - - -$$ -H_1 = 0 \geq \eta^2_p < \Delta -$$ - -In TOSTER, I have gone a tad farther than @Campbell_2021, and have included a calculation for a generalization of the non-centrality parameter that allows the equivalence test for *F*-tests to be applied to variety of designs. - -@Campbell_2021 calculate the *p*-value as: - -$$ -p = p_f(F; J-1, N-J, \frac{N \cdot \Delta}{1-\Delta}) -$$ - -The non-centrality parameter (ncp = $\lambda$) can be calculated with the equivalence bound and the degrees of freedom: - -$$ -\lambda_{eq} = \frac{\Delta}{1-\Delta} \cdot(df_1 + df_2 +1) -$$ - -\newpage - -The *p*-value for the equivalence test ($p_{eq}$) could then be calculated from traditional ANOVA results and the distribution function: - -$$ -p_{eq} = p_f(F; df_1, df_2, \lambda_{eq}) -$$ - -## Example of Equivalence ANOVA Testing - -Using the `InsectSprays` data set in R and the base R `aov` function, I can demonstrate how this omnibus equivalence testing can be applied with TOSTER. From the initial analysis we an see a clear "significant" effect (very small p-value) of the inspect spray. However, we *may* be interested in testing if the effect is practically equivalent. I will arbitrarily set the equivalence bound to a partial eta-squared of 0.35 ($H_0: \eta^2_p > 0.35$). +Using the `InsectSprays` data and the `equ_ftest` function, which accepts summary *F*-test statistics directly: ```{r warning=FALSE, message=FALSE} data("InsectSprays") @@ -715,30 +536,24 @@ anova(aovtest) ``` -We can then use the information in the table above to perform an equivalence test using the `equ_ftest` function. This function returns an object of the S3 class `htest` and the output will look very familiar to that of the t-test. The main difference is the estimates, and confidence interval, are for partial $\eta^2_p$. - ```{r} equ_ftest(Fstat = 34.70228, df1 = 5, df2 = 66, eqb = 0.35) ``` -Based on the results above we would conclude there is a significant effect of "spray" and the differences due to spray are *not* statistically equivalent. In essence, we reject the traditional null hypothesis of "no effect" but accept the null hypothesis of the equivalence test. +Based on these results, there is a significant effect of "spray" and the differences are *not* statistically equivalent --- we reject the traditional null hypothesis but fail to reject the equivalence null hypothesis. \newpage -The `equ_ftest` function is very useful because all you need is very basic summary statistics. However, if you are doing all your analyses in R then you can use the `equ_anova` function. This function accepts objects produced from `stats::aov`, `car::Anova` and `afex::aov_car` (or any ANOVA from derived from `afex`). +## Example with R ANOVA Objects -As a second example, we can use the afex package's data and ANOVA [@afex]. -Again, we will use the equivalence bound of 0.35, -which is a completely arbitrary (and baseless) equivalence bound. -Notice that the output contains 2 p-values: -one for the significance (`p.null`) and another for the equivalence test (`p.equ`). +The `equ_anova` function accepts objects from `stats::aov`, `car::Anova`, and `afex::aov_car`, avoiding the need to extract summary statistics manually: ```{r} -# Example using a purely within-subjects design +# Example using a purely within-subjects design # (Maxwell & Delaney, 2004, Chapter 12, Table 12.5, p. 578): library(afex) data(md_12.1) -aovtest2 = aov_ez("id", "rt", md_12.1, within = c("angle", "noise"), +aovtest2 = aov_ez("id", "rt", md_12.1, within = c("angle", "noise"), anova_table=list(correction = "none", es = "none")) equ_anova(aovtest2, eqb = 0.35) @@ -746,101 +561,67 @@ equ_anova(aovtest2, \newpage -# Equivalence Between Replication Studies - -During the development of this TOSTER update, I was helping advise a team of researchers on a massive replication project for sport and exercise science [@repSES]. -How to determine whether a direct^[Defined as being a as-close-as possible replication to the original study, in contrast to "conceptual" replications.] replication was a successful replication of the original study was contentious topic of conversation among the team. -Inspired by these discussions, I created 2 functions that would utilize the basic principles of SMDs^[The textbook by @borenstein and the some of the works of Wolfgang Vietchbauer, metafor R package author, were a large source of information for developing these functions.] to test for differences between two studies. - -Overall, the concept is simple: if we have estimates of SMDs from two very similar studies we can use the large-sample approximation to compute the sampling variances^[Users can also supply their own sampling variances using the `se1` and `se2` arguments.] to estimate the degree to which the two studies differ from one another (i.e., calculate p-values). The users of TOSTER then have the option to test whether the two SMDs significantly differ, or use TOST to estimate if they are practically equivalent. Additionally, there are two options for comparing SMDs: using the summary statistics or using bootstrapping (assuming original data is available). +# Power Analysis and Sample Size Planning -## Example using Summary Statistics +Researchers planning an equivalence study will immediately need to determine the required sample size. Power for equivalence tests behaves differently than for standard tests: larger samples are generally needed, and power depends on both the true effect size and the width of the equivalence bounds. -In this example, let us imagine an "original" study that reports an effect of Cohen's dz -= 0.95 in a paired samples design with 25 subjects. However, a -replication doubled the sample size, found a non-significant effect at -an SMD of 0.2. Are these two studies compatible (the lower the p-value the lower the compatibility)? Or, to put it another -way, should the replication be considered a "failure" to replicate the original study? +## Power for TOST t-tests -We can use the `compare_smd` function to at least measure how often we -would expect a discrepancy between the original and replication study if -the same underlying effect was being measured (also assuming no -publication bias). +The `power_t_TOST` function computes exact power for TOST with one-sample, two-sample, and paired t-tests. It can solve for any one of `n`, `power`, or `alpha` given the others. The `delta` argument specifies the true mean difference (set to 0 if the true effect is exactly zero) and `eqb` specifies the equivalence bounds. -We can see from the results below that, if the null hypothesis were -true, we would only expect to see a discrepancy in SMDs between studies at -least this large \~1% of the time. +For example, to determine the sample size needed to achieve 80% power to detect equivalence within bounds of $\pm$ 0.5 (assuming the true difference is zero and $SD$ = 1): ```{r} -compare_smd(smd1 = 0.95, - n1 = 25, - smd2 = 0.23, - n2 = 50, - paired = TRUE) +power_t_TOST(delta = 0, sd = 1, eqb = 0.5, + alpha = 0.05, power = 0.8, + type = "two.sample") ``` -Let us also imagine a scenario where a replication team considers a replication successful if the SMDs are within 0.25 units of each other. We can set the `TOST` argument to TRUE, and then set the equivalence bound using `null` argument. +## Power for Correlations + +The `power_z_cor` function provides power analysis for equivalence tests on Pearson correlations using Fisher's z-transformation: ```{r} -compare_smd(smd1 = 0.95, n1 = 25, smd2 = 0.23,n2 = 50, - paired = TRUE, TOST = TRUE, null = .25) +power_z_cor(rho = 0, power = 0.8, + null = 0.3, alpha = 0.05, + alternative = "equivalence") ``` -Based on the imaginary studies we outlined above, we would not reject the null equivalence hypothesis, but reject the null significance hypothesis. Therefore, we would could conclude that there are significant differences between the studies that are not practically equivalent. - -## Example using Bootstrapping - -The above results are only based on an approximating the differences -between the SMDs. If the raw data is available, then the optimal -solution is the bootstrap. This can be accomplished with the -`boot_compare_smd` function. The only drawback to this function is that TOST is -currently not available, and users would instead have to run 2 one-sided tests -manually using the `null` and `alternative` arguments. +## Extensions: PowerTOST and Superpower -For this example, we will simulate some data. As an alternative approach to TOST, -we can just set the `alpha` to 0.1, and then check to see if the -confidence interval is within the preset equivalence bounds. - -```{r} -set.seed(4522) -boot_test = boot_compare_smd(x1 = rnorm(25,.95), x2 = rnorm(50), - paired = TRUE, alpha = .1) -boot_test -``` +For more complex designs --- crossover studies, replicate designs, group-sequential equivalence trials --- the PowerTOST package [@PowerTOST] provides extensive functionality beyond what TOSTER offers. For factorial designs where equivalence contrasts are tested via estimated marginal means, simulation-based power analysis via the Superpower package is recommended. Both packages complement TOSTER's built-in power tools for standard designs. \newpage # Conclusions -In this manuscript I have demonstrated most of the newest functions and features within the TOSTER R package. -I hope that updates to the package builds upon the original impact of the TOSTER package^[In my opinion, the impact of the @lakens_ori cannot be overstated considering it is cited by over 1000 other papers!], -and has been made TOST more accessible to the average researcher. -In addition, I have added a number of other functions that offer robust alternatives to -the t-test for performing TOST analyses. -I would strongly recommend users of TOSTER to explore these functions, and, at the very least, -compare the robust results to the t-test results to ensure that the conclusions do not change -due to the chosen analysis. If your interpretation of the results substantially changes depending on the test, then it would be prudent to explore what features in the data, and the underlying data generating process, might explain this discrepancy. -Lastly, to my knowledge, this is the first package to offer equivalence testing options for ANOVAs or for comparing SMDs between studies. -Overall, this package and its functions offer an easily accessible option for researchers to explore equivalence testing, and hopefully improve their statistical analyses. +This paper has described the major updates to the TOSTER R package, expanding its scope from the original t-test and correlation equivalence tests to a comprehensive toolkit for frequentist equivalence testing. The update adds a unified t-test interface with minimal effect testing support, a simplified hypothesis testing interface that integrates with the standard R ecosystem, tools for standardized effect sizes and correlations, robust and resampling-based alternatives, ANOVA equivalence testing, and power analysis for sample size planning. + +Several limitations should be acknowledged. TOSTER is designed for standard research designs --- it does not handle multilevel or mixed-effects models, latent variable models, or survival data. For these settings, researchers should consider packages like emmeans or marginaleffects, which can accommodate non-zero null hypotheses in more complex modeling frameworks. TOSTER also does not provide Bayesian equivalence testing; researchers interested in Bayes factors for the null should consult packages such as BayesFactor. + +I hope that the updated package continues to build upon the original impact of @lakens_ori and makes equivalence testing more accessible to researchers across disciplines. At the very least, I would encourage users to compare results across methods (e.g., parametric vs. robust) to ensure that conclusions do not depend on the chosen analysis. If the interpretation substantially changes depending on the test, this suggests features of the data --- violations of assumptions, outliers, distributional shape --- that warrant further investigation. \newpage # Additional Information -All analyses/code in this manuscript are from TOSTER v0.9.0: +The TOSTER package can be installed from CRAN or from GitHub: ``` -# Install the exact release with this code -devtools::install_github("Lakens/TOSTER@v0.9.0") +# Install from CRAN +install.packages("TOSTER") + +# Install the development version from GitHub +devtools::install_github("Lakens/TOSTER") ``` ## Acknowledgement(s) {-} -I'd would like to thank everyone from the Lakens' laboratory group for their input and suggestions. +I would like to thank everyone from the Lakens laboratory group for their input and suggestions. ## Disclosure statement {-} -The author of this manuscript is the author of the TOSTER package. +The author of this manuscript is the author of the TOSTER package. Citations of this manuscript will benefit his citation count. ## Funding {-} @@ -862,7 +643,7 @@ Without his help and encouragement, the TOSTER package and this update would not - ncp: non-centrality parameter - SESOI: Smallest Effect Size of Interest - SMD: Standardized Mean Difference (e.g., Cohen's d) -- TOST: Two-One Sided Tests +- TOST: Two One-Sided Tests - WMW: Wilcoxon-Mann-Whitney ## Notes {-} @@ -872,4 +653,3 @@ The R package is also (partially) implemented in jamovi as the TOSTER module. \newpage # References - diff --git a/papers/Avocado_Update/Avocado_Update.docx b/papers/Avocado_Update/Avocado_Update.docx index 9a0e6d094058a0b1b4d87245d8b1ad75cc459685..f73731e3ce2d0dd049188bdfcb5991230c9743ed 100644 GIT binary patch delta 97352 zcmZ6yQ*@xs7rhyq9d>Nnw$ZVjbZm8;H@2OQJGO1xwrx9^@Asd%n3=n}tFzWwwQJXN zcAcgZge5>BDawLFpo74`z<@O5dg{g_kpr7{>r4m|27@=3D74a{dAJD=#yb9+HjpX& zek8NuOtR==ZG!fKYx>c7nkLPTGZ`SYBlq~TKXw)uJFDw-qz;%W?!}p>Sz(|m;on`q 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b/papers/Avocado_Update/toster_assumption_checks_supplement.qmd new file mode 100644 index 00000000..f02dfb1d --- /dev/null +++ b/papers/Avocado_Update/toster_assumption_checks_supplement.qmd @@ -0,0 +1,810 @@ +--- +title: "Diagnostic Checks for Equivalence Tests" +subtitle: "Supplement to the 'Updated TOSTER' Manuscript" +author: "Aaron R. Caldwell" +format: + pdf: + toc: true + toc-depth: 3 + number-sections: true + colorlinks: true + geometry: + - margin=1in +execute: + warning: false + message: false + fig-width: 6 + fig-height: 4 +--- + +# Introduction + +Equivalence testing and its companion procedures are often described as robust or assumption-light, but the tests implemented in TOSTER rely on conditions that can fail in practice. This supplement provides diagnostic workflows for checking those assumptions, covering parametric t-tests, bootstrap and permutation resamples, parametric and bootstrap correlations, rank-based tests (Wilcoxon-Mann-Whitney and Brunner-Munzel), and a brief pointer to ANOVA-style designs. + +These diagnostics are intentionally not built into TOSTER itself. Keeping the package dependency-light is a deliberate design choice, and several of the most useful diagnostics (notably the jackknife-after-bootstrap, ECDF overlays, and Brant's test of proportional odds) require specialized packages or the `boot` package's internal data structures rather than TOSTER's `htest` output. Readers who want these checks should run them separately, as demonstrated below. For ANOVA-type designs, `performance::check_model()` provides a one-call suite of diagnostic plots that we recommend rather than re-implementing here. + +The examples use several datasets: `sleep` for the paired and one-sample cases, `ToothGrowth` for the two-sample and rank-based cases, and `mtcars` (columns `mpg` and `wt`) for the correlation section. For each, we show diagnostics on the observed data and, where instructive, compare with a deliberately problematic simulated case so readers can see what failing diagnostics look like. + +```{r setup} +# setup -------- +library(TOSTER) +library(boot) +library(lawstat) +library(ggplot2) +library(dplyr) +library(MASS) +library(brant) +library(performance) +library(halfmoon) + +set.seed(42) +``` + +# Parametric t-Test Diagnostics + +The parametric t-test assumes independent observations, approximately normal residuals, and (for Student's form) equal variances across groups. The normality requirement is most critical at small `n`, where the central limit theorem offers little protection; by `n` in the mid-twenties per group the t distribution is already a reasonable approximation for moderately non-normal data. Welch's form relaxes the equal-variance assumption and is the default in most modern workflows. The diagnostics below apply regardless of whether the test is used for standard null/nil-hypothesis significance testing or for TOST equivalence testing; equivalence bounds do not alter the distributional assumptions of the test statistic. + +## Two-Sample Parametric t-Test + +We use the same `ToothGrowth` subset (dose of 1 mg, comparing `VC` with `OJ`) that later sections reuse. + +```{r} +# two sample data -------- +tg <- ToothGrowth |> filter(dose == 1) +tg_vc <- tg$len[tg$supp == "VC"] +tg_oj <- tg$len[tg$supp == "OJ"] +``` + +### Residuals from the equivalent linear model + +The two-sample t-test is equivalent to a linear model with a single dichotomous predictor, so residuals from that fit are the right target for the normality check. + +```{r} +# two sample residuals -------- +tg_fit <- lm(len ~ supp, data = tg) + +ggplot(data.frame(resid = resid(tg_fit)), aes(sample = resid)) + + stat_qq() + stat_qq_line() + + labs(title = "QQ plot of residuals (two-sample t-test)", + x = "Theoretical quantiles", y = "Residuals") + + theme_minimal() +``` + +### Shapiro-Wilk on residuals + +```{r} +# shapiro residuals -------- +shapiro.test(resid(tg_fit)) +``` + +Shapiro-Wilk is a useful supplement to the QQ plot but not a substitute. In very small samples it has low power and rarely rejects even for clearly non-normal data; in large samples it becomes over-sensitive and flags minor, inconsequential deviations. The visual check should drive the decision, with the formal test as a secondary input. + +### Variance homogeneity + +```{r} +# levene two sample -------- +lawstat::levene.test(tg$len, tg$supp, location = "median") +``` + +The median-centered Brown-Forsythe variant is preferred over the classical mean-centered Levene test because it is itself robust to non-normality. A small p-value here argues against Student's t-test and for Welch's form; it does not invalidate the test in any deeper sense. + +### Cook's distance + +```{r, fig.height=4} +# cooks d two sample -------- +cd_tg <- cooks.distance(tg_fit) +plot(cd_tg, type = "h", + main = "Cook's distance (two-sample)", + ylab = "Cook's distance", xlab = "Observation index") +abline(h = 4 / length(cd_tg), col = "red", lty = 2) +``` + +Cook's distance measures how much the fitted values change when an observation is removed, combining leverage and residual magnitude into a single number. In the balanced two-sample design used here, leverage is constant across observations, so Cook's D reduces to a monotone transformation of the standardized residual and adds little beyond a residual plot. It becomes more informative in unbalanced designs and in regression settings where leverage varies meaningfully (see the correlation section). + +The `4/n` cutoff plotted above is a common heuristic, not a formal threshold, and should be read as a flag for closer inspection rather than a test. + +A critical caveat: for `n` below roughly 10 per group, Cook's D cutoffs are essentially meaningless. With so few observations every point has substantial influence on the estimate and there is no stable reference distribution to judge unusualness against. At these sample sizes the diagnostic is best understood as a sanity check for gross data-entry errors rather than a formal outlier test. + +## Paired-Sample Parametric t-Test + +The paired t-test is equivalent to a one-sample t-test on the difference scores, so the relevant normality assumption applies to the differences themselves. With `lm(d ~ 1)`, residuals equal deviations from the mean, which is why the QQ plot is shown directly on `d_paired`. + +```{r} +# paired data -------- +d_paired <- sleep$extra[sleep$group == 2] - sleep$extra[sleep$group == 1] +``` + +```{r, fig.height=4} +# paired parametric diagnostics -------- +ggplot(data.frame(d = d_paired), aes(sample = d)) + + stat_qq() + stat_qq_line() + + labs(title = "QQ plot of paired differences", + x = "Theoretical quantiles", y = "Differences") + + theme_minimal() + +shapiro.test(d_paired) + +d_fit <- lm(d_paired ~ 1) +cd_d <- cooks.distance(d_fit) +plot(cd_d, type = "h", + main = "Cook's distance (paired differences)", + ylab = "Cook's distance", xlab = "Observation index") +abline(h = 4 / length(cd_d), col = "red", lty = 2) +``` + +The same small-`n` caveat from the two-sample case applies with extra force here: paired designs often have sample sizes well below 20, in which case the Cook's D plot is best read as a flag for extreme values rather than a formal influence test. + +## One-Sample Parametric t-Test + +Applied to `x_one`, the diagnostics mirror the paired case. + +```{r} +# one sample data -------- +x_one <- sleep$extra[sleep$group == 1] +``` + +```{r, fig.height=4} +# one sample parametric diagnostics -------- +ggplot(data.frame(x = x_one), aes(sample = x)) + + stat_qq() + stat_qq_line() + + labs(title = "QQ plot of one-sample observations", + x = "Theoretical quantiles", y = "Observations") + + theme_minimal() + +shapiro.test(x_one) + +x_fit <- lm(x_one ~ 1) +cd_x <- cooks.distance(x_fit) +plot(cd_x, type = "h", + main = "Cook's distance (one-sample)", + ylab = "Cook's distance", xlab = "Observation index") +abline(h = 4 / length(cd_x), col = "red", lty = 2) +``` + +## Note on Log Transformation + +When residual diagnostics reveal right-skewness, as is common for reaction times, concentrations, and count-like outcomes, a log transformation often resolves skewness, stabilizes variance, and reduces the leverage of the most extreme observations simultaneously. Where that combination holds, analyzing on the log scale is usually preferable to switching to a rank-based or resampling-based test, because it preserves a parametric model with familiar properties. + +The interpretive cost is that the mean of the logged data is the log of the geometric mean, not of the arithmetic mean. Back-transforming the estimate yields a geometric mean (and, for two-group comparisons, a ratio of geometric means), which is usually the appropriate summary on the original scale when the log has stabilized the distribution. For equivalence testing, bounds specified on the log scale correspond to ratio bounds on the original scale. The classical bioequivalence bounds of 80 to 125 percent correspond to $\pm \log(1.25) \approx \pm 0.223$ on the log scale; users should specify `eqb` on the log scale and interpret results as ratios of geometric means. + +Other transformations (square-root, Box-Cox) share similar motivations, but the clean geometric-mean interpretation is specific to the log. + +```{r} +# log transform example -------- +set.seed(99) +x_lognormal <- rlnorm(30, meanlog = 1, sdlog = 0.5) +y_lognormal <- rlnorm(30, meanlog = 1.1, sdlog = 0.5) + +# On log scale: bounds interpreted as log ratios +# log(1.25) ~ 0.223 corresponds to the 80-125% bioequivalence bounds +log_bound <- log(1.25) + +TOSTER::t_TOST(x = log(x_lognormal), + y = log(y_lognormal), + eqb = log_bound, + hypothesis = "EQU") +``` + +# Bootstrap t-Test Diagnostics + +The bootstrap t-test is asymptotically valid under mild conditions — primarily that the empirical distribution is a reasonable stand-in for the population distribution and that observations are independent within groups. These conditions are difficult to verify directly, but several diagnostics help identify cases where the bootstrap approximation is unreliable: influential observations that dominate resampling, extreme skewness in the bootstrap distribution, and small-sample instability. + +## Two-Sample Bootstrap + +We reuse the `ToothGrowth` subset (`tg`) introduced in the parametric section, comparing supplement types (`VC` vs `OJ`) at the 1 mg dose. + +### Check 1: Group-wise distributional inspection + +The studentized two-sample bootstrap tolerates unequal variances but still assumes shapes are comparable beyond location and scale. Dramatic shape differences (e.g., one group skewed, the other symmetric) are a warning sign. + +```{r} +# group distributions -------- +ggplot(tg, aes(x = len, fill = supp)) + + geom_density(alpha = 0.5) + + labs(title = "Distribution of tooth length by supplement type", + x = "Length", y = "Density") + + theme_minimal() + +# Sample size, mean, sd, skewness by group +tg |> + group_by(supp) |> + summarise(n = n(), + mean = mean(len), + sd = sd(len), + skewness = mean((len - mean(len))^3) / sd(len)^3) +``` + +### Check 2: Outlier identification + +Outliers are particularly problematic for the bootstrap because resampling with replacement means an extreme observation can be drawn multiple times within a single bootstrap sample, distorting the replicate distribution. + +```{r} +# outlier check -------- +boxplot(len ~ supp, data = tg, + main = "Boxplots by group (outliers flagged)", + ylab = "Length") +``` + +If outliers are present, either investigate them substantively or consider the trimmed-means option (`tr` argument in `boot_t_test`), which reduces their influence while preserving the mean-based interpretation. + +Cook's distance from the equivalent linear model (covered in the parametric section above) provides a complementary leverage-based assessment, while the jackknife-after-bootstrap described next is specifically diagnostic of bootstrap-distribution sensitivity. The two checks answer related but distinct questions: Cook's D asks how much an observation moves the point estimate under a parametric fit, whereas the jackknife-after-bootstrap asks how much removing an observation shifts the resampled sampling distribution itself. + +### Check 3: Bootstrap distribution and jackknife-after-bootstrap + +The jackknife-after-bootstrap assesses how sensitive the bootstrap distribution is to individual observations. An observation whose removal dramatically shifts the bootstrap distribution is a leverage point that undermines bootstrap inference. This diagnostic requires running the bootstrap through the `boot` package directly. + +```{r, fig.height=5} +# jackknife after bootstrap -------- +# Define the statistic: Welch t-statistic for mean difference +welch_t <- function(data, indices) { + d <- data[indices, ] + x <- d$len[d$supp == "VC"] + y <- d$len[d$supp == "OJ"] + t_num <- mean(x) - mean(y) + t_den <- sqrt(var(x)/length(x) + var(y)/length(y)) + t_num / t_den +} + +# Stratified bootstrap preserves group sizes +boot_out <- boot(data = tg, + statistic = welch_t, + R = 2000, + strata = as.factor(tg$supp)) + +# Bootstrap distribution +plot(boot_out) + +# Jackknife-after-bootstrap +jack.after.boot(boot_out, + main = "Jackknife-after-bootstrap") +``` + +In the jackknife-after-bootstrap plot, each vertical strip shows how the bootstrap distribution shifts when a particular observation is removed. Strips that visibly differ from the others flag influential points. Mild fluctuations across strips are expected; dramatic shifts are cause for concern. + +### Check 4: Replication stability + +Run the bootstrap twice with different seeds and a reasonable `R`. If intervals or p-values differ meaningfully, `R` is too small. + +```{r} +# replication stability -------- +set.seed(1) +r1 <- boot_t_test(formula = len ~ supp, data = tg, R = 2000) +set.seed(2) +r2 <- boot_t_test(formula = len ~ supp, data = tg, R = 2000) + +data.frame(run = c("seed 1", "seed 2"), + p_value = c(r1$p.value, r2$p.value), + ci_low = c(r1$conf.int[1], r2$conf.int[1]), + ci_high = c(r1$conf.int[2], r2$conf.int[2])) +``` + +For inference-grade results with tail-sensitive quantities (e.g., small p-values, narrow equivalence bounds), `R` in the range of 5,000–10,000 is often safer than the 999 used in quick demonstrations. + +## Paired-Sample Bootstrap + +For paired data, the bootstrap resamples the difference scores (`d_paired`, defined in the parametric section). The same checks apply, but on the differences rather than on the raw groups. + +### Distribution and outliers + +```{r} +# paired diagnostics -------- +par(mfrow = c(1, 2)) +hist(d_paired, breaks = 8, main = "Difference scores", + xlab = "Group 2 - Group 1") +abline(v = mean(d_paired), col = "red", lwd = 2) +boxplot(d_paired, main = "Boxplot of differences") +par(mfrow = c(1, 1)) +``` + +### Jackknife-after-bootstrap on differences + +```{r, fig.height=5} +# paired jackknife -------- +paired_t <- function(data, indices) { + d <- data[indices] + mean(d) / (sd(d) / sqrt(length(d))) +} + +boot_paired <- boot(data = d_paired, + statistic = paired_t, + R = 2000) + +plot(boot_paired) +jack.after.boot(boot_paired, + main = "Jackknife-after-bootstrap (paired differences)") +``` + +## One-Sample Bootstrap + +The one-sample case uses the same machinery as the paired case, applied to the raw observations (`x_one`, defined in the parametric section) rather than differences. + +### Distribution, outliers, and jackknife + +```{r, fig.height=5} +# one sample diagnostics -------- +par(mfrow = c(1, 2)) +hist(x_one, breaks = 6, main = "One-sample observations", xlab = "extra") +abline(v = mean(x_one), col = "red", lwd = 2) +boxplot(x_one, main = "Boxplot") +par(mfrow = c(1, 1)) + +one_t <- function(data, indices) { + d <- data[indices] + mean(d) / (sd(d) / sqrt(length(d))) +} + +boot_one <- boot(data = x_one, + statistic = one_t, + R = 2000) + +plot(boot_one) +jack.after.boot(boot_one, main = "Jackknife-after-bootstrap (one-sample)") +``` + +With `n = 10`, leave-one-out influence is expected to be visible — this is the nature of small-sample bootstrap and one of the reasons permutation-based inference is often preferred in this regime. + +# Permutation t-Test Diagnostics + +The permutation t-test has different assumptions depending on design. For two independent samples, the studentized permutation test requires exchangeability under the null and provides asymptotic validity under unequal variances. For paired and one-sample designs, the sign-flip permutation requires **symmetry** of the differences (or observations, in the one-sample case) around the hypothesized value. Symmetry is the key diagnostic target for paired and one-sample permutation tests and is often overlooked. + +## Two-Sample Permutation + +Using the same `ToothGrowth` subset: + +### Check 1: Shape comparability across groups + +```{r} +# group shapes -------- +ggplot(tg, aes(sample = len, color = supp)) + + stat_qq() + stat_qq_line() + + facet_wrap(~ supp) + + labs(title = "QQ plots by group") + + theme_minimal() +``` + +Strong departures in shape (e.g., one group skewed, the other symmetric, or markedly different tail behavior) are a concern because the studentized permutation test handles scale differences but not dramatic shape differences. + +### Check 2: Variance comparison + +While the studentized version tolerates heteroscedasticity, it is still useful to know how different the variances are, both for interpretation and for deciding whether the studentized form is necessary. + +```{r} +# variance comparison -------- +tg |> + group_by(supp) |> + summarise(n = n(), variance = var(len)) + +# Levene's test from lawstat for robust variance comparison +lawstat::levene.test(tg$len, tg$supp, location = "median") +``` + +The median-centered Levene test (Brown–Forsythe) is preferred over the classical version because it is itself robust to non-normality. + +## Paired-Sample Permutation + +This is the case where the symmetry assumption bites hardest. The sign-flip permutation test for paired data is exact under the null **only** if the difference distribution is symmetric around zero (or around the hypothesized mean difference). Asymmetry makes the test approximate rather than exact, though studentization provides asymptotic protection. + +Paired designs frequently produce asymmetric difference distributions — floor or ceiling effects, multiplicative treatment effects, or a subset of strong responders can all induce skew. Checking symmetry is therefore essential. + +### Check 1: Histogram with reference line + +```{r} +# histogram symmetry -------- +ggplot(data.frame(d = d_paired), aes(x = d)) + + geom_histogram(bins = 8, fill = "grey70", color = "white") + + geom_vline(xintercept = mean(d_paired), color = "red", linewidth = 1) + + geom_vline(xintercept = median(d_paired), color = "blue", + linewidth = 1, linetype = "dashed") + + labs(title = "Differences with mean (red) and median (blue)", + x = "Difference", y = "Count") + + theme_minimal() +``` + +A visible gap between the mean and median, or a tail that is clearly longer on one side, flags asymmetry. + +### Check 2: Symmetry plot + +A symmetry plot pairs order statistics equidistant from the median: for each pair, plot the upper deviation against the lower deviation. Under symmetry, points fall along the 45-degree line. + +```{r} +# symmetry plot -------- +symmetry_plot <- function(x) { + m <- median(x) + sorted <- sort(x) + n <- length(sorted) + upper <- sorted[sorted > m] - m + lower <- m - sorted[sorted < m] + k <- min(length(upper), length(lower)) + upper <- sort(upper)[1:k] + lower <- sort(lower)[1:k] + data.frame(lower = lower, upper = upper) +} + +sp <- symmetry_plot(d_paired) +ggplot(sp, aes(x = lower, y = upper)) + + geom_point(size = 2) + + geom_abline(intercept = 0, slope = 1, linetype = "dashed") + + coord_equal() + + labs(title = "Symmetry plot of paired differences", + x = "Distance below median", + y = "Distance above median") + + theme_minimal() +``` + +Points clustered near the 45-degree line support symmetry; systematic departure (all points above or below the line, or strong curvature) indicates asymmetry. + +### Check 3: QQ plot against normal + +Normality is stronger than symmetry, but a normal QQ plot is a familiar tool and distinguishes symmetry failure (systematic curvature) from heavy tails (fanning at the ends with points still roughly straight in the middle). + +```{r} +# qq normal -------- +ggplot(data.frame(d = d_paired), aes(sample = d)) + + stat_qq() + stat_qq_line() + + labs(title = "Normal QQ plot of differences") + + theme_minimal() +``` + +### Check 4: Formal symmetry test via `lawstat` + +```{r} +# formal symmetry test -------- +lawstat::symmetry.test(d_paired, option = "MGG") +lawstat::symmetry.test(d_paired, option = "CM") +``` + +The Miao–Gel–Gastwirth ("MGG") and Cabilio–Masaro ("CM") tests provide formal assessment. With small samples these tests have limited power, so a non-significant result should not be taken as strong evidence of symmetry — the visual checks remain more informative. Conversely, a significant result with small `n` is worth taking seriously. + +### Check 5: Skewness with bootstrap confidence interval + +```{r} +# skewness CI -------- +skew_stat <- function(x, i) { + xi <- x[i] + mean((xi - mean(xi))^3) / sd(xi)^3 +} +skew_boot <- boot(d_paired, skew_stat, R = 2000) +boot.ci(skew_boot, type = "perc") +``` + +A confidence interval for skewness that comfortably covers zero supports symmetry; an interval clearly on one side of zero indicates asymmetry. + +### What to do if symmetry fails + +If symmetry is implausible, options in rough order of preference are: (1) use the bootstrap instead, which does not require symmetry; (2) report the studentized permutation test with an explicit note that it is approximate rather than exact under the observed asymmetry; (3) consider whether the mean is the right estimand, and if a transformation (e.g., log) produces a symmetric difference distribution on a scientifically meaningful scale. + +## One-Sample Permutation + +The one-sample permutation test assumes the observations are symmetric around the hypothesized mean (often zero). Every diagnostic from the paired case applies, substituting the raw observations for the difference scores. + +```{r, fig.height=4} +# one sample symmetry -------- +# Histogram +ggplot(data.frame(x = x_one), aes(x = x)) + + geom_histogram(bins = 6, fill = "grey70", color = "white") + + geom_vline(xintercept = mean(x_one), color = "red", linewidth = 1) + + geom_vline(xintercept = median(x_one), color = "blue", + linewidth = 1, linetype = "dashed") + + labs(title = "One-sample observations", + x = "Value", y = "Count") + + theme_minimal() + +# Symmetry plot +sp_one <- symmetry_plot(x_one) +ggplot(sp_one, aes(x = lower, y = upper)) + + geom_point(size = 2) + + geom_abline(intercept = 0, slope = 1, linetype = "dashed") + + coord_equal() + + labs(title = "Symmetry plot (one-sample)", + x = "Distance below median", + y = "Distance above median") + + theme_minimal() + +# Formal test +lawstat::symmetry.test(x_one, option = "MGG") +``` + +# Correlation Diagnostics + +The three common correlation coefficients make different assumptions about the underlying data. Pearson assumes bivariate normality and a linear relationship between the two variables. Spearman assumes a monotonic relationship and is checkable by applying linear-regression diagnostics to the ranked data: once ranks are in place, the rank-rank scatter should look linear for Spearman to be interpretable as a measure of monotonic association. Kendall's tau shares Spearman's monotonicity requirement but counts concordant and discordant pairs rather than using rank differences, which makes it somewhat more robust to ties and outliers; it generally does not require additional diagnostics beyond those used for Spearman. The bootstrap correlation relaxes distributional assumptions but adds its own: the empirical joint distribution must be a reasonable stand-in for the population joint distribution, and no single observation should dominate the bootstrap variability. + +```{r} +# correlation data -------- +cor_data <- mtcars[, c("mpg", "wt")] +``` + +## Pearson Correlation + +### Linearity and marginal distributions + +```{r, fig.height=3.5} +# pearson scatter -------- +ggplot(cor_data, aes(x = wt, y = mpg)) + + geom_point() + + geom_smooth(method = "loess", se = FALSE) + + labs(title = "mpg vs wt with loess smoother", + x = "Weight (1000 lbs)", y = "Miles per gallon") + + theme_minimal() +``` + +The loess curve should look roughly linear for Pearson to be appropriate. Pronounced curvature suggests either a monotonic-but-nonlinear relationship (in which case Spearman or Kendall is preferable) or a genuinely nonmonotonic pattern (in which case no summary correlation is meaningful). + +```{r, fig.height=3.5} +# pearson marginals -------- +par(mfrow = c(1, 2)) +qqnorm(cor_data$mpg, main = "QQ plot: mpg"); qqline(cor_data$mpg) +qqnorm(cor_data$wt, main = "QQ plot: wt"); qqline(cor_data$wt) +par(mfrow = c(1, 1)) +``` + +Symmetric marginals are a reasonable proxy for bivariate normality in most applied settings. Marginal normality does not guarantee bivariate normality, but marked skew in either marginal is sufficient to reject it. + +### Influence via Cook's distance + +```{r, fig.height=4} +# pearson cooks d -------- +pearson_fit <- lm(mpg ~ wt, data = cor_data) +cd_pearson <- cooks.distance(pearson_fit) +plot(cd_pearson, type = "h", + main = "Cook's distance (Pearson regression)", + ylab = "Cook's distance", xlab = "Observation index") +abline(h = 4 / length(cd_pearson), col = "red", lty = 2) +``` + +Unlike the balanced two-sample t-test case, leverage varies meaningfully in regression, so Cook's D is genuinely informative here: observations with extreme `wt` values (high leverage) combined with large residuals will show up as candidates for driving the estimated correlation. A Pearson correlation that rests on one or two high-leverage points is fragile, and the bootstrap jackknife in the next subsection provides a complementary check. + +Fisher's z-transformation, $z = \tfrac{1}{2}\log\left((1+r)/(1-r)\right)$, is the standard route to parametric confidence intervals on Pearson `r`. It is sometimes treated as assumption-free but in fact still relies on bivariate normality; its coverage can degrade substantially when the bivariate distribution has heavy tails or strong skew. + +## Spearman Correlation + +Spearman's correlation applies Pearson's formula to ranks, so the relevant diagnostics operate on the ranked data. + +```{r, fig.height=3.5} +# spearman ranks -------- +cor_data$mpg_rank <- rank(cor_data$mpg) +cor_data$wt_rank <- rank(cor_data$wt) + +ggplot(cor_data, aes(x = wt_rank, y = mpg_rank)) + + geom_point() + + geom_smooth(method = "loess", se = FALSE) + + labs(title = "Ranked scatter: mpg_rank vs wt_rank", + x = "Rank of wt", y = "Rank of mpg") + + theme_minimal() +``` + +If the original relationship is monotonic, the rank-rank loess curve should look roughly linear. Ranks are uniform by construction, so marginal normality is not a concern; the diagnostic target is linearity of the rank-rank relationship. + +```{r, fig.height=3.5} +# spearman residuals -------- +spearman_fit <- lm(mpg_rank ~ wt_rank, data = cor_data) + +ggplot(data.frame(resid = resid(spearman_fit)), aes(sample = resid)) + + stat_qq() + stat_qq_line() + + labs(title = "QQ plot of residuals (ranked regression)", + x = "Theoretical quantiles", y = "Residuals") + + theme_minimal() +``` + +```{r, fig.height=4} +# spearman cooks d -------- +cd_spearman <- cooks.distance(spearman_fit) +plot(cd_spearman, type = "h", + main = "Cook's distance (Spearman ranked regression)", + ylab = "Cook's distance", xlab = "Observation index") +abline(h = 4 / length(cd_spearman), col = "red", lty = 2) +``` + +The interpretation mirrors the Pearson case: observations with high Cook's D in the ranked regression are candidates for driving the Spearman estimate. Because the regression is on ranks, extreme original-scale values are pulled toward the centre, so Cook's D in this context flags observations whose rank combination is unusual rather than whose raw value is extreme. + +## Kendall's Tau + +Kendall's tau uses the same monotonicity assumption as Spearman but is less sensitive to the magnitude of rank differences because it counts concordant and discordant pairs rather than computing correlation on ranks. This makes it generally more robust to outliers and better-behaved in the presence of ties. The ranked scatterplot from the Spearman subsection serves as the primary visual check for Kendall as well, and additional diagnostics are rarely needed. + +## Bootstrap Correlation Diagnostics + +The bootstrap correlation provides confidence intervals without appealing to bivariate normality or Fisher's z. Its own assumptions can be checked with the same toolkit used for the bootstrap t-test: the shape of the bootstrap distribution and the jackknife-after-bootstrap. + +```{r, fig.height=5} +# bootstrap correlation -------- +cor_stat <- function(data, indices) { + d <- data[indices, ] + cor(d$mpg, d$wt) +} + +boot_cor <- boot(data = cor_data, statistic = cor_stat, R = 2000) + +plot(boot_cor) +jack.after.boot(boot_cor, main = "Jackknife-after-bootstrap (correlation)") +``` + +The bootstrap distribution should be reasonably symmetric. Near $r = \pm 1$, boundary effects make the distribution skewed and Fisher's z helps but does not fully solve the problem; interpret the intervals cautiously in that regime. In the jackknife-after-bootstrap plot, strips that visibly differ from the others flag observations that drive the bootstrap variability of the correlation estimate. + +# Rank-Based Tests: WMW and Brunner-Munzel + +The Wilcoxon-Mann-Whitney (WMW) test is commonly presented as a nonparametric test of medians, but that interpretation requires the **location shift assumption**: the two group distributions differ only in location, not in shape. When shapes differ (for example when one group has greater variance, or the groups have different skewness), WMW tests a hypothesis about stochastic ordering, $P(X < Y)$, rather than a difference in medians. Users who report WMW as a median test when shapes are clearly different are over-interpreting the result. + +**Brunner-Munzel** drops the location-shift requirement and directly tests the stochastic-superiority hypothesis $P(X < Y) + 0.5 \cdot P(X = Y) = 0.5$. The tradeoff is analogous to Welch's correction of Student's t-test: Brunner-Munzel gives up a cleaner location-difference interpretation in exchange for validity under unequal shapes and variances. + +The **proportional odds** assumption is a stronger condition used in ordinal regression: the log odds ratio between groups is constant across all cutpoints of the outcome. WMW can be derived as the score test from a proportional odds model, so when proportional odds holds, WMW has a clean odds-ratio interpretation in Harrell's framework. When it fails, WMW still provides a valid test of $P(X < Y)$ but its interpretation as an odds ratio breaks down. + +The diagnostics below cover (a) location shift, the condition under which the median-difference interpretation is licensed; (b) proportional odds, the condition under which the odds-ratio interpretation is licensed; and (c) the symmetry requirement for the one-sample and paired signed-rank tests. + +```{r} +# rank test data -------- +rank_data <- ToothGrowth |> dplyr::filter(dose == 1) +``` + +## Location Shift Check (Two-Sample) + +The cleanest visual check for location shift is a pair of overlaid empirical CDFs (ECDFs). Under location shift, the two ECDFs are horizontal translations of each other: the horizontal distance between the curves is constant across the range of the outcome. + +```{r, fig.height=3.5} +# location shift ecdf -------- +ggplot(rank_data, aes(x = len, color = supp)) + + halfmoon::geom_ecdf() + + labs(title = "Empirical CDFs by group", + subtitle = "Parallel (horizontally shifted) curves support location shift", + x = "Length", y = "Proportion <= x") + + theme_minimal() +``` + +`halfmoon::geom_ecdf()` is preferred over `ggplot2::stat_ecdf()` because it signals the diagnostic intent and supports weighted ECDFs if the workflow is later extended to more complex designs. + +```{r, fig.height=3.5} +# density overlay -------- +ggplot(rank_data, aes(x = len, fill = supp)) + + geom_density(alpha = 0.5) + + labs(title = "Density overlay by group", + x = "Length", y = "Density") + + theme_minimal() +``` + +If the ECDFs cross, or if the horizontal gap between them varies substantially across the range, location shift is violated. In that case the difference-in-medians interpretation of WMW should not be used; report stochastic superiority or switch to Brunner-Munzel. + +## Proportional Odds Check (Two-Sample) + +Two diagnostics are useful: a visual check of the empirical logit curves and a formal test via ordinal regression. + +### Visual: parallel empirical logit curves + +For each observed value of the outcome, compute the empirical cumulative proportion in each group at or below that value, and transform to logits: $\log(F_g(x)/(1 - F_g(x)))$ for each group $g$. Plotting the two logit curves on the same axes against the outcome value, proportional odds implies that the curves are vertically parallel, with constant vertical distance equal to the log odds ratio. Crossing curves, or non-constant gaps, indicate violation. + +We use a hand-rolled helper here rather than `halfmoon::geom_ecdf()` because that function returns ECDF values on the probability scale. The proportional-odds check requires the logit transformation of those values, where parallel becomes visually meaningful; on the probability scale the sigmoid shape of the ECDF obscures whether the gap is constant or varying. + +```{r, fig.height=4} +# proportional odds visual -------- +po_logit_data <- function(data, outcome, group) { + vals <- sort(unique(data[[outcome]])) + groups <- unique(data[[group]]) + out <- expand.grid(value = vals, group = groups, stringsAsFactors = FALSE) + out$logit <- NA_real_ + for (i in seq_len(nrow(out))) { + g <- out$group[i] + v <- out$value[i] + sub <- data[data[[group]] == g, outcome, drop = TRUE] + p <- mean(sub <= v) + # Clamp to avoid infinities at the extremes + p <- pmin(pmax(p, 1 / (2 * length(sub))), 1 - 1 / (2 * length(sub))) + out$logit[i] <- log(p / (1 - p)) + } + out +} + +po_df <- po_logit_data(rank_data, "len", "supp") +ggplot(po_df, aes(x = value, y = logit, color = group)) + + geom_line(linewidth = 1) + + geom_point() + + labs(title = "Empirical logit curves by group", + subtitle = "Parallel curves support proportional odds", + x = "Outcome value", y = "Empirical logit") + + theme_minimal() +``` + +With small samples the logit curves can look jagged even when proportional odds holds in the population. The visual is most informative when `n` is at least 30 per group; below that threshold it should be read as suggestive rather than conclusive. + +### Formal: Brant test via ordinal regression + +The Brant test formalizes the proportional-odds check by fitting an ordinal regression and testing whether the log-odds slopes are equal across cutpoints. For continuous outcomes the data must first be discretized, which we do here with quartile-based cuts. + +```{r} +# brant test -------- +# Discretize the outcome into ordinal categories for polr +rank_data$len_ord <- cut(rank_data$len, + breaks = quantile(rank_data$len, probs = seq(0, 1, 0.25)), + include.lowest = TRUE, + ordered_result = TRUE) +polr_fit <- MASS::polr(len_ord ~ supp, data = rank_data, Hess = TRUE) +brant::brant(polr_fit) +``` + +The Brant test shares the usual caveats about power in small samples: a non-significant result with small `n` is weak evidence for proportional odds. The discretization of a continuous outcome is also somewhat arbitrary; different cutpoints can yield different conclusions. In practice the visual check and the formal test are complementary, and agreement between them is more informative than either alone. + +## Symmetry Check for Signed-Rank (One-Sample/Paired) + +The Wilcoxon signed-rank test assumes that the distribution of the one-sample values, or of the paired differences, is symmetric around the hypothesized value. This is the same assumption discussed in the permutation section for sign-flip tests. The full diagnostic suite from the paired permutation section (histogram with mean and median reference lines, symmetry plot, normal QQ plot, `lawstat::symmetry.test`, and the bootstrap confidence interval for skewness) applies directly to the signed-rank test. Rather than duplicate the code here, we refer readers to the "Paired-Sample Permutation" section above; the same checks, with the same interpretive thresholds, are the relevant diagnostics for the signed-rank test. + +## When to Use What + +- **Location shift plausible, proportional odds plausible, symmetric differences (for paired or one-sample):** WMW and the signed-rank test are interpretable as tests of the location parameter, with a clean odds-ratio interpretation. +- **Location shift violated but independence and exchangeability hold:** Brunner-Munzel is the more defensible two-sample test. Report stochastic superiority rather than a difference in medians. +- **Proportional odds violated:** WMW still provides a valid test of $P(X < Y)$ but the odds-ratio interpretation is not clean. Consider whether the mean, estimated via a bootstrap or permutation t-test, is a more interpretable estimand for the question at hand. + +# ANOVA Diagnostics + +For ANOVA-style designs (one-way, factorial, repeated-measures, mixed), the `performance` package's `check_model()` function provides a comprehensive diagnostic suite in a single call, covering linearity, homogeneity of variance, normality of residuals, influential observations, and where applicable collinearity. The function accepts `aov`, `lm`, `lmer`, `glm`, and several other model object classes, so the same workflow scales from a simple one-way ANOVA to a mixed model with random effects. For equivalence testing of ANOVA contrasts (for example via TOSTER's ANOVA support), diagnostics should be run on the underlying model fit rather than on the contrast output, because the distributional assumptions live at the model level. + +```{r, fig.height=7} +# anova diagnostics -------- +aov_fit <- aov(len ~ supp * factor(dose), data = ToothGrowth) +performance::check_model(aov_fit) +``` + +When `check_model()` flags a violation, the remedies are the usual ones: consider a transformation (log for right-skew, square-root for count-like variance scaling), use a robust variant such as Welch ANOVA or a mixed model with heteroscedastic residuals, or switch to a resampling-based test on the contrasts of interest. + +# Illustration: What Failing Diagnostics Look Like + +To show readers what problematic data looks like, we simulate a deliberately skewed paired difference distribution (exponential) and apply the same paired diagnostic suite. + +```{r, fig.height=4} +# skewed example -------- +set.seed(2026) +d_skewed <- rexp(20, rate = 1) - 1 # shifted exponential, right-skewed + +# Histogram +ggplot(data.frame(d = d_skewed), aes(x = d)) + + geom_histogram(bins = 10, fill = "grey70", color = "white") + + geom_vline(xintercept = mean(d_skewed), color = "red", linewidth = 1) + + geom_vline(xintercept = median(d_skewed), color = "blue", + linewidth = 1, linetype = "dashed") + + labs(title = "Deliberately skewed differences", + x = "Difference", y = "Count") + + theme_minimal() + +# Symmetry plot +sp_skew <- symmetry_plot(d_skewed) +ggplot(sp_skew, aes(x = lower, y = upper)) + + geom_point(size = 2) + + geom_abline(intercept = 0, slope = 1, linetype = "dashed") + + coord_equal() + + labs(title = "Symmetry plot (skewed example)", + x = "Distance below median", + y = "Distance above median") + + theme_minimal() + +# Formal test +lawstat::symmetry.test(d_skewed, option = "MGG") +``` + +Note the visible mean–median gap, the points systematically above the 45-degree line in the symmetry plot, and (with adequate `n`) a significant symmetry test. In this situation, the sign-flip permutation test would be unreliable as an exact procedure, and the bootstrap would be the preferred resampling approach. + +# Summary + +The table below collects the diagnostics covered in this supplement. It is organized into two groups: tests of means (t-tests and their resampling variants, plus ANOVA), and tests of distributions and associations (rank-based tests and correlations). + +**Tests of means** + +| Design / Test | Key assumption(s) | Primary diagnostics | +|---------------|-------------------|---------------------| +| Two-sample parametric t-test | Independent observations, normal residuals, equal variance (Student) or not (Welch) | QQ plot of residuals, Shapiro-Wilk, Brown-Forsythe, Cook's distance | +| Paired / one-sample parametric t-test | Normality of differences (paired) or observations (one-sample) | QQ plot, Shapiro-Wilk, Cook's D from intercept-only lm | +| Two-sample studentized bootstrap | Independence, stable empirical distribution, no severe outliers | Group densities, boxplots, Cook's D (leverage), jackknife-after-bootstrap, replication stability | +| Paired / one-sample studentized bootstrap | Independence, stable empirical distribution | Histogram, boxplot, jackknife-after-bootstrap | +| Two-sample studentized permutation | Exchangeability; comparable shape (scale allowed to differ) | QQ plots by group, Brown-Forsythe test for variance | +| Paired sign-flip permutation | Symmetry of differences around null value | Histogram w/ mean-median, symmetry plot, normal QQ, `lawstat::symmetry.test`, bootstrap CI for skewness | +| One-sample sign-flip permutation | Symmetry of observations around null value | Same as paired, on raw observations | +| ANOVA (one-way, factorial, mixed) | Independence, normal residuals, homogeneous variance | `performance::check_model()` suite | + +**Tests of distributions and associations** + +| Test | Key assumption(s) | Primary diagnostics | +|------|-------------------|---------------------| +| Pearson correlation | Bivariate normality, linear relationship | Scatter with loess, marginal QQ plots, Cook's D from `lm` | +| Spearman / Kendall correlation | Monotonic relationship | Ranked scatter with loess, QQ of residuals on ranked lm, Cook's D on ranked fit | +| Bootstrap correlation | Stable empirical joint distribution, no single-observation dominance | Bootstrap distribution plot, jackknife-after-bootstrap | +| Wilcoxon-Mann-Whitney | Location shift (for median interpretation); proportional odds (for odds-ratio interpretation) | Overlaid ECDFs, empirical logit curves, Brant test via `MASS::polr()` | +| Brunner-Munzel | Independence, exchangeability (no location-shift required) | ECDF inspection as sanity check; no additional distributional assumptions | +| Wilcoxon signed-rank (paired/one-sample) | Symmetry around null value | Same diagnostics as paired sign-flip permutation | + +Across all designs, replication stability (running resampling tests twice with different seeds) and sensitivity analysis (trying `tr > 0` for trimmed means, or comparing bootstrap, permutation, and parametric results) provide additional robustness checks. When diagnostics conflict, for example when paired differences appear mildly asymmetric, reporting both bootstrap and permutation results and noting any disagreement is more informative than defending a single choice. + + From 91c46f30653b5738c2ba31c732e3382f20d641b8 Mon Sep 17 00:00:00 2001 From: Aaron Caldwell Date: Fri, 17 Apr 2026 10:21:20 -0500 Subject: [PATCH 09/40] Update Avocado_Update.Rmd --- papers/Avocado_Update/Avocado_Update.Rmd | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index 186ca5b7..402930ae 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -398,6 +398,10 @@ boot_smd_calc(formula = extra ~ group, ``` +### Checking Assumptions + +The SMD functions inherit the assumptions of the underlying t-test: independent observations, approximately normal residuals or differences, and, for two-sample designs, comparable variances across groups. Visual diagnostics from the t-test discussion above apply directly. Two considerations are specific to standardized effect sizes, however. First, outliers and skew affect both the numerator (mean difference) and the denominator (standard deviation) of the SMD, which can amplify their influence on the estimate; the `tr` trimmed-means argument available in the `boot_smd_calc` function is particularly worth considering when the raw data show extreme values. Second, the parametric z-test in `smd_calc` relies on an approximation to the sampling distribution of the SMD that can be very inaccurate in small samples, which is the primary motivation for preferring `boot_smd_calc` when hypothesis testing is the goal. + ## Equivalence Tests for Correlations Researchers sometimes need to test that a correlation is practically zero. The `z_cor_test` function provides a parametric approach using the Fisher z-transformation. However, this is only an approximate method and is generally *not recommended* as the primary inferential tool for equivalence testing on correlations. From c00072f9833ec1f544e62927fc0c471448917316 Mon Sep 17 00:00:00 2001 From: Aaron Caldwell Date: Fri, 17 Apr 2026 12:38:01 -0500 Subject: [PATCH 10/40] rendering appropriately Now, need to fix a few things then keep adding checking assumptions sections. --- papers/Avocado_Update/Avocado_Update.R | 130 + papers/Avocado_Update/Avocado_Update.Rmd | 35 +- papers/Avocado_Update/Avocado_Update.docx | Bin 100616 -> 150849 bytes papers/Avocado_Update/Avocado_Update.log | 767 ++++++ papers/Avocado_Update/Avocado_Update.pdf | Bin 635116 -> 390406 bytes papers/Avocado_Update/Avocado_Update.tex | 2189 ++++++++++------- .../figure-docx/cdplot-1.png | Bin 0 -> 27099 bytes .../figure-docx/genericplot-1.png | Bin 0 -> 32209 bytes .../figure-docx/hypplot-1.png | Bin 0 -> 20487 bytes .../figure-docx/shplot1-1.png | Bin 0 -> 20562 bytes .../figure-docx/symmplot-1.png | Bin 0 -> 19722 bytes .../figure-latex/cdplot-1.pdf | Bin 0 -> 18842 bytes .../figure-latex/fig-symmplot-1.pdf | Bin 0 -> 4534 bytes 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b/papers/Avocado_Update/Avocado_Update.R index 28f12184..f1b9053f 100644 --- a/papers/Avocado_Update/Avocado_Update.R +++ b/papers/Avocado_Update/Avocado_Update.R @@ -214,3 +214,133 @@ boot_smd_calc(formula = extra ~ group, alternative = "equivalence", null.value = c(-0.5,0.5)) +## ----------------------------------------------------------------------------- +z_cor_test( + x = bugs$LDLF, + y = bugs$LDHF, + method = "pearson", + alternative = "t") + +## ----------------------------------------------------------------------------- +res_boot_cor = boot_cor_test( + x = bugs$LDLF, + y = bugs$LDHF, + method = "pearson", + alternative = "equivalence", + null = 0.4) + +print(res_boot_cor) + +describe_htest(res_boot_cor) + +## ----------------------------------------------------------------------------- +compare_smd(smd1 = 0.95, + n1 = 25, + smd2 = 0.23, + n2 = 50, + paired = TRUE, + null = .25, + alternative = "equivalence") + +## ----------------------------------------------------------------------------- +compare_cor(r1 = 0.45, + df1 = 48, + r2 = 0.25, + df2 = 78, + alternative = "equivalence", + null = 0.25) + +## ----------------------------------------------------------------------------- +test1 = wilcox_TOST(formula = extra ~ group, + data = sleep, + paired = TRUE, + eqb = .5) +print(test1) + +## ----------------------------------------------------------------------------- +# symmetry plot for sleep paired differences +d_sleep <- sleep$extra[sleep$group == 2] - sleep$extra[sleep$group == 1] +m <- median(d_sleep) +sorted <- sort(d_sleep) +upper <- sorted[sorted > m] - m +lower <- m - sorted[sorted < m] +k <- min(length(upper), length(lower)) +sp_df <- data.frame(lower = sort(lower)[1:k], upper = sort(upper)[1:k]) + +ggplot(sp_df, aes(x = lower, y = upper)) + + geom_point(size = 2) + + geom_abline(intercept = 0, slope = 1, linetype = "dashed") + + coord_equal() + + labs(x = "Distance below median", + y = "Distance above median") + + theme_minimal() + +## ----------------------------------------------------------------------------- +bm_test = brunner_munzel(formula = extra ~ group, + data = sleep, + alternative = "equivalence", + mu = c(0.3, 0.7)) +print(bm_test) + +## ----------------------------------------------------------------------------- +set.seed(4522) +boot_t_test(formula = extra ~ group, + data = sleep, + paired = TRUE, + alternative = "equivalence", + mu = c(-0.5, 0.5), + R = 999) + +## ----------------------------------------------------------------------------- +set.seed(891111) +test1 = boot_t_TOST(formula = extra ~ group, + data = sleep, + paired = TRUE, + eqb = .5, + R = 999) +print(test1) + +## ----------------------------------------------------------------------------- +set.seed(8812) +perm_t_test(formula = extra ~ group, + data = sleep, + paired = TRUE, + alternative = "equivalence", + mu = c(-0.5, 0.5), + R = 999) + +## ----error=FALSE-------------------------------------------------------------- +log_TOST(mpg ~ am, data = mtcars) + +## ----error=FALSE-------------------------------------------------------------- +boot_log_TOST(mpg ~ am, data = mtcars, R=999) + +## ----warning=FALSE, message=FALSE--------------------------------------------- +data("InsectSprays") +aovtest = aov(count ~ spray, data = InsectSprays) +anova(aovtest) + + +## ----------------------------------------------------------------------------- +equ_ftest(Fstat = 34.70228, df1 = 5, df2 = 66, eqb = 0.35) + +## ----------------------------------------------------------------------------- +# Example using a purely within-subjects design +# (Maxwell & Delaney, 2004, Chapter 12, Table 12.5, p. 578): +library(afex) +data(md_12.1) +aovtest2 = aov_ez("id", "rt", md_12.1, within = c("angle", "noise"), + anova_table=list(correction = "none", es = "none")) +equ_anova(aovtest2, + eqb = 0.35) + +## ----------------------------------------------------------------------------- +power_t_TOST(delta = 0, sd = 1, eqb = 0.5, + alpha = 0.05, power = 0.8, + type = "two.sample") + +## ----------------------------------------------------------------------------- +power_z_cor(rho = 0, power = 0.8, + null = 0.3, alpha = 0.05, + alternative = "equivalence") + diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index 402930ae..7bee11bb 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -8,9 +8,9 @@ author: email: caldwellaaron@uams.edu type: PREPRINT output: - papaja::apa6_docx: default rticles::tf_article: extra_dependencies: ["float"] + papaja::apa6_docx: default bibliography: interactcadsample.bib #appendix: appendix.tex abstract: | @@ -35,6 +35,7 @@ header-includes: | \usepackage{hyperref} \usepackage[utf8]{inputenc} \def\tightlist{} + \providecommand{\pandocbounded}[1]{#1} \hypersetup{ colorlinks=true, citecolor = cyan, @@ -464,6 +465,10 @@ compare_cor(r1 = 0.45, null = 0.25) ``` +When raw data are available for both studies, `boot_compare_cor` provides a bootstrap alternative that avoids the distributional approximations underlying the Fisher z and Kraatz methods. + +The comparison functions `compare_smd` and `compare_cor` inherit the assumptions of the underlying effect size estimates they combine. Because both studies contribute to the comparison, violations in either study propagate into the test. When raw data are available, the bootstrap alternatives (`boot_compare_smd` and `boot_compare_cor`) are preferred because they avoid the Fisher z and related parametric approximations, which can be inaccurate particularly when effect sizes are large or sample sizes are modest. When only summary statistics are available, readers should report the diagnostic context of the original and replication studies where possible. + \newpage **A note on argument names for equivalence bounds.** The equivalence bound argument is named differently across TOSTER functions for historical and compatibility reasons: `eqb` in `t_TOST`, `wilcox_TOST`, `equ_ftest`, and `equ_anova`; `mu` in `simple_htest`, `brunner_munzel`, `boot_t_test`, and `perm_t_test` (for compatibility with base R's `t.test`); and `null` in `compare_smd` and `z_cor_test`. Despite the different names, the interpretation is the same: the bound(s) defining the region of practical equivalence. @@ -494,6 +499,34 @@ print(test1) Based on these results, we would conclude there is a significant difference and the results are also not equivalent. But again, the estimate we are testing is the pseudo-median, not the mean or median difference, so this conclusion may not be relevant to the research question at hand. +#### Checking Assumptions + +The WMW tests assume independent observations and exchangeability under the null, the latter meaning that group labels carry no information about outcomes if there is no effect. For the interpretations discussed above to be clean, two additional assumptions matter. The location-shift assumption — that the two group distributions differ only in location, not shape — is what allows WMW results to be interpreted as a test of the median difference via the Hodges-Lehmann estimator. Violations are best diagnosed by overlaying the empirical cumulative density functions of the two groups; under location shift, the curves are horizontal translations of each other, whereas crossing curves or varying horizontal gaps indicate violation. For the paired and one-sample signed-rank variants used in `wilcox_TOST` or `simple_htest`, the signed-rank procedure additionally requires that the difference distribution is symmetric around the null value for the pseudo-median to coincide with the mean or median difference. Asymmetry is readily checked with a symmetry plot of the paired differences (@fig-symmplot). As with the location-shift case, visible departures suggest that the test's output — the pseudo-median and associated p-value — does not map cleanly onto a hypothesis about central tendency. + + +```{r } +#| label: fig-symmplot +#| echo: false +#| fig.cap: "Symmetry plot of the paired differences from the sleep dataset, pairing each observation's distance above the median with the corresponding distance below. Under symmetry, points fall along the 45-degree reference line (dashed); systematic departures — points consistently above or below the line — indicate asymmetry. The signed-rank variant of WMW requires this symmetry for the pseudo-median to coincide with the mean or median of the differences." +# symmetry plot for sleep paired differences +d_sleep <- sleep$extra[sleep$group == 2] - sleep$extra[sleep$group == 1] +m <- median(d_sleep) +sorted <- sort(d_sleep) +upper <- sorted[sorted > m] - m +lower <- m - sorted[sorted < m] +k <- min(length(upper), length(lower)) +sp_df <- data.frame(lower = sort(lower)[1:k], upper = sort(upper)[1:k]) + +ggplot(sp_df, aes(x = lower, y = upper)) + + geom_point(size = 2) + + geom_abline(intercept = 0, slope = 1, linetype = "dashed") + + coord_equal() + + labs(x = "Distance below median", + y = "Distance above median") + + theme_minimal() +``` + + ### Brunner-Munzel Test The Brunner-Munzel test [@brunner2000] estimates the relative effect $\hat{p} = P(X > Y) + 0.5 \cdot P(X = Y)$, sometimes referred to as the probability of superiority --- the probability that a randomly chosen observation from one group exceeds a randomly chosen observation from the other, with ties split evenly. 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At the desired error rate, it cannot be stated that the true correlation is between 0.4 and -0.4."[] + [] + +[18] +Underfull \hbox (badness 7273) in paragraph at lines 972--976 +[]\OT1/cmr/m/n/10.95 For com-par-ing cor-re-la-tion co-ef-fi-cients be-tween in-de-pen-dent stud-ies, the + [] + + +Overfull \hbox (17.4029pt too wide) in paragraph at lines 1001--1001 +[]\OT1/cmtt/m/n/10.95 ## Difference between two independent correlations (Fisher's z transform)[] + [] + +[19] [20] +Overfull \hbox (34.64896pt too wide) in paragraph at lines 1112--1112 +[]\OT1/cmtt/m/n/10.95 ## NHST: reject null significance hypothesis that the effect is equal to zero[] + [] + + +Package natbib Warning: Citation `fig-symmplot' on page 21 undefined on input line 1138. + + +File: Avocado_Update_files/Avocado_Update_files/figure-latex/fig-symmplot-1.pdf Graphic file (type pdf) + +Package pdftex.def Info: Avocado_Update_files/Avocado_Update_files/figure-latex/fig-symmplot-1.pdf used on input line 1145. +(pdftex.def) Requested size: 469.75386pt x 325.2142pt. + +Overfull \hbox (61.75386pt too wide) in paragraph at lines 1145--1152 + [][] + [] + +[21] [22 +pdfTeX warning (ext4): destination with the same identifier (name{figure.5}) has been already used, duplicate ignored + ...shipout:D \box_use:N \l_shipout_box + \__shipout_drop_firstpage_... +l.1167 + <./Avocado_Update_files/Avocado_Update_files/figure-latex/fig-symmplot-1.pdf>] [23] [24] [25] +Overfull \hbox (34.64896pt too wide) in paragraph at lines 1401--1401 +[]\OT1/cmtt/m/n/10.95 ## NHST: reject null significance hypothesis that the effect is equal to zero[] + [] + +[26] [27] +Overfull \hbox (28.90027pt too wide) in paragraph at lines 1521--1521 +[]\OT1/cmtt/m/n/10.95 ## The equivalence test was non-significant, t(23.96) = -1.363, p = 9.07e-01[] + [] + + +Overfull \hbox (28.90027pt too wide) in paragraph at lines 1521--1521 +[]\OT1/cmtt/m/n/10.95 ## The null hypothesis test was significant, t(23.96) = -3.826, p = 8.19e-04[] + [] + + +Overfull \hbox (28.90027pt too wide) in paragraph at lines 1521--1521 +[]\OT1/cmtt/m/n/10.95 ## NHST: reject null significance hypothesis that the effect is equal to one[] + [] + +[28] +Overfull \hbox (28.90027pt too wide) in paragraph at lines 1556--1556 +[]\OT1/cmtt/m/n/10.95 ## The equivalence test was non-significant, t(23.96) = -1.363, p = 9.54e-01[] + [] + + +Overfull \hbox (11.6542pt too wide) in paragraph at lines 1556--1556 +[]\OT1/cmtt/m/n/10.95 ## The null hypothesis test was significant, t(23.96) = -3.826, p = 0e+00[] + [] + + +Overfull \hbox (17.4029pt too wide) in paragraph at lines 1556--1556 +[]\OT1/cmtt/m/n/10.95 ## NHST: reject null significance hypothesis that the effect is equal to 1[] + [] + +[29] [30] +Underfull \hbox (badness 2221) in paragraph at lines 1636--1639 +\OT1/cmr/m/n/10.95 The \OT1/cmtt/m/n/10.95 equ[]anova \OT1/cmr/m/n/10.95 func-tion ac-cepts ob-jects from \OT1/cmtt/m/n/10.95 stats::aov\OT1/cmr/m/n/10.95 , \OT1/cmtt/m/n/10.95 car::Anova\OT1/cmr/m/n/10.95 , and + [] + + +Overfull \hbox (23.15158pt too wide) in paragraph at lines 1659--1659 +[]\OT1/cmtt/m/n/10.95 ## effect df1 df2 F.value p.null pes eqbound p.equ[] + [] + + +Overfull \hbox (23.15158pt too wide) in paragraph at lines 1659--1659 +[]\OT1/cmtt/m/n/10.95 ## 1 (Intercept) 1 9 598.44917 1.526600e-09 0.9851839 0.35 0.9999997[] + [] + + +Overfull \hbox (23.15158pt too wide) in paragraph at lines 1659--1659 +[]\OT1/cmtt/m/n/10.95 ## 2 angle 2 18 40.71910 2.086763e-07 0.8189831 0.35 0.9992557[] + [] + + +Overfull \hbox (23.15158pt too wide) in paragraph at lines 1659--1659 +[]\OT1/cmtt/m/n/10.95 ## 3 noise 1 9 33.76596 2.559737e-04 0.7895522 0.35 0.9763228[] + [] + + +Overfull \hbox (23.15158pt too wide) in paragraph at lines 1659--1659 +[]\OT1/cmtt/m/n/10.95 ## 4 angle:noise 2 18 45.31034 9.424093e-08 0.8342857 0.35 0.9996103[] + [] + +[31] +Overfull \hbox (11.6542pt too wide) in paragraph at lines 1733--1733 +[]\OT1/cmtt/m/n/10.95 ## Approximate Power for Pearson Product-Moment Correlation (z-test)[] + [] + +[32] [33] [34] [35] [36] (./Avocado_Update.bbl +Underfull \hbox (badness 10000) in paragraph at lines 53--54 +[]\OT1/cmr/m/n/10 Food and Drug Ad-min-is-tra-tion. 2014. \OT1/cmr/m/it/10 Bioavail-abil-ity and Bioe-quiv-a-lence + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 53--54 +\OT1/cmr/m/it/10 Stud-ies Sub-mit-ted in NDAs or INDs | Gen-eral Con-sid-er-a-tions\OT1/cmr/m/n/10 . Vol. + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 53--54 +\OT1/cmr/m/n/10 FDA-2014-D-0204. Cen-ter for Drug Eval-u-a-tion and Re-search. [][]$\OT1/cmtt/m/n/10 https : / / + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 53--54 +\OT1/cmtt/m/n/10 www . fda . gov / regulatory-[]information / search-[]fda-[]guidance-[]documents / + [] + +[37] [38] +Underfull \hbox (badness 10000) in paragraph at lines 119--120 +[]\OT1/cmr/m/n/10 Thulin, M^^Wans. 2024. ``Mod-ern Statis-tics with R.'' 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\newcommand{\AnnotationTok}[1]{\textcolor[rgb]{0.56,0.35,0.01}{\textbf{\textit{#1}}}} -\newcommand{\AttributeTok}[1]{\textcolor[rgb]{0.77,0.63,0.00}{#1}} +\newcommand{\AttributeTok}[1]{\textcolor[rgb]{0.13,0.29,0.53}{#1}} \newcommand{\BaseNTok}[1]{\textcolor[rgb]{0.00,0.00,0.81}{#1}} \newcommand{\BuiltInTok}[1]{#1} \newcommand{\CharTok}[1]{\textcolor[rgb]{0.31,0.60,0.02}{#1}} \newcommand{\CommentTok}[1]{\textcolor[rgb]{0.56,0.35,0.01}{\textit{#1}}} \newcommand{\CommentVarTok}[1]{\textcolor[rgb]{0.56,0.35,0.01}{\textbf{\textit{#1}}}} -\newcommand{\ConstantTok}[1]{\textcolor[rgb]{0.00,0.00,0.00}{#1}} +\newcommand{\ConstantTok}[1]{\textcolor[rgb]{0.56,0.35,0.01}{#1}} \newcommand{\ControlFlowTok}[1]{\textcolor[rgb]{0.13,0.29,0.53}{\textbf{#1}}} \newcommand{\DataTypeTok}[1]{\textcolor[rgb]{0.13,0.29,0.53}{#1}} \newcommand{\DecValTok}[1]{\textcolor[rgb]{0.00,0.00,0.81}{#1}} @@ -51,7 +51,7 @@ \newcommand{\ErrorTok}[1]{\textcolor[rgb]{0.64,0.00,0.00}{\textbf{#1}}} \newcommand{\ExtensionTok}[1]{#1} \newcommand{\FloatTok}[1]{\textcolor[rgb]{0.00,0.00,0.81}{#1}} -\newcommand{\FunctionTok}[1]{\textcolor[rgb]{0.00,0.00,0.00}{#1}} +\newcommand{\FunctionTok}[1]{\textcolor[rgb]{0.13,0.29,0.53}{\textbf{#1}}} \newcommand{\ImportTok}[1]{#1} \newcommand{\InformationTok}[1]{\textcolor[rgb]{0.56,0.35,0.01}{\textbf{\textit{#1}}}} \newcommand{\KeywordTok}[1]{\textcolor[rgb]{0.13,0.29,0.53}{\textbf{#1}}} @@ -60,7 +60,7 @@ \newcommand{\OtherTok}[1]{\textcolor[rgb]{0.56,0.35,0.01}{#1}} \newcommand{\PreprocessorTok}[1]{\textcolor[rgb]{0.56,0.35,0.01}{\textit{#1}}} \newcommand{\RegionMarkerTok}[1]{#1} -\newcommand{\SpecialCharTok}[1]{\textcolor[rgb]{0.00,0.00,0.00}{#1}} +\newcommand{\SpecialCharTok}[1]{\textcolor[rgb]{0.81,0.36,0.00}{\textbf{#1}}} \newcommand{\SpecialStringTok}[1]{\textcolor[rgb]{0.31,0.60,0.02}{#1}} \newcommand{\StringTok}[1]{\textcolor[rgb]{0.31,0.60,0.02}{#1}} \newcommand{\VariableTok}[1]{\textcolor[rgb]{0.00,0.00,0.00}{#1}} @@ -76,11 +76,12 @@ \usepackage{hyperref} \usepackage[utf8]{inputenc} \def\tightlist{} +\providecommand{\pandocbounded}[1]{#1} \hypersetup{ colorlinks=true, citecolor = cyan, linkcolor=blue, - filecolor=magenta, + filecolor=magenta, urlcolor=blue } @@ -95,132 +96,184 @@ \author{\name{Aaron R. Caldwell$^{a}$} -\affil{$^{a}$Natick, MA, \url{https://orcid.org/0000-0002-4541-6283}} +\affil{$^{a}$Department of Biostatistics, Fay W. Boozman College of +Public Health, University of Arkansas for Medical Sciences Northwest, +Springdale, AR \url{https://orcid.org/0000-0002-4541-6283}} } -\thanks{CONTACT Aaron R. -Caldwell. Email: \href{mailto:arcaldwell49@gmail.com}{\nolinkurl{arcaldwell49@gmail.com}}} \maketitle \begin{abstract} -Equivalence testing is arguably under utilized by experimental -researchers. Due to limited software support for such analyses, and -little education on the topic in graduate programs, the utilization of -equivalence testings still appares to be low. One option for equivalence -testing is the use of two one-sided tests (TOST). The TOSTER R package -and jamovi module, originally developed by Daniel Lakens in 2017, was -created to make TOST more accessible to the average researcher. In the -past two years, I have made significant changes to the TOSTER package in -order to increase its accessibility and provide more robust analysis -options for researchers. In this paper, I will detail the changes to the -package and highlight new analysis options that will make TOST easier -for the average quantitative researcher. +Despite decades of methodological guidance, researchers continue to +misinterpret non-significant results as evidence for the absence of an +effect. Equivalence testing via two one-sided tests (TOST) provides a +principled alternative, but adoption remains limited by software +accessibility and lack of training. The TOSTER R package, originally +developed by Lakens (2017), made TOST accessible to experimental +psychologists. This paper describes a major update to the package that +expands its scope with (1) a unified t-test interface supporting both +equivalence and minimal effect testing, (2) a simplified hypothesis +testing interface returning standard R objects, (3) tools for computing +and testing standardized effect sizes and correlations, (4) robust and +resampling-based alternatives including bootstrap and permutation tests, +(5) equivalence testing for ANOVAs, and (6) power analysis and sample +size planning. Each capability is demonstrated with worked examples. +TOSTER is also available as a jamovi module for researchers who prefer a +graphical interface. \end{abstract} \begin{keywords} -statistics, bootstrap, minimal effects test, NHST, TOST +statistics, bootstrap, minimal effects test, NHST, TOST, R package, +robust statistics \end{keywords} -\hypertarget{introduction}{% -\section{Introduction}\label{introduction}} - -Researchers often erroneously declare that no statistical effect exists -based on a single ``non-significant'' p-value \citep{blandaltman95}. In -many of these cases, the data may corroborate the researcher's claim, -but the interpretation of a null hypothesis significance test (NHST), -wherein the lack of significance is considered evidence of ``no -effect'', is nonetheless incorrect. In order to statistically test for -whether there is practically no effect, researchers could use -equivalence testing. Equivalence testing is used when the goal of a -statistical test is to demonstrate that the difference between two -conditions is too small to be meaningful. For example, if a researcher -wanted to test whether a new drug was no worse than a standard drug, the -null hypothesis would be that the new drug is worse than the standard -drug by more than a meaningful amount, and the alternative hypothesis -would be that the difference between the two drugs is small enough to be -meaningless. A very simple equivalence testing approach is the use of -``two one-sided tests'' (TOST) \citep{schuirmann1987}. - -The TOST procedure is a statistical test of whether a parameter (e.g., -mean difference) is within a specified interval. The TOST procedure can -be used to test the equivalence of two means, two proportions, two -regression coefficients, and even two variances. An upper (\(\Delta_U\)) -and lower (\(\Delta_L\)) equivalence bound is specified based on the -smallest effect size of interest (SESOI). If the TOST is below a -pre-specified alpha level, then the effect can be considered close -enough to zero to be practically equivalent \citep{lakens_ori}. - -Both the complaints about erroneous conclusions regarding equivalence -\citep{blandaltman95} and proposed statistical solutions -\citep{schuirmann1987} have existed for decades now. Yet, the problem -appears to persist in many applied disciplines. I believe the continued -dissonance is due to a general lack of education on equivalence testing -and a struggle for many applied researchers to implement equivalence -testing. In my experience, most researchers have received some degree of -statistical training in their doctoral or master's studies, but it is -rare that any have idea of how to use TOST. It may also be difficult to -implement equivalence testing for many researchers. This may be caused -by most statistical software defaulting to a null hypothesis of zero, or -even completely lacking an ability to change the null hypothesis. -Therefore, I feel the continued development of educational content on -TOST, and software to help with such analyses, would be beneficial to -many quantitative researchers. +\section{Introduction}\label{introduction} + +Researchers routinely interpret non-significant p-values as evidence +that no effect exists --- a logical error that has been recognized for +decades \citep{blandaltman95}. A non-significant result means only that +the data are compatible with the null hypothesis; it does not mean the +null hypothesis is true. + +Equivalence testing provides a principled framework for evaluating +whether an effect is small enough to be considered negligible. The two +one-sided tests (TOST) procedure is the most widely used approach +\citep{schuirmann1987}. Researchers specify an upper (\(\Delta_U\)) and +lower (\(\Delta_L\)) equivalence bound based on the smallest effect size +of interest (SESOI). Two one-sided tests then evaluate whether the +observed effect falls within those bounds. If both tests reject at a +pre-specified alpha level, the effect can be considered practically +equivalent to zero \citep{lakens_ori}. The TOSTER R package\footnote{All updates to the package can be found on the package's website \url{https://aaroncaldwell.us/TOSTERpkg}} was -originally developed in by \citet{lakens_ori} to introduce experimental +originally developed by \citet{lakens_ori} to introduce experimental psychologists to the concept of equivalence testing and provide an -easy-to-use implementation in R. In the years since that publication, I -have made a significant update to the package in order to improve the -user interface and expand the tools available within the package. An -experienced R programmer may have no problem performing equivalence -testing within R, but beginners may struggle with both writing the code -and interpreting the output. If you fall into that category, I would -suggest using jamovi, an open-source statistical software, that has a -TOSTER module to perform equivalence/TOST analyses. Not all the features -listed in this manuscript are available in the jamovi module, but it is -a good starting point for most researchers without statistical -programming experience. - -In this manuscript, I will detail the updates to the TOSTER package, and -give some basic usage examples of some of the new functions. This is -meant to just be an introduction to \emph{how} to perform such analyses, -and provide a little bit of context for when such analyses are -appropriate. For a greater introduction to equivalence testing, I would -suggest reading other methodological tutorials -\citep{lakens_ori, lakens2018equivalence, lakens2020improving, mazzolari2022myths}. - -\hypertarget{tost-with-t-tests}{% -\section{TOST with t-tests}\label{tost-with-t-tests}} - -In an effort to make TOSTER more informative and easier to use, a new -function \texttt{t\_TOST} was created. This function operates very -similarly to base R's \texttt{t.test} function, but performs 3 t-tests -(one two-tailed and two one-tailed tests). In addition, this function -has a generic method where two vectors can be supplied or a formula can -be given (e.g.,\texttt{y\ \textasciitilde{}\ group}). This function also -makes it easier to switch between types of t-tests. All three types (two -sample, one sample, and paired samples) can be performed/calculated from -the same function. Moreover, the output from this function is verbose, -and should make the decisions derived from the function more informative -and user-friendly. - -Also, \texttt{t\_TOST} is not limited to equivalence tests. Minimal -effects testing (MET) is possible. MET is useful for situations where -the hypothesis is about a minimal effect and the \emph{null hypothesis -is equivalence} (see Figure 1) \citep{mazzolari2022myths}. +accessible implementation in R. The package provided dedicated functions +for independent, paired, and one-sample t-tests, as well as correlations +and meta-analytic equivalence tests. It has been widely adopted, with +over 1000 citations to date. + +However, the original package had a number of limitations. It offered a +function-per-design interface (separate functions for each t-test type), +lacked robust alternatives to parametric tests, provided no support for +ANOVA equivalence testing, and had no tools for comparing effect sizes +across studies. The package also did not integrate with standard R +workflows and usually just printed results to the console rather than +returning objects that could be manipulated or plotted. + +This paper provides a tutorial-style introduction to the updated TOSTER +package, organized around its major capabilities: (1) a unified t-test +interface (\texttt{t\_TOST}) that pairs a standard significance test +with an equivalence test and supports minimal effect testing, (2) a +simplified hypothesis testing interface (\texttt{simple\_htest}) that +returns standard \texttt{htest} objects, (3) tools for computing and +testing standardized effect sizes --- including standardized mean +differences, rank-based effect sizes, and correlations, (4) robust and +resampling-based alternatives including the Brunner-Munzel test, +bootstrap and permutation t-tests, and log-transformed methods, (5) +equivalence testing for ANOVAs, and (6) power analysis and sample size +planning. Each section motivates when and why a researcher would use the +tool before demonstrating how. + +TOSTER occupies a specific niche in the R ecosystem in that it is a +frequentist toolkit for equivalence testing of standard designs. +Researchers interested in Bayesian approaches to evidence for the null +have excellent options (e.g., \texttt{bayestestR}). For +equivalence-style contrasts in more complex models --- mixed models, +marginal means --- the \texttt{emmeans} and \texttt{marginaleffects} +packages can accommodate non-zero null hypotheses. TOSTER fills the +space of accessible, self-contained equivalence testing for common, +albeit simple, research designs. For a broader introduction to the +methodology, see \citet{lakens_ori}, \citet{lakens2018equivalence}, +\citet{lakens2020improving}, and \citet{mazzolari2022myths}. TOSTER is +also available as a \href{https://www.jamovi.org/}{jamovi} module for +researchers who prefer a graphical interface although it has limited +capabilities compared to the R package. + +\section{Setting Equivalence Bounds}\label{setting-equivalence-bounds} + +Before applying any equivalence test, the researcher must specify the +equivalence bounds which defines the range of effect sizes considered +too small to be meaningful. There is no single correct method for +choosing these bounds; the appropriate approach depends entirely on the +research question and domain of research. + +If a researcher has an equivalence hypothesis, they should be able to +articulate what magnitude of effect would be too small to matter in +context. The SESOI must flow from the hypothesis formulated by theory +and context not from a formula. However, there are several general +strategies that have been evaluated \citep{delta2}. Anchor-based methods +tie the bound to an externally meaningful criterion, such as a +clinically meaningful difference or a ``just-noticeable'' difference +\citep{sesoi}. Distribution-based methods define the bound as a fraction +of the measurement error \citep{Jacobson_Truax_1991}, though this +approach is generally not recommended \citep{delta2}. Cost-benefit, also +referred to as health economic methods \citep{delta2}, reasoning sets +the bound at the point where there is a net benefit relative to the cost +of an intervention. Researchers can draw on existing literature or +normative data to identify effect magnitudes that prior work suggests +are negligible. When theory is sufficiently precise, however, the bound +may require no external benchmark at all, and the hypothesis itself +implies what magnitude of effect would be too small to matter. + +Using ``default'' bounds on the standardized mean difference (SMD; also +known as Cohen's d) based on Cohen's benchmarks (e.g., SMD = 0.2, 0.5, +0.8 to imply small, medium, or large effects, respectively) without +domain-specific justification is strongly discouraged. Cohen himself +stated that these conventions were ``recommended for use only when no +better basis for estimating the index is available'' (p.~25, as cited in +\citet{panzarella2021}) and reportedly came to regret having proposed +them at all \citep{funder2019}. These benchmarks were never intended as +universal thresholds; applied uncritically, they can produce equivalence +bounds that are too wide, too narrow, and likely entirely disconnected +from the substantive research question. + +If a researcher cannot specify meaningful bounds, this itself is +informative --- it suggests the equivalence hypothesis is not well +enough developed to test such a hypothesis. In such cases, it may be +more productive to focus on effect estimation and description rather +than leaping to hypothesis tests \citep{scheel_hypothesis}. Throughout +the examples in this paper, equivalence bounds, and the data sets +themselves, are chosen for illustrative purposes; readers should not +treat them as direct recommendations for their own research. + +\section{TOST with t-tests}\label{tost-with-t-tests} + +The \texttt{t\_TOST} function provides a unified interface for +equivalence and minimal effect testing with t-tests. It runs a standard +nil significance test (\$H\_0: \$) alongside the equivalence (or minimal +effect) test, because in practice researchers may want both: ``Is there +a significant effect?'' and ``Can I rule out a meaningful effect?'' The +function handles independent, paired, and one-sample designs through the +\texttt{paired} argument and accepts data via either a formula interface +or raw vectors, making it a direct analog to base R's \texttt{t.test}. +Additionally, the output includes an SMD calculation alongside the +unstandarized mean difference with appropriate, albeit approximate, +confidence intervals. For researchers who only want to test a single +hypothesis at a time, the \texttt{simple\_htest} function (described in +the next section) provides a lighter alternative. + +Minimal effect testing (MET) is also supported. MET is useful when the +hypothesis is that an effect \emph{exceeds} a meaningful threshold --- +the null hypothesis is equivalence, and rejection implies a practically +meaningful effect (see Figure 1) \citep{mazzolari2022myths}. \begin{figure} \centering -\includegraphics{Avocado_Update_files/figure-latex/hypplot-1.pdf} -\caption{Type of Hypothesis} +\pandocbounded{\includegraphics[keepaspectratio]{Avocado_Update_files/Avocado_Update_files/figure-latex/hypplot-1.pdf}} +\caption{Types of hypotheses tests supported by two one-sided tests +(TOST) procedures.} \end{figure} \newpage -In these examples of \texttt{t\_TOST}, we will use the \texttt{bugs} -data from the \texttt{jmv} R package and the \texttt{sleep} data. +\subsection{Independent Groups +Example}\label{independent-groups-example} + +In these examples we will use the \texttt{sleep} data and the +\texttt{bugs} data from the \texttt{jmv} R package. \begin{Shaded} \begin{Highlighting}[] @@ -230,66 +283,27 @@ \section{TOST with t-tests}\label{tost-with-t-tests}} \end{Highlighting} \end{Shaded} -\hypertarget{independent-groups}{% -\subsection{Independent Groups}\label{independent-groups}} - -For this example, we will use the sleep data. In this data, there is a -\texttt{group} variable and an outcome \texttt{extra}. - -\begin{Shaded} -\begin{Highlighting}[] -\FunctionTok{head}\NormalTok{(sleep,}\DecValTok{2}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## extra group ID -## 1 0.7 1 1 -## 2 -1.6 1 2 -\end{verbatim} - -We will assume the data are independent (in reality this is paired -data), and that we have equivalence bounds of +/- 0.5 units of -\texttt{extra}. All we need to do is provide the \texttt{formula}, -\texttt{data}, and \texttt{eqb} arguments for the function to run -appropriately. In addition, we can set the \texttt{var.equal} argument -(to assume equal variance), and the \texttt{paired} argument (sets if -the data is paired or not). Both are logical indicators that can be set -to TRUE or FALSE. The \texttt{alpha} is automatically set to 0.05 but -this can also be adjusted by the user depending on the desired -alpha-level\footnote{I strongly recommend users ``justify their alpha'' - \citep{jya1, jya2}, and the justification process can be aided by my - other R package \href{https://aaroncaldwell.us/Superpower}{Superpower}}. - -Standardize mean differences (SMDs) are provided in the output for any -t-test based TOST analysis (e.g., Cohen's d). The Hedges's corrected SMD -\citep{hedges_bias} is automatically calculated, but this can be -overridden with the \texttt{bias\_correction} argument\footnote{Glass's - delta can also be produced in the output by using the \texttt{glass} - argument}. In previous versions of this package, the equivalence -bounds could be set by the SMD (e.g., equivalence bound of 0.5 SD), but -this is an erroneous approach since the bound would be dependent upon -the \emph{sample} variance. However, users can opt for such an analysis -by setting \texttt{eqbound\_type} to SMD, which will produce a -noticeable warning to the R console. - -The \texttt{hypothesis} argument is automatically set to ``EQU'' for -equivalence, but if a minimal effect is of interest then ``MET'' can be -supplied. +For the first example, we use the \texttt{sleep} data with equivalence +bounds (set by the \texttt{eqb} argument) of \(\pm\) 0.5 units. The +\texttt{smd\_ci} argument can also be set to use one of four possible +calculations for the SMD confidence interval. In the example below, I +have opted to go with the normal approximation for computational ease; +such approximations tend provide adequate coverage +\citep{Viechtbauer_2007}. The formula interface works almost identically +to \texttt{t.test}: \begin{Shaded} \begin{Highlighting}[] -\CommentTok{\# Formula Interface} -\NormalTok{res1 }\OtherTok{=} \FunctionTok{t\_TOST}\NormalTok{(}\AttributeTok{formula =}\NormalTok{ extra }\SpecialCharTok{\textasciitilde{}}\NormalTok{ group, }\AttributeTok{data =}\NormalTok{ sleep, } - \AttributeTok{eqb =}\NormalTok{ .}\DecValTok{5}\NormalTok{, }\AttributeTok{smd\_ci =} \StringTok{"t"}\NormalTok{)} -\CommentTok{\# x \& y Interface} -\NormalTok{res1a }\OtherTok{=} \FunctionTok{t\_TOST}\NormalTok{(}\AttributeTok{x =} \FunctionTok{subset}\NormalTok{(sleep,group}\SpecialCharTok{==}\DecValTok{1}\NormalTok{)}\SpecialCharTok{$}\NormalTok{extra,} - \AttributeTok{y =} \FunctionTok{subset}\NormalTok{(sleep,group}\SpecialCharTok{==}\DecValTok{2}\NormalTok{)}\SpecialCharTok{$}\NormalTok{extra, }\AttributeTok{eqb =}\NormalTok{.}\DecValTok{5}\NormalTok{)} +\NormalTok{res1 }\OtherTok{=} \FunctionTok{t\_TOST}\NormalTok{(}\AttributeTok{formula =}\NormalTok{ extra }\SpecialCharTok{\textasciitilde{}}\NormalTok{ group, }\CommentTok{\# difference in "group" of outcome "extra"} + \AttributeTok{data =}\NormalTok{ sleep, }\CommentTok{\# data} + \AttributeTok{eqb =}\NormalTok{ .}\DecValTok{5}\NormalTok{, }\CommentTok{\# equivalence bound} + \AttributeTok{smd\_ci =} \StringTok{"z"}\NormalTok{) }\CommentTok{\# sets type of SMD confidence interval calculation} \end{Highlighting} \end{Shaded} -Once the function has run, we can print the results with the -\texttt{print} method. This provides a verbose summary of the results. +The \texttt{print} method provides a verbose summary including the +two-tailed test, both one-sided equivalence tests, and the standardized +mean difference (Hedges' \(g\) by default): \begin{Shaded} \begin{Highlighting}[] @@ -301,8 +315,8 @@ \subsection{Independent Groups}\label{independent-groups}} ## ## Welch Two Sample t-test ## -## The equivalence test was non-significant, t(17.78) = -1.272, p = 8.9e-01 -## The null hypothesis test was non-significant, t(17.78) = -1.861, p = 7.94e-02 +## The equivalence test was non-significant, t(17.78) = -1.3, p = 0.89 +## The null hypothesis test was non-significant, t(17.78) = -1.86, p = 0.08 ## NHST: don't reject null significance hypothesis that the effect is equal to zero ## TOST: don't reject null equivalence hypothesis ## @@ -315,382 +329,762 @@ \subsection{Independent Groups}\label{independent-groups}} ## Effect Sizes ## Estimate SE C.I. Conf. Level ## Raw -1.5800 0.8491 [-3.0534, -0.1066] 0.9 -## Hedges's g(av) -0.7965 0.5992 [-1.8362, 0.2433] 0.9 +## Hedges's g(av) -0.7965 0.4900 [-1.6024, 0.0094] 0.9 ## Note: SMD confidence intervals are an approximation. See vignette("SMD_calcs"). \end{verbatim} \newpage -Another nice feature is the generic \texttt{plot} method that can -provide a visual summary of the results. Most of the plots in this -package were inspired by the -\href{https://cran.r-project.org/package=concurve}{concurve} R package -\citep{rafi2020}. There are two types of plots that can be produced. The -first, and default, is the consonance density plot -(\texttt{type\ =\ "cd"}). +The default visualization provided by the generic \texttt{plot} method +for \texttt{t\_TOST} provides a simple visualization of both effect +sizes and their respective confidence intervals alongside the +equivalence bounds. Test information is also printed at the top of the +plot. \begin{Shaded} \begin{Highlighting}[] -\FunctionTok{plot}\NormalTok{(res1, }\AttributeTok{type =} \StringTok{"cd"}\NormalTok{)} +\FunctionTok{plot}\NormalTok{(res1)} \end{Highlighting} \end{Shaded} \begin{figure} \centering -\includegraphics{Avocado_Update_files/figure-latex/cdplot-1.pdf} -\caption{Example of consonance density plot.} +\pandocbounded{\includegraphics[keepaspectratio]{Avocado_Update_files/Avocado_Update_files/figure-latex/genericplot-1.pdf}} +\caption{Results of the independent-samples equivalence test visualized +with the default plot method. The top panel displays the raw mean +difference with its 90\% confidence interval, and the bottom panel +displays the standardized effect size (Hedges's g\textsubscript{av}) +with its 90\% confidence interval. In both panels, the dashed vertical +lines indicate the equivalence bounds (raw: -0.5 to 0.5; standardized: +approximately -0.3 to 0.3), and the point estimate is marked by a filled +circle. A 90\% confidence interval is used because it corresponds to the +two one-sided tests (TOST) procedure at an alpha level of 0.05.} \end{figure} -\newpage - -The shading pattern can be modified with the \texttt{ci\_shades}. +Also when the generic \texttt{plot} method has the \texttt{type} +argument set to ``cd'' this produces a consonance density plot, inspired +by the ``concurve'' R package \citep{rafi2020}, that shows the estimate, +confidence intervals, and equivalence bounds simultaneously: \begin{Shaded} \begin{Highlighting}[] -\FunctionTok{plot}\NormalTok{(res1, }\AttributeTok{type =} \StringTok{"cd"}\NormalTok{,} - \AttributeTok{ci\_shades =} \FunctionTok{c}\NormalTok{(.}\DecValTok{9}\NormalTok{,.}\DecValTok{95}\NormalTok{))} +\FunctionTok{plot}\NormalTok{(res1, }\AttributeTok{type =} \StringTok{"cd"}\NormalTok{)} \end{Highlighting} \end{Shaded} \begin{figure} \centering -\includegraphics{Avocado_Update_files/figure-latex/shadeplot-1.pdf} -\caption{Demonstrating the shading in plot method.} +\pandocbounded{\includegraphics[keepaspectratio]{Avocado_Update_files/Avocado_Update_files/figure-latex/cdplot-1.pdf}} +\caption{Consonance density plot for the independent-samples equivalence +test. The top panel displays the standardized effect size (Hedges's +g\textsubscript{av}) and the bottom panel displays the raw mean +difference. In each panel, the consonance density curve represents the +distribution of confidence intervals across all levels, with color-coded +regions corresponding to the 68\%, 90\%, 95\%, and 99.9\% confidence +intervals. The point estimate is marked by a filled circle along the +x-axis. The dashed vertical lines indicate the equivalence bounds +(standardized: -0.26 to 0.26; raw: -0.5 to 0.5). This visualization +conveys the full range of effect sizes compatible with the data at +varying confidence levels, rather than relying on a single confidence +interval. Wider, less certain intervals (e.g., 99.9\%) extend further +from the point estimate, while narrower intervals (e.g., 68\%) cluster +near it.} \end{figure} +Other plot types are available: \texttt{type\ =\ "c"} produces a +consonance plot showing multiple confidence interval levels, and shading +can be customized with the \texttt{ci\_shades} argument. + \newpage -Consonance plots, where all confidence intervals can be simultaneous -plotted, can also be produced. The advantage here is multiple confidence -interval lines can plotted at once. +\subsection{One-Sample t-test with Asymmetric +Bounds}\label{one-sample-t-test-with-asymmetric-bounds} + +When the equivalence bounds are not symmetric around the null, two +values can be supplied to \texttt{eqb}. This is common when the research +question implies asymmetric consequences. Similarly, we may want to test +the mean itself rather than a mean difference. This harkens back to the +original use of the t-test by Gosset at Guinness Brewery, where the +concern was whether a single sample mean fell within acceptable limits. +Here we test whether the mean of LDHF falls within the bounds of 5.5 and +8.5 around a hypothesized value of 7.5: \begin{Shaded} \begin{Highlighting}[] -\FunctionTok{plot}\NormalTok{(res1, }\AttributeTok{type =} \StringTok{"c"}\NormalTok{,} - \AttributeTok{ci\_lines =} \FunctionTok{c}\NormalTok{(.}\DecValTok{9}\NormalTok{,.}\DecValTok{95}\NormalTok{))} +\NormalTok{res4 }\OtherTok{=} \FunctionTok{t\_TOST}\NormalTok{(}\AttributeTok{x =}\NormalTok{ bugs}\SpecialCharTok{$}\NormalTok{LDHF, }\CommentTok{\# simple sample, vector from which to estimate mean} + \AttributeTok{mu =} \FloatTok{7.5}\NormalTok{, }\CommentTok{\# set nil significance test to 7.5} + \AttributeTok{eqb =} \FunctionTok{c}\NormalTok{(}\FloatTok{5.5}\NormalTok{,}\FloatTok{8.5}\NormalTok{) }\CommentTok{\# set TOST bounds to 5.5 and 8.5} +\NormalTok{ )} +\NormalTok{res4} \end{Highlighting} \end{Shaded} -\begin{figure} -\centering -\includegraphics{Avocado_Update_files/figure-latex/conplot-1.pdf} -\caption{Example of consonance plot.} -\end{figure} +\begin{verbatim} +## +## One Sample t-test +## +## The equivalence test was significant, t(90) = -4.2, p < 0.01 +## The null hypothesis test was non-significant, t(90) = -0.458, p = 0.65 +## NHST: don't reject null significance hypothesis that the effect is equal to 7.5 +## TOST: reject null equivalence hypothesis +## +## TOST Results +## t df p.value +## t-test -0.4577 90 0.648 +## TOST Lower 7.1156 90 < 0.001 +## TOST Upper -4.2444 90 < 0.001 +## +## Effect Sizes +## Estimate SE C.I. Conf. Level +## Raw -0.12088 0.2641 [6.9402, 7.818] 0.9 +## Hedges's g -0.04758 0.1049 [-0.2185, 0.1236] 0.9 +## Note: SMD confidence intervals are an approximation. See vignette("SMD_calcs"). +\end{verbatim} + +The same interface handles paired designs by setting +\texttt{paired\ =\ TRUE}. Standardized mean differences are reported +automatically; Hedges's correction is applied by default but can be +toggled with \texttt{bias\_correction}, and Glass's delta can be +requested via the \texttt{glass} argument. Setting equivalence bounds in +SMD units (via \texttt{eqbound\_type\ =\ "SMD"}) is possible but +discouraged because the bound becomes dependent on sample variance (see +the Standardized Effect Sizes section below). + +\subsection{Checking Assumptions}\label{checking-assumptions} + +The parametric t-tests implemented in \texttt{t\_TOST} assume +independent observations, approximately normal residuals (or +differences, in the paired and one-sample cases), and, for Student's +form, equal variances between groups; Welch's correction relaxes the +equal-variance assumption and is the default. The residual normality +assumption matters most in small samples, where the central limit +theorem provides limited protection. Visual diagnostics are generally +more informative than formal tests: QQ plots of residuals or differences +for normality, side-by-side boxplots or density overlays for variance +comparison, and inspection of residual plots or boxplots for extreme or +influential observations. A worked diagnostic workflow for each design +is provided in the online supplement. When assumptions are seriously in +doubt, the bootstrap and permutation alternatives discussed in later +sections provide valid inference under relaxed ----- though not +eliminated ----- distributional requirements. \newpage -\hypertarget{paired-sample}{% -\subsection{Paired Sample}\label{paired-sample}} +\subsection{Using Summary Statistics}\label{using-summary-statistics} -To perform TOST on paired samples, the process does not change much. We -could process the test the same way by providing a formula. All we would -need to then is change \texttt{paired} to TRUE. +When only summary statistics are available (e.g., from a published +article), the \texttt{tsum\_TOST} function performs the same tests using +sample means, standard deviations, sample sizes, and (for paired +designs) the correlation between measures\footnote{The + \texttt{extract\_r\_paired} function can be used if the correlation + between paired observations is not readily available.}: \begin{Shaded} \begin{Highlighting}[] -\NormalTok{res2 }\OtherTok{=} \FunctionTok{t\_TOST}\NormalTok{(}\AttributeTok{formula =}\NormalTok{ extra }\SpecialCharTok{\textasciitilde{}}\NormalTok{ group,} - \AttributeTok{data =}\NormalTok{ sleep,} - \AttributeTok{paired =} \ConstantTok{TRUE}\NormalTok{,} - \AttributeTok{eqb =}\NormalTok{ .}\DecValTok{5}\NormalTok{)} -\NormalTok{res2} +\NormalTok{res\_tsum }\OtherTok{=} \FunctionTok{tsum\_TOST}\NormalTok{(} + \AttributeTok{m1 =} \FunctionTok{mean}\NormalTok{(bugs}\SpecialCharTok{$}\NormalTok{LDHF, }\AttributeTok{na.rm=}\ConstantTok{TRUE}\NormalTok{), } + \AttributeTok{sd1 =} \FunctionTok{sd}\NormalTok{(bugs}\SpecialCharTok{$}\NormalTok{LDHF, }\AttributeTok{na.rm=}\ConstantTok{TRUE}\NormalTok{),} + \AttributeTok{n1 =} \FunctionTok{length}\NormalTok{(}\FunctionTok{na.omit}\NormalTok{(bugs}\SpecialCharTok{$}\NormalTok{LDHF)),} + \AttributeTok{hypothesis =} \StringTok{"EQU"}\NormalTok{, }\AttributeTok{smd\_ci =} \StringTok{"t"}\NormalTok{, } + \AttributeTok{eqb =} \FunctionTok{c}\NormalTok{(}\FloatTok{5.5}\NormalTok{, }\FloatTok{8.5}\NormalTok{)} +\NormalTok{)} + +\NormalTok{res\_tsum} \end{Highlighting} \end{Shaded} \begin{verbatim} ## -## Paired t-test +## One-sample t-test ## -## The equivalence test was non-significant, t(9) = -2.777, p = 9.89e-01 -## The null hypothesis test was significant, t(9) = -4.062, p = 2.83e-03 +## The equivalence test was significant, t(90) = -4.244, p = 2.66e-05 +## The null hypothesis test was significant, t(90) = 27.942, p = 3.91e-46 ## NHST: reject null significance hypothesis that the effect is equal to zero -## TOST: don't reject null equivalence hypothesis +## TOST: reject null equivalence hypothesis ## ## TOST Results ## t df p.value -## t-test -4.062 9 0.003 -## TOST Lower -2.777 9 0.989 -## TOST Upper -5.348 9 < 0.001 +## t-test 27.942 90 < 0.001 +## TOST Lower 7.116 90 < 0.001 +## TOST Upper -4.244 90 < 0.001 ## ## Effect Sizes -## Estimate SE C.I. Conf. Level -## Raw -1.580 0.389 [-2.293, -0.867] 0.9 -## Hedges's g(z) -1.174 0.411 [-1.8046, -0.4977] 0.9 +## Estimate SE C.I. Conf. Level +## Raw 7.379 0.2641 [6.9402, 7.818] 0.9 +## Hedges's g 2.905 0.2395 [2.4289, 3.3804] 0.9 ## Note: SMD confidence intervals are an approximation. See vignette("SMD_calcs"). \end{verbatim} -\newpage - -However, we may have two vectors of data that are paired. So instead we -may want to just provide those separately rather than using a data set -and setting the formula. This can be demonstrated with the ``bugs'' -data. +\section{Simplified Hypothesis +Testing}\label{simplified-hypothesis-testing} + +The \texttt{t\_TOST} function is designed to be comprehensive in that it +runs three tests, computes standardized effect sizes, and produces +verbose output. This is useful for exploration or situations where +researchers \emph{always} want to pair a traditional nil significance +test with an equivalence test. However, sometimes a researcher has a +single specific hypothesis to test, or simply wants less verbose output +(e.g., SMD calculation is not wanted). The \texttt{simple\_htest} +function provides a lighter interface for this purpose. It returns a +standard R \texttt{htest} object --- the same class returned by +\texttt{t.test()} and \texttt{wilcox.test()} --- but with expanded +\texttt{alternative} options that include \texttt{"equivalence"} and +\texttt{"minimal.effect"}. This means researchers can perform +equivalence testing with the same familiar interface they already use +for standard hypothesis tests\footnote{In fact, users can mostly replace + \texttt{t.test()} and \texttt{wilcox.test()} with + \texttt{simple\_htest} as it also allows for the standard alternative + hypotheses of two-sided, greater, less. If you have a hypothesis that + would traditionally utilize a t-test or Wilcoxon-Mann-Whitney (WMW) + test, then you can jump right to using \texttt{simple\_htest} in + TOSTER}. + +Alongside \texttt{simple\_htest}, a set of helper functions makes it +easy to work with any \texttt{htest} object: \texttt{as\_htest} converts +verbose TOST output to the simpler format, \texttt{describe\_htest} +generates formatted text for reporting, \texttt{df\_htest} converts +results to data frames for tabulation, and \texttt{plot\_htest\_est} +produces estimate plots with equivalence bounds. Checking assumptions +would follow the same workflow as \texttt{t\_TOST}, as both are using +t-tests, as discussed in the online supplement. + +\subsection{The simplified function}\label{the-simplified-function} + +The \texttt{simple\_htest} function supports \texttt{t.test} and +\texttt{wilcox.test} as the underlying test (via the \texttt{test} +argument) and all five alternative hypotheses: \texttt{"two.sided"}, +\texttt{"less"}, \texttt{"greater"}, \texttt{"equivalence"}, and +\texttt{"minimal.effect"}. The \texttt{mu} parameter specifies +equivalence bounds --- a single value creates symmetric bounds (e.g., if +1 is provided at \texttt{mu} with \texttt{alternative\ =\ "equivalence"} +then the equivalence region will be {[}-1,1{]}), or two values can be +given for asymmetric bounds. \begin{Shaded} \begin{Highlighting}[] -\NormalTok{res3 }\OtherTok{=} \FunctionTok{t\_TOST}\NormalTok{(}\AttributeTok{x =}\NormalTok{ bugs}\SpecialCharTok{$}\NormalTok{LDHF,} - \AttributeTok{y =}\NormalTok{ bugs}\SpecialCharTok{$}\NormalTok{LDLF,} - \AttributeTok{paired =} \ConstantTok{TRUE}\NormalTok{,} - \AttributeTok{eqb =} \DecValTok{1}\NormalTok{)} -\NormalTok{res3} +\NormalTok{res\_sh1 }\OtherTok{\textless{}{-}} \FunctionTok{simple\_htest}\NormalTok{(extra }\SpecialCharTok{\textasciitilde{}}\NormalTok{ group, } + \AttributeTok{data =}\NormalTok{ sleep,} + \AttributeTok{alternative =} \StringTok{"equivalence"}\NormalTok{, } + \AttributeTok{mu =} \FloatTok{0.5}\NormalTok{)} +\NormalTok{res\_sh1} \end{Highlighting} \end{Shaded} \begin{verbatim} ## -## Paired t-test -## -## The equivalence test was non-significant, t(90) = 2.655, p = 9.95e-01 -## The null hypothesis test was significant, t(90) = 6.649, p = 2.22e-09 -## NHST: reject null significance hypothesis that the effect is equal to zero -## TOST: don't reject null equivalence hypothesis -## -## TOST Results -## t df p.value -## t-test 6.649 90 < 0.001 -## TOST Lower 10.642 90 < 0.001 -## TOST Upper 2.655 90 0.995 +## Welch Two Sample t-test ## -## Effect Sizes -## Estimate SE C.I. Conf. Level -## Raw 1.6648 0.2504 [1.2487, 2.081] 0.9 -## Hedges's g(z) 0.6911 0.1167 [0.4987, 0.8802] 0.9 -## Note: SMD confidence intervals are an approximation. See vignette("SMD_calcs"). +## data: extra by group +## t = -1.2719, df = 17.776, p-value = 0.8901 +## alternative hypothesis: equivalence +## null values: +## difference in means difference in means +## -0.5 0.5 +## 90 percent confidence interval: +## -3.0533815 -0.1066185 +## sample estimates: +## mean of group '1' mean of group '2' +## 0.75 2.33 +## mean difference ('1' - '2') +## -1.58 +\end{verbatim} + +Notice how the output is a single clean \texttt{htest} object. This is +much more concise than the verbose \texttt{t\_TOST} output shown +previously on purpose. If you want to perform additional tests --- such +as a complementary nil significance tests --- then that has to be done +separately. However, there are some methods (described below) that make +this \texttt{htest} format easy to deal with. + +\subsection{Describing and Tabulating +Results}\label{describing-and-tabulating-results} + +For writing up results, \texttt{describe\_htest} generates a formatted +text string following typical scientific reporting conventions: + +\begin{Shaded} +\begin{Highlighting}[] +\FunctionTok{describe\_htest}\NormalTok{(res\_sh1)} +\end{Highlighting} +\end{Shaded} + +\begin{verbatim} +## [1] "The Welch Two Sample t-test is not statistically significant (t(17.776) = -1.27, p = 0.89, mean of group '1' = 0.75, mean of group '2' = 2.33, mean difference ('1' - '2') = -1.58, 90% C.I.[-3.05, -0.107]) at a 0.05 alpha-level. The null hypothesis cannot be rejected. At the desired error rate, it cannot be stated that the true difference in means is between -0.5 and 0.5." \end{verbatim} +\subsection{Plotting Results}\label{plotting-results} + +A visual summary of the estimate and confidence interval can be produced +with \texttt{plot\_htest\_est}. For equivalence and minimal effect +tests, the equivalence bounds are shown as dashed vertical lines: + +\begin{Shaded} +\begin{Highlighting}[] +\FunctionTok{plot\_htest\_est}\NormalTok{(res\_sh1)} +\end{Highlighting} +\end{Shaded} + +\begin{figure} +\centering +\pandocbounded{\includegraphics[keepaspectratio]{Avocado_Update_files/Avocado_Update_files/figure-latex/shplot1-1.pdf}} +\caption{Estimate plot for an equivalence test via simple\_htest.} +\end{figure} + \newpage -Additionally, a MET, instead of equivalence testing, can be performed -with the \texttt{hypothesis} argument set to ``MET''. With this setting, -the hypothesis being tested is whether the effect is \emph{greater} than -the equivalence bound. +\section{Standardized Effect Sizes}\label{standardized-effect-sizes} + +This section provides a unified treatment of the standardized effect +size tools in TOSTER --- standardized mean differences (SMDs), +correlation coefficients, and tools for comparing both between studies. +Before demonstrating these tools, it is worth discussing the limitations +of standardized effect sizes, because these limitations are especially +important for equivalence testing. + +\subsection{Problems with Standardized Effect +Sizes}\label{problems-with-standardized-effect-sizes} + +Standardized effect sizes --- Cohen's \(d\), correlation coefficients, +standardized regression coefficients --- are appealing because they +allow comparison across studies and scales, are required by many +journals, and serve as the primary currency of meta-analysis. However, +they have well-documented problems that are especially relevant to +equivalence testing. + +Standardized mean differences are sensitive to the sample variance. The +``same'' real-world effect produces different \(d\) values across +populations and study designs. The ubiquitously reported Cohen's \(d\) +conflates the magnitude of the mean difference with the precision of +measurement, which means that a narrow equivalence bound on \(d\) may +correspond to a very different raw-scale bound depending on the +variability of the sample \citep{Caldwell2020, baguley2009}. This makes +it unclear what, substantively, is being declared ``equivalent'' --- are +we claiming the means are close, or that the means are close +\emph{relative to the variance}? The choice of denominator (pooled SD, +control SD, within-subject SD) further complicates that same dubious +interpretation. + +Correlations have analogous issues. \citet{tukey1969} noted that +correlations are influenced by the range of both variables, so the +``same'' bivariate relationship produces different coefficients +depending on sample selection. A correlation of 0.3 in a +restricted-range sample may correspond to a much larger association in +the full population. Standardized regression coefficients share these +vulnerabilities. These depend on the variances of both the predictor and +outcome, which makes them unstable across samples and difficult to +interpret substantively \citep{greenland1991}. + +The implication for equivalence testing is straightforward. When the +measurement scale is meaningful, researchers should generally prefer +\emph{unstandardized} (raw-scale) equivalence bounds. When standardized +bounds are used, the choice should be justified with reference to the +specific population and measurement context, not borrowed from generic +benchmarks. + +With these caveats in mind, the following subsections describe the tools +TOSTER provides for computing and testing standardized effects assuming +they are appropriate. + +\subsection{Standardized Mean Difference +Functions}\label{standardized-mean-difference-functions} + +The \texttt{smd\_calc} function computes standardized mean differences +--- Cohen's \(d\), Hedges' \(g\) (bias-corrected), or Glass's \(\Delta\) +--- with confidence intervals and \emph{optional} hypothesis testing +(utilizing a z-test: \(z = (SMD - \mu)/SE\)). It supports one-sample, +paired, and two-sample designs, and can return either an \texttt{htest} +object or a data frame. The SMD is already reported in \texttt{t\_TOST} +output, but \texttt{smd\_calc} allows standalone computation and +supports equivalence testing directly on the effect size. This may be +useful as a complementary function when using \texttt{simple\_htest} as +the primary statistical testing function. + +When estimation rather than testing is the goal, \texttt{smd\_calc} can +be called with \texttt{alternative\ =\ "none"} (the default), which +returns only the point estimate and confidence interval without +performing a hypothesis test. This is the recommended usage when the +primary objective is to report the magnitude of the effect. \begin{Shaded} \begin{Highlighting}[] -\NormalTok{res3a }\OtherTok{=} \FunctionTok{t\_TOST}\NormalTok{(}\AttributeTok{x =}\NormalTok{ bugs}\SpecialCharTok{$}\NormalTok{LDHF,} - \AttributeTok{y =}\NormalTok{ bugs}\SpecialCharTok{$}\NormalTok{LDLF,} - \AttributeTok{paired =} \ConstantTok{TRUE}\NormalTok{,} - \AttributeTok{hypothesis =} \StringTok{"MET"}\NormalTok{,} - \AttributeTok{eqb =} \DecValTok{1}\NormalTok{)} -\NormalTok{res3a} +\FunctionTok{smd\_calc}\NormalTok{(}\AttributeTok{formula =}\NormalTok{ extra }\SpecialCharTok{\textasciitilde{}}\NormalTok{ group,} + \AttributeTok{data =}\NormalTok{ sleep,} + \AttributeTok{paired =} \ConstantTok{TRUE}\NormalTok{,} + \AttributeTok{bias\_correction =} \ConstantTok{TRUE}\NormalTok{,} + \AttributeTok{rm\_correction =} \ConstantTok{TRUE}\NormalTok{,} + \AttributeTok{output =} \StringTok{"htest"}\NormalTok{)} \end{Highlighting} \end{Shaded} \begin{verbatim} ## -## Paired t-test -## -## The minimal effect test was significant, t(90) = 10.642, p = 4.69e-03 -## The null hypothesis test was significant, t(90) = 6.649, p = 2.22e-09 -## NHST: reject null significance hypothesis that the effect is equal to zero -## TOST: reject null MET hypothesis +## Paired Sample Standardized Mean Difference (SMD; Hedges's +## g[rm]=('1'-'2')/SD_rm) ## -## TOST Results -## t df p.value -## t-test 6.649 90 < 0.001 -## TOST Lower 10.642 90 1 -## TOST Upper 2.655 90 0.005 +## data: extra by group ## -## Effect Sizes -## Estimate SE C.I. Conf. Level -## Raw 1.6648 0.2504 [1.2487, 2.081] 0.9 -## Hedges's g(z) 0.6911 0.1167 [0.4987, 0.8802] 0.9 -## Note: SMD confidence intervals are an approximation. See vignette("SMD_calcs"). +## alternative hypothesis: none +## 95 percent confidence interval: +## -1.39802439 -0.07479266 +## sample estimates: +## SMD (g[rm]) +## -0.7513666 \end{verbatim} -The data would indicate that we should accept the MET hypothesis. +When hypothesis testing on the SMD is desired, \texttt{smd\_calc} +supports equivalence and minimal-effect alternatives directly: -\newpage +\begin{Shaded} +\begin{Highlighting}[] +\FunctionTok{smd\_calc}\NormalTok{(}\AttributeTok{formula =}\NormalTok{ extra }\SpecialCharTok{\textasciitilde{}}\NormalTok{ group,} + \AttributeTok{data =}\NormalTok{ sleep,} + \AttributeTok{paired =} \ConstantTok{TRUE}\NormalTok{,} + \AttributeTok{bias\_correction =} \ConstantTok{TRUE}\NormalTok{,} + \AttributeTok{rm\_correction =} \ConstantTok{TRUE}\NormalTok{,} + \AttributeTok{alternative =} \StringTok{"equivalence"}\NormalTok{,} + \AttributeTok{null.value =} \FunctionTok{c}\NormalTok{(}\SpecialCharTok{{-}}\FloatTok{0.5}\NormalTok{,}\FloatTok{0.5}\NormalTok{),} + \AttributeTok{output =} \StringTok{"htest"}\NormalTok{)} +\end{Highlighting} +\end{Shaded} -\hypertarget{one-sample-t-test}{% -\subsection{One Sample t-test}\label{one-sample-t-test}} +\begin{verbatim} +## +## Paired Sample Standardized Mean Difference (SMD; Hedges's +## g[rm]=('1'-'2')/SD_rm) +## +## data: extra by group +## z = -0.78495, p-value = 0.7838 +## alternative hypothesis: equivalence +## null values: +## lower bound upper bound +## -0.5 0.5 +## 90 percent confidence interval: +## -1.2894564 -0.1789204 +## sample estimates: +## SMD (g[rm]) +## -0.7513666 +\end{verbatim} -In other cases we may have a one sample test. If that is the case, only -\texttt{x} argument for the data is needed. This is useful in situations -where you may have hypotheses to test about a single samples mean. In -order for the two-sample test to be correct, we also need to supply the -\texttt{mu} argument. In the example below, we hypothesize that the mean -of \texttt{LDHF} is not more than 1.5 points greater or less than 7. -With the way the \texttt{mu} and \texttt{eqb} arguments are set, we are -testing whether the mean of \texttt{LDHF} is significantly different -from 7.5 (two-tailed tests) and (\(\pm\)) than 1.5 points 7.5 as well -(equivalence bounds at 5.5 and 8.5). +However, the parametric z-test used by \texttt{smd\_calc} relies on +distributional assumptions about the sampling distribution of the SMD +that may not hold in practice. When a hypothesis test on the SMD is the +goal, bootstrapping via \texttt{boot\_smd\_calc} is recommended because +it provides more robust confidence intervals and p-values without +requiring these assumptions. \begin{Shaded} \begin{Highlighting}[] -\NormalTok{res4 }\OtherTok{=} \FunctionTok{t\_TOST}\NormalTok{(}\AttributeTok{x =}\NormalTok{ bugs}\SpecialCharTok{$}\NormalTok{LDHF,} - \AttributeTok{hypothesis =} \StringTok{"EQU"}\NormalTok{,} - \AttributeTok{mu =} \FloatTok{7.5}\NormalTok{,} - \AttributeTok{eqb =} \FunctionTok{c}\NormalTok{(}\FloatTok{5.5}\NormalTok{,}\FloatTok{8.5}\NormalTok{))} -\NormalTok{res4} +\FunctionTok{boot\_smd\_calc}\NormalTok{(}\AttributeTok{formula =}\NormalTok{ extra }\SpecialCharTok{\textasciitilde{}}\NormalTok{ group,} + \AttributeTok{data =}\NormalTok{ sleep,} + \AttributeTok{paired =} \ConstantTok{TRUE}\NormalTok{,} + \AttributeTok{bias\_correction =} \ConstantTok{TRUE}\NormalTok{,} + \AttributeTok{rm\_correction =} \ConstantTok{TRUE}\NormalTok{,} + \AttributeTok{alternative =} \StringTok{"equivalence"}\NormalTok{,} + \AttributeTok{null.value =} \FunctionTok{c}\NormalTok{(}\SpecialCharTok{{-}}\FloatTok{0.5}\NormalTok{,}\FloatTok{0.5}\NormalTok{))} \end{Highlighting} \end{Shaded} \begin{verbatim} ## -## One Sample t-test +## Paired Sample bootstrapped Standardized Mean Difference (SMD; Hedges's +## g[rm]=('1'-'2')/SD_rm) ## -## The equivalence test was significant, t(90) = -4.244, p = 2.66e-05 -## The null hypothesis test was non-significant, t(90) = -0.458, p = 6.48e-01 -## NHST: don't reject null significance hypothesis that the effect is equal to 7.5 -## TOST: reject null equivalence hypothesis +## data: extra by group +## z-observed = -0.78495, p-value = 0.908 +## alternative hypothesis: equivalence +## null values: +## lower bound upper bound +## -0.5 0.5 +## 90 percent confidence interval: +## -1.269159 -0.449214 +## sample estimates: +## SMD (g[rm]) +## -0.7513666 +\end{verbatim} + +\subsubsection{Checking Assumptions}\label{checking-assumptions-1} + +The SMD functions inherit the assumptions of the underlying t-test: +independent observations, approximately normal residuals or differences, +and, for two-sample designs, comparable variances across groups. Visual +diagnostics from the t-test discussion above apply directly. Two +considerations are specific to standardized effect sizes, however. +First, outliers and skew affect both the numerator (mean difference) and +the denominator (standard deviation) of the SMD, which can amplify their +influence on the estimate; the \texttt{tr} trimmed-means argument +available in the \texttt{boot\_smd\_calc} function is particularly worth +considering when the raw data show extreme values. Second, the +parametric z-test in \texttt{smd\_calc} relies on an approximation to +the sampling distribution of the SMD that can be very inaccurate in +small samples, which is the primary motivation for preferring +\texttt{boot\_smd\_calc} when hypothesis testing is the goal. + +\subsection{Equivalence Tests for +Correlations}\label{equivalence-tests-for-correlations} + +Researchers sometimes need to test that a correlation is practically +zero. The \texttt{z\_cor\_test} function provides a parametric approach +using the Fisher z-transformation. However, this is only an approximate +method and is generally \emph{not recommended} as the primary +inferential tool for equivalence testing on correlations. + +\begin{Shaded} +\begin{Highlighting}[] +\FunctionTok{z\_cor\_test}\NormalTok{(} + \AttributeTok{x =}\NormalTok{ bugs}\SpecialCharTok{$}\NormalTok{LDLF,} + \AttributeTok{y =}\NormalTok{ bugs}\SpecialCharTok{$}\NormalTok{LDHF,} + \AttributeTok{method =} \StringTok{"pearson"}\NormalTok{,} + \AttributeTok{alternative =} \StringTok{"t"}\NormalTok{)} +\end{Highlighting} +\end{Shaded} + +\begin{verbatim} ## -## TOST Results -## t df p.value -## t-test -0.4577 90 0.648 -## TOST Lower 7.1156 90 < 0.001 -## TOST Upper -4.2444 90 < 0.001 +## Pearson's product-moment correlation with approximate SE ## -## Effect Sizes -## Estimate SE C.I. Conf. Level -## Raw -0.1209 0.2641 [6.9402, 7.818] 0.9 -## Hedges's g 2.9047 0.2395 [2.5058, 3.2949] 0.9 -## Note: SMD confidence intervals are an approximation. See vignette("SMD_calcs"). +## data: bugs$LDLF and bugs$LDHF +## z = NA, N = 91, p-value = NA +## alternative hypothesis: true correlation is not equal to 0 +## 95 percent confidence interval: +## NA NA +## sample estimates: +## r +## NA \end{verbatim} -We would conclude that \texttt{LDHF} is practically equivalent to the -hypothesized mean (7.5). +Because the z-test relies on distributional approximations that can be +inaccurate --- particularly with small samples --- bootstrapping via +\texttt{boot\_cor\_test} is recommended for more robust inference. The +\texttt{boot\_cor\_test} function supports all three major correlation +methods (Pearson, Kendall, and Spearman) as well as the Winsorized +\citep{Wilcox_1993} and percentage bend \citep{Wilcox_1994} correlation +coefficients as robust alternatives, and returns standard \texttt{htest} +objects that integrate with the helper functions described above. -\newpage +\begin{Shaded} +\begin{Highlighting}[] +\NormalTok{res\_boot\_cor }\OtherTok{=} \FunctionTok{boot\_cor\_test}\NormalTok{(} + \AttributeTok{x =}\NormalTok{ bugs}\SpecialCharTok{$}\NormalTok{LDLF,} + \AttributeTok{y =}\NormalTok{ bugs}\SpecialCharTok{$}\NormalTok{LDHF,} + \AttributeTok{method =} \StringTok{"pearson"}\NormalTok{,} + \AttributeTok{alternative =} \StringTok{"equivalence"}\NormalTok{,} + \AttributeTok{null =} \FloatTok{0.4}\NormalTok{)} + +\FunctionTok{print}\NormalTok{(res\_boot\_cor)} +\end{Highlighting} +\end{Shaded} -\hypertarget{using-summary-statistics}{% -\subsection{Using Summary Statistics}\label{using-summary-statistics}} +\begin{verbatim} +## +## Bootstrapped Pearson's product-moment correlation (BCa) +## +## data: bugs$LDLF and bugs$LDHF +## N = 91, p-value = 0.9827 +## alternative hypothesis: equivalence +## null values: +## correlation correlation +## 0.4 -0.4 +## 90 percent confidence interval: +## 0.4398575 0.6899351 +## sample estimates: +## r +## 0.5767783 +\end{verbatim} -In some cases you may only have access to the summary statistics (e.g., -when reviewing an article or attempting to perform a meta-analysis). -Therefore, I created a function, \texttt{tsum\_TOST}, to perform the -same tests just based on the summary statistics. This involves providing -the function with a number of different arguments. +\begin{Shaded} +\begin{Highlighting}[] +\FunctionTok{describe\_htest}\NormalTok{(res\_boot\_cor)} +\end{Highlighting} +\end{Shaded} -\begin{itemize} -\tightlist -\item - \texttt{n1\ \&\ n2} the sample sizes (only n1 needs to be provided for - one sample case) -\item - \texttt{m1\ \&\ m2} the sample means -\item - \texttt{sd1\ \&\ sd2} the sample standard deviation -\item - \texttt{r12} the correlation between each if paired is set to - TRUE\footnote{The \texttt{extract\_r\_paired} function can be used if - the correlation between paired observations is not readily - available.} -\end{itemize} +\begin{verbatim} +## [1] "The Bootstrapped Pearson's product-moment correlation (BCa) is not statistically significant (p = 0.983, r = 0.577, 90% C.I.[0.44, 0.69]) at a 0.05 alpha-level. The null hypothesis cannot be rejected. At the desired error rate, it cannot be stated that the true correlation is between 0.4 and -0.4." +\end{verbatim} -The results from the \texttt{bugs} example can be replicated with the -\texttt{tsum\_TOST}: +If only summary statistics are available, an approximate method is +provided through \texttt{corsum\_test}, which uses the Fisher +z-transformation to compute confidence intervals and p-values. + +\subsubsection{Checking Assumptions}\label{checking-assumptions-2} + +Correlation-based inference in \texttt{z\_cor\_test} and +\texttt{boot\_cor\_test} assumes independent paired observations and, +for Pearson's coefficient, a roughly linear relationship between +variables with approximately symmetric marginal distributions; Spearman +and Kendall relax the linearity requirement to monotonicity, making them +appropriate when the relationship is non-linear but directionally +consistent. The bootstrap version in \texttt{boot\_cor\_test} further +relaxes distributional assumptions about the marginals but remains +sensitive to observations that exert disproportionate influence on the +estimate. Visual diagnostics are the primary tools: scatterplots with a +loess smoother to assess linearity or monotonicity, marginal density +plots or QQ plots to check distributional shape, and +jackknife-after-bootstrap plots to identify individual observations +driving bootstrap variability. A worked diagnostic workflow is provided +in the online supplement. + +\subsection{Comparing Standardized Effect Sizes Between +Studies}\label{comparing-standardized-effect-sizes-between-studies} + +When evaluating whether a replication study produced results consistent +with the original, researchers need a way to formally compare effect +sizes. The \texttt{compare\_smd} function tests whether two SMD +estimates --- from an original study and a replication, for example --- +differ more than would be expected by sampling variability alone. + +Consider an original study reporting Cohen's \(d_z\) = 0.95 with 25 +participants and a replication with \(d\) = 0.23 and 50 participants. +Are these estimates compatible? \begin{Shaded} \begin{Highlighting}[] -\NormalTok{res\_tsum }\OtherTok{=} \FunctionTok{tsum\_TOST}\NormalTok{(} - \AttributeTok{m1 =} \FunctionTok{mean}\NormalTok{(bugs}\SpecialCharTok{$}\NormalTok{LDHF, }\AttributeTok{na.rm=}\ConstantTok{TRUE}\NormalTok{), }\AttributeTok{sd1 =} \FunctionTok{sd}\NormalTok{(bugs}\SpecialCharTok{$}\NormalTok{LDHF, }\AttributeTok{na.rm=}\ConstantTok{TRUE}\NormalTok{),} - \AttributeTok{n1 =} \FunctionTok{length}\NormalTok{(}\FunctionTok{na.omit}\NormalTok{(bugs}\SpecialCharTok{$}\NormalTok{LDHF)),} - \AttributeTok{hypothesis =} \StringTok{"EQU"}\NormalTok{, }\AttributeTok{smd\_ci =} \StringTok{"t"}\NormalTok{, }\AttributeTok{eqb =} \FunctionTok{c}\NormalTok{(}\FloatTok{5.5}\NormalTok{, }\FloatTok{8.5}\NormalTok{)} -\NormalTok{)} - -\NormalTok{res\_tsum} +\FunctionTok{compare\_smd}\NormalTok{(}\AttributeTok{smd1 =} \FloatTok{0.95}\NormalTok{,} + \AttributeTok{n1 =} \DecValTok{25}\NormalTok{,} + \AttributeTok{smd2 =} \FloatTok{0.23}\NormalTok{,} + \AttributeTok{n2 =} \DecValTok{50}\NormalTok{,} + \AttributeTok{paired =} \ConstantTok{TRUE}\NormalTok{,} + \AttributeTok{null =}\NormalTok{ .}\DecValTok{25}\NormalTok{,} + \AttributeTok{alternative =} \StringTok{"equivalence"}\NormalTok{)} \end{Highlighting} \end{Shaded} \begin{verbatim} ## -## One-sample t-Test +## Difference in Cohen's dz (paired) ## -## The equivalence test was significant, t(90) = -4.244, p = 2.66e-05 -## The null hypothesis test was significant, t(90) = 27.942, p = 3.91e-46 -## NHST: reject null significance hypothesis that the effect is equal to zero -## TOST: reject null equivalence hypothesis +## data: Summary Statistics +## z = 1.6767, p-value = 0.9532 +## alternative hypothesis: equivalence +## null values: +## difference in SMDs difference in SMDs +## 0.25 -0.25 +## sample estimates: +## difference in SMDs +## 0.72 +\end{verbatim} + +Based on these results, we would reject the null significance hypothesis +(the SMDs differ significantly) but fail to reject the equivalence null +hypothesis (we cannot conclude the SMDs are practically equivalent). +When raw data are available, \texttt{boot\_compare\_smd} provides a +bootstrap alternative (see the package vignettes for details). + +For comparing correlation coefficients between independent studies, the +\texttt{compare\_cor} function provides an analogous test. It uses +either the Fisher z-transformation (default) or the Kraatz +(Anderson-Hauck) method to compare two correlation coefficients. + +\begin{Shaded} +\begin{Highlighting}[] +\FunctionTok{compare\_cor}\NormalTok{(}\AttributeTok{r1 =} \FloatTok{0.45}\NormalTok{,} + \AttributeTok{df1 =} \DecValTok{48}\NormalTok{,} + \AttributeTok{r2 =} \FloatTok{0.25}\NormalTok{,} + \AttributeTok{df2 =} \DecValTok{78}\NormalTok{,} + \AttributeTok{alternative =} \StringTok{"equivalence"}\NormalTok{,} + \AttributeTok{null =} \FloatTok{0.25}\NormalTok{)} +\end{Highlighting} +\end{Shaded} + +\begin{verbatim} ## -## TOST Results -## t df p.value -## t-test 27.942 90 < 0.001 -## TOST Lower 7.116 90 < 0.001 -## TOST Upper -4.244 90 < 0.001 +## Difference between two independent correlations (Fisher's z transform) ## -## Effect Sizes -## Estimate SE C.I. Conf. Level -## Raw 7.379 0.2641 [6.9402, 7.818] 0.9 -## Hedges's g 2.905 0.2395 [2.4289, 3.3804] 0.9 -## Note: SMD confidence intervals are an approximation. See vignette("SMD_calcs"). +## data: Summary Statistics +## z = -0.026125, p-value = 0.4439 +## alternative hypothesis: equivalence +## null values: +## difference between correlations difference between correlations +## 0.25 -0.25 +## sample estimates: +## difference between correlations +## 0.2 \end{verbatim} -\newpage - -\hypertarget{robust-methods-for-equivalence-testing}{% -\section{Robust Methods for Equivalence -Testing}\label{robust-methods-for-equivalence-testing}} - -In some cases, the use of t-test may be less than ideal. Any serious -violation to the assumptions of a t-test (e.g., normality or -homoscedasticity) could greatly inflate the type 1 error rate of TOST. -Therefore, it may be useful to explore alternatives to the t-test for -TOST that either do not have those assumptions or are robust to -violating those assumptions. - -The TOSTER package currently provides 4 robust alternatives to the -t-test for TOST. First, there is the \texttt{wilcox\_TOST} function -which uses the Wilcoxon-Mann-Whitney (WMW) type tests (i.e., -\texttt{wilcox.test}) to perform TOST as a test of symmetry. Second, -there is the \texttt{boot\_t\_TOST} function which uses the bootstrap -method outlined by \citet{efron93}. Third, there is the -\texttt{log\_TOST} function which performs log-transformed t-tests, -which is a parametric approach commonly used in pharmaceutical -bioequivalence studies on ratio data \citep{he2022}. Fourth, there is -the \texttt{boot\_log\_TOST} function which uses the same bootstrap -method outlined by \citet{efron93} but on the log-transformed data, -which is more robust than parametric log t-test \citep{he2022}. - -In the following sections, I will briefly outline the available robust -TOST functions within the TOSTER package. - -\hypertarget{tests-of-symmetry-rank-based-tests}{% -\subsection{Tests of Symmetry (rank based -tests)}\label{tests-of-symmetry-rank-based-tests}} - -The WMW group of tests (e.g., Mann-Whitney U-test) provide a -non-parametric test of differences between groups, or within samples, -based on \emph{ranks}. This provides a test of location shift, which is -a fancy way of saying differences in the center of the distribution -(i.e., in parametric tests the location is mean). Within the TOST -framework, there are two separate tests of directional location shift to -determine if the location shift is within (equivalence) or outside -(minimal effect) the equivalence bounds. Many researchers mistakenly -think these are tests of medians, but this is not the case (See -\citet{median_test} for details). Using a WMW-based TOST is useful for -testing whether the differences between groups/conditions is symmetric -around the equivalence bounds\footnote{Care should be taken when - considering paired samples; a test on the rank transformed data - \citep{kornbrot1990rank} or another robust test may be more prudent.}. -For equivalence testing, the TOST would be testing whether there is -asymmetry towards no effect with a null hypothesis of symmetry at the -equivalence bound. - -In the TOSTER package, we accomplish this ``test of symmetry'' with the -\texttt{wilcox\_TOST} function. This function operates in an extremely -similar implementation to the \texttt{t\_TOST} function. The exact -calculations utilized in this function can be explored via the -documentation of the \texttt{wilcox.test} function. A standardized mean -difference (SMD) is \emph{not} calculated in this function since this -would be an inappropriate measure of effect size alongside the -non-parametric test statistics. Instead, a standardized effect size -(SES) is calculated for \emph{all} types of comparisons (e.g., two -sample, one sample, and paired samples). The function can produce a -rank-biserial correlation \citep{Kerby_2014}, a WMW Odds -\citep{wmwodds}, or a ``common language effect size'' \citep{Kerby_2014} -(Also known as the non-parametric probability of superiority, or -concordance probability).\footnote{There is no plotting capability at - this time for the output of this function.} +When raw data are available for both studies, +\texttt{boot\_compare\_cor} provides a bootstrap alternative that avoids +the distributional approximations underlying the Fisher z and Kraatz +methods. + +The comparison functions \texttt{compare\_smd} and \texttt{compare\_cor} +inherit the assumptions of the underlying effect size estimates they +combine. Because both studies contribute to the comparison, violations +in either study propagate into the test. When raw data are available, +the bootstrap alternatives (\texttt{boot\_compare\_smd} and +\texttt{boot\_compare\_cor}) are preferred because they avoid the Fisher +z and related parametric approximations, which can be inaccurate +particularly when effect sizes are large or sample sizes are modest. +When only summary statistics are available, readers should report the +diagnostic context of the original and replication studies where +possible. \newpage -As an example, we can use the sleep data to make a non-parametric -comparison of equivalence. +\textbf{A note on argument names for equivalence bounds.} The +equivalence bound argument is named differently across TOSTER functions +for historical and compatibility reasons: \texttt{eqb} in +\texttt{t\_TOST}, \texttt{wilcox\_TOST}, \texttt{equ\_ftest}, and +\texttt{equ\_anova}; \texttt{mu} in \texttt{simple\_htest}, +\texttt{brunner\_munzel}, \texttt{boot\_t\_test}, and +\texttt{perm\_t\_test} (for compatibility with base R's +\texttt{t.test}); and \texttt{null} in \texttt{compare\_smd} and +\texttt{z\_cor\_test}. Despite the different names, the interpretation +is the same: the bound(s) defining the region of practical equivalence. + +\section{Robust Methods for Equivalence +Testing}\label{robust-methods-for-equivalence-testing} + +When t-test assumptions (normality, homoscedasticity) are seriously +violated, or when the mean difference is a poor summary of the effect +(e.g., with heavy skew or ordinal psychometric scales) alternative +approaches are worth considering. TOSTER organizes these into three +categories: rank-based tests, resampling methods, and log-transformed +methods. + +\subsection{Rank-Based Tests}\label{rank-based-tests} + +Rank-based tests operate on the ranks of the data rather than the raw +values, which is why they are typically described as non-parametric. It +is worth emphasizing that this label applies to the test, not the data. +The underlying observations may still come from any distribution, and +rank-based methods should not be treated as automatic substitutes for a +t-test when normality is in doubt. These tests are not tests of means or +medians \citep{median_test, karch2021}; they target different estimands +(i.e., the quantity a study is attempting to estimate), and the choice +between a rank-based test and a t-test should be driven by which +estimand is of scientific interest. + +\subsubsection{Wilcoxon-Mann-Whitney +Tests}\label{wilcoxon-mann-whitney-tests} + +The WMW group of tests provide non-parametric tests based on ranks. +Within the TOST framework, two directional location-shift tests +determine if the shift is within (equivalence) or outside (minimal +effect) the equivalence bounds. + +However, there are important caveats. Many researchers mistakenly +believe these are tests of medians, but without a location-shift +assumption, the WMW test is a test of stochastic equality. As +\citet{Fay_Malinovsky_2018} demonstrate, the effect size can even be +presented as stochastic superiority/dominance. Only under a +location-shift assumption does it become a test of central tendency +through the Hodges-Lehmann estimator. \citet{median_test} demonstrate +striking counterexamples: groups can have equal medians yet produce a +significant WMW test, very different medians yet a non-significant WMW +test, or even significance in the opposite direction from the median +comparison. + +For equivalence testing, bounds are applied to the pseudo-median, which +corresponds to the mean or median difference only under the +location-shift assumption. For testing hypotheses about mean +differences, \texttt{perm\_t\_test} or \texttt{boot\_t\_test} are more +appropriate. For stochastic superiority, \texttt{brunner\_munzel} +provides a clearer framework. \begin{Shaded} \begin{Highlighting}[] \NormalTok{test1 }\OtherTok{=} \FunctionTok{wilcox\_TOST}\NormalTok{(}\AttributeTok{formula =}\NormalTok{ extra }\SpecialCharTok{\textasciitilde{}}\NormalTok{ group,} \AttributeTok{data =}\NormalTok{ sleep,} - \AttributeTok{paired =} \ConstantTok{FALSE}\NormalTok{,} + \AttributeTok{paired =} \ConstantTok{TRUE}\NormalTok{,} \AttributeTok{eqb =}\NormalTok{ .}\DecValTok{5}\NormalTok{)} \FunctionTok{print}\NormalTok{(test1)} \end{Highlighting} @@ -698,101 +1092,279 @@ \subsection{Tests of Symmetry (rank based \begin{verbatim} ## -## Wilcoxon rank sum test with continuity correction +## Wilcoxon signed rank test with continuity correction ## -## The equivalence test was non-significant W = 20.000, p = 8.94e-01 -## The null hypothesis test was non-significant W = 25.500, p = 6.93e-02 -## NHST: don't reject null significance hypothesis that the effect is equal to zero +## The equivalence test was non-significant V = 0.000, p = 9.96e-01 +## The null hypothesis test was significant V = 0.000, p = 9.09e-03 +## NHST: reject null significance hypothesis that the effect is equal to zero ## TOST: don't reject null equivalence hypothesis ## ## TOST Results ## Test Statistic p.value -## NHST 25.5 0.069 -## TOST Lower 34.0 0.894 -## TOST Upper 20.0 0.013 +## NHST 0 0.009 +## TOST Lower 2 0.996 +## TOST Upper 0 0.003 ## ## Effect Sizes -## Estimate C.I. Conf. Level -## Median of Differences -1.346 [-3.4, -0.1] 0.9 -## Rank-Biserial Correlation -0.490 [-0.7493, -0.1005] 0.9 +## Estimate C.I. Conf. Level +## Median of Differences -1.4 [-2.7, -1.15] 0.9 +## Rank-Biserial Correlation -1.0 [-1, -0.4492] 0.9 \end{verbatim} -Based on these results, we would have conclude there is no significant -difference but not equivalent differences either (i.e., inconclusive -result). - -\newpage - -\hypertarget{bootstrap-tost}{% -\subsection{Bootstrap TOST}\label{bootstrap-tost}} +Based on these results, we would conclude there is a significant +difference and the results are also not equivalent. But again, the +estimate we are testing is the pseudo-median, not the mean or median +difference, so this conclusion may not be relevant to the research +question at hand. + +\paragraph{Checking Assumptions}\label{checking-assumptions-3} + +The WMW tests assume independent observations and exchangeability under +the null, the latter meaning that group labels carry no information +about outcomes if there is no effect. For the interpretations discussed +above to be clean, two additional assumptions matter. The location-shift +assumption --- that the two group distributions differ only in location, +not shape --- is what allows WMW results to be interpreted as a test of +the median difference via the Hodges-Lehmann estimator. Violations are +best diagnosed by overlaying the empirical cumulative density functions +of the two groups; under location shift, the curves are horizontal +translations of each other, whereas crossing curves or varying +horizontal gaps indicate violation. For the paired and one-sample +signed-rank variants used in \texttt{wilcox\_TOST} or +\texttt{simple\_htest}, the signed-rank procedure additionally requires +that the difference distribution is symmetric around the null value for +the pseudo-median to coincide with the mean or median difference. +Asymmetry is readily checked with a symmetry plot of the paired +differences (\citet{fig-symmplot}). As with the location-shift case, +visible departures suggest that the test's output --- the pseudo-median +and associated p-value --- does not map cleanly onto a hypothesis about +central tendency. -The bootstrap refers to resampling with replacement and can be used for -statistical estimation and inference. Bootstrapping techniques are very -useful because they are considered somewhat robust to the violations of -assumptions for a simple t-test and provide better estimations of SMDs -\citep{Kirby2013}. Therefore, I added a bootstrapping function, -\texttt{boot\_t\_TOST}, to the package to provide another robust -alternative to the \texttt{t\_TOST} function. - -In this function we provide a percentile bootstrap solution outlined by -\citet{efron93} (see chapter 16, page 220). The bootstrapped p-values -are derived from the ``studentized'' version of a test of mean -differences \citep{efron93}. Overall, the results should be similar to -the results of \texttt{t\_TOST}. \textbf{However}, for paired samples, -the Cohen's d(rm) effect size \emph{cannot} be calculated by this -function. +\begin{figure} +\centering +\pandocbounded{\includegraphics[keepaspectratio]{Avocado_Update_files/Avocado_Update_files/figure-latex/fig-symmplot-1.pdf}} +\caption{Symmetry plot of the paired differences from the sleep dataset, +pairing each observation's distance above the median with the +corresponding distance below. Under symmetry, points fall along the +45-degree reference line (dashed); systematic departures --- points +consistently above or below the line --- indicate asymmetry. The +signed-rank variant of WMW requires this symmetry for the pseudo-median +to coincide with the mean or median of the differences.} +\end{figure} -\hypertarget{two-sample-algorithm}{% -\subsubsection{Two Sample Algorithm}\label{two-sample-algorithm}} +\subsubsection{Brunner-Munzel Test}\label{brunner-munzel-test} + +The Brunner-Munzel test \citep{brunner2000} estimates the relative +effect \(\hat{p} = P(X > Y) + 0.5 \cdot P(X = Y)\), sometimes referred +to as the probability of superiority --- the probability that a randomly +chosen observation from one group exceeds a randomly chosen observation +from the other, with ties split evenly. This estimand is interpretable +without distributional assumptions, unlike the WMW test which requires a +location-shift assumption to say anything about location differences. +\citet{karch2021} argues it should be preferred as the default +non-parametric procedure because it is valid under heteroscedasticity +and provides an interpretable effect size. + +Stochastic superiority tests may be particularly well-suited to +Likert-type scales which are ubiquitous in psychological research. For a +single ordinal item, the spacing between response categories carries no +guaranteed meaning, and a mean difference therefore has a strained +interpretation. For example, what, concretely, does a 0.4-unit mean +difference on a five-point scale communicate? The stochastic superiority +estimand sidesteps this entirely. The probability that a randomly +selected participant from one group outscores a randomly selected +participant from the other is interpretable on its own terms, with no +knowledge of the scale required. We are not suggesting that t-tests will +\emph{always} mislead in these contexts but it is a concern that should +not be quickly dismissed either \citep{Liddell_Kruschke_2018}. +Additionally, even if we assume traditional parametric methods are +robust to the ordinal nature of the data, the preserved error rate is +not the same as interpretive clarity. For researchers whose primary +outcomes live on ordinal scales, the Brunner-Munzel test and its +associated effect size --- as well as ordinal models more generally +\citep{Burkner_Vuorre_2019} --- may offer a more natural and defensible +default. + +The \texttt{brunner\_munzel} function supports equivalence and minimal +effect testing via \texttt{alternative}, with bounds on the probability +scale via \texttt{mu}. Results can be reported on different scales +(\texttt{"probability"}, \texttt{"difference"}, \texttt{"logodds"}, or +\texttt{"odds"}), and three test methods are available: \texttt{"t"} +(default), \texttt{"logit"} (range-preserving CIs), and \texttt{"perm"} +(studentized permutation; \citet{neubert2007}). -The steps by which the bootstrapping occurs are fairly simple. +\begin{Shaded} +\begin{Highlighting}[] +\NormalTok{bm\_test }\OtherTok{=} \FunctionTok{brunner\_munzel}\NormalTok{(}\AttributeTok{formula =}\NormalTok{ extra }\SpecialCharTok{\textasciitilde{}}\NormalTok{ group,} + \AttributeTok{data =}\NormalTok{ sleep,} + \AttributeTok{alternative =} \StringTok{"equivalence"}\NormalTok{,} + \AttributeTok{mu =} \FunctionTok{c}\NormalTok{(}\FloatTok{0.3}\NormalTok{, }\FloatTok{0.7}\NormalTok{))} +\FunctionTok{print}\NormalTok{(bm\_test)} +\end{Highlighting} +\end{Shaded} -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - Form B bootstrap data sets from x* and y* wherein x* is sampled with - replacement from \(\tilde x_1,\tilde x_2, ... \tilde x_n\) and y* is - sampled with replacement from - \(\tilde y_1,\tilde y_2, ... \tilde y_n\) -\item - t is then evaluated on each sample, but the mean of each sample (y or - x) and the overall average (z) are subtracted from each (i.e., null - distribution is formed) -\end{enumerate} - -\[ -t(z^{*b}) = \frac {(\bar x^*-\bar x - \bar z) - (\bar y^*-\bar y - \bar z)}{\sqrt {sd_y^*/n_y + sd_x^*/n_x}} -\] - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\setcounter{enumi}{2} -\tightlist -\item - An approximate p-value can then be calculated as the number of - bootstrapped results greater than the observed t-statistic from the - sample. -\end{enumerate} +\begin{verbatim} +## +## Two-sample Brunner-Munzel test +## +## data: extra by group +## t = -0.39392, df = 16.898, p-value = 0.6507 +## alternative hypothesis: equivalence +## null values: +## lower bound upper bound +## 0.3 0.7 +## 90 percent confidence interval: +## 0.0562039 0.4537961 +## sample estimates: +## P('1'>'2') + .5*P('1'='2') +## 0.255 +\end{verbatim} -\[ -p_{boot} = \frac {\#t(z^{*b}) \ge t_{sample}}{B} -\] +\newpage -The same process is completed for the one sample case but with the one -sample solution for the equation outlined by \(t(z^{*b})\). The paired -sample case in this bootstrap procedure is equivalent to the one sample -solution because the test is based on the difference scores. +\subsection{Resampling Methods}\label{resampling-methods} + +When the goal is to test hypotheses about means while relaxing +distributional assumptions, resampling methods provide powerful tools. +TOSTER implements both bootstrap and permutation approaches through +\texttt{boot\_t\_test} and \texttt{perm\_t\_test}, each with distinct +inferential logic but a shared motivation to retain the mean as the +estimand while obtaining reference distributions empirically --- through +resampling --- rather than relying on the normal-theory derivations +(i.e., Student's t-test). + +Both approaches are natural extensions of the traditional t-test rather +than alternatives to it. The test statistic itself is unchanged; what +changes is how the null distribution is constructed. In the parametric +t-test, the null distribution of the standardized mean difference is +derived analytically under the assumption that the data are drawn from a +population with a normal distribution (and, for Student's version, with +equal variances). The bootstrap approximates the sampling distribution +of the same statistic by resampling from the observed data with +replacement, whereas the permutation approach constructs an exact or +near-exact null distribution by shuffling group labels (two-sample) or +flipping signs (one-sample and paired). In both cases, the researcher +continues to ask the same question about means; only the calibration of +the reference distribution differs. This is important for researchers +who may view resampling methods as a departure from familiar testing +logic: the estimand, the null hypothesis, and the test statistic remain +the same. + +Because the reference distribution is generated from the data rather +than assumed, distributional requirements are relaxed, though not +eliminated. The bootstrap relies on asymptotic arguments and assumes +that the empirical distribution is a reasonable stand-in for the +population distribution; it performs well with moderate-to-large samples +and skewed or heavy-tailed data, but can struggle with very small +samples or extreme outliers. The permutation approach is exact under the +null hypothesis of exchangeability, meaning group labels carry no +information about outcomes. In its basic form this exactness requires +identical distributions under the null, which implicitly assumes equal +variances. The studentized permutation test +\citep{janssen1997, chung2013} addresses this limitation by permuting a +scale-invariant statistic (essentially the Welch t-statistic), yielding +asymptotically valid inference even under heteroscedasticity. +\citet{Arboretti_Pesarin_Salmaso_2020} extended this logic to the +equivalence testing setting, showing that studentized permutation TOST +maintains nominal Type I error rates when variances differ across +groups. In short, these methods are assumption-lighter rather than +assumption-free, and users should still consider whether independence +and exchangeability are reasonable for their design. + +A common reflex when normality seems implausible is to switch to the WMW +or other rank-based tests. However, these tests answer a different +question. As discussed earlier in this manuscript, rank-based procedures +test hypotheses about stochastic ordering or pseudo-medians rather than +means, and under heteroscedasticity they can reject the null even when +the groups have identical means. When the scientific question is +genuinely about means ----- as it often is in applied research where +effect sizes and interventions are naturally expressed on the original +scale ----- bootstrap and permutation t-tests should generally be +preferred over rank-based methods for handling such violations of +assumptions. The rank-based tests remain useful, but as answers to +different questions, not as robust substitutes for tests of means. + +Choosing between the bootstrap and permutation approaches involves +tradeoffs rather than a clear winner in most situations. The studentized +permutation test has the strongest theoretical footing for two-sample +mean comparisons under heteroscedasticity, and is particularly +attractive for equivalence testing given the +\citet{Arboretti_Pesarin_Salmaso_2020} results. The bootstrap offers +greater flexibility because it extends naturally to statistics beyond +the mean difference, provides readily interpretable confidence intervals +via CI inversion \citep{Thulin_2024}, and ensures consistency between +reported intervals and p-values. In small samples, permutation +approaches tend to have better finite-sample calibration because the +null distribution is constructed exactly (or nearly so) from the data at +hand, whereas the bootstrap's asymptotic justification can produce +liberal tests with very small sample sizes. When outliers are a concern, +both methods accept the \texttt{tr} argument for trimmed means, which +reduces the influence of extreme observations while preserving the +mean-based interpretation. As a rough heuristic, researchers comparing +two independent groups with suspected unequal variances may find the +studentized permutation test most defensible, while those wanting a +single framework that yields coherent intervals and p-values across a +range of designs may prefer the bootstrap. Both are substantial +improvements over ignoring assumption violations, and both return +standard \texttt{htest} objects that integrate with the TOST framework +for equivalence and minimal effect testing. + +\subsubsection{Bootstrap t-test}\label{bootstrap-t-test} + +The bootstrap t-test in TOSTER defaults to the studentized bootstrap +(\texttt{boot\_ci\ =\ "stud"}) because this matches the original +bootstrap t-test proposed by \citet{efron93} (Chapter 16), which is the +procedure \texttt{boot\_t\_TOST} implements. In this approach, the +t-statistic itself is resampled rather than the raw mean difference, and +confidence intervals are constructed from the bootstrap distribution of +these pivotal quantities. Other CI methods (percentile, basic, and BCa) +remain available via the \texttt{boot\_ci} argument, but the studentized +default preserves consistency with the canonical Efron and Tibshirani +formulation. It also ensures that the reported p-value and confidence +interval are derived from the same bootstrap distribution, so the two +will always agree; previous versions of the package used the studentized +p-value regardless of the selected CI method, which could produce +disagreement between reported intervals and p-values. -\newpage +\begin{Shaded} +\begin{Highlighting}[] +\FunctionTok{set.seed}\NormalTok{(}\DecValTok{4522}\NormalTok{)} +\FunctionTok{boot\_t\_test}\NormalTok{(}\AttributeTok{formula =}\NormalTok{ extra }\SpecialCharTok{\textasciitilde{}}\NormalTok{ group,} + \AttributeTok{data =}\NormalTok{ sleep,} + \AttributeTok{paired =} \ConstantTok{TRUE}\NormalTok{,} + \AttributeTok{alternative =} \StringTok{"equivalence"}\NormalTok{,} + \AttributeTok{mu =} \FunctionTok{c}\NormalTok{(}\SpecialCharTok{{-}}\FloatTok{0.5}\NormalTok{, }\FloatTok{0.5}\NormalTok{),} + \AttributeTok{R =} \DecValTok{999}\NormalTok{)} +\end{Highlighting} +\end{Shaded} -\hypertarget{example-of-bootsrapping}{% -\subsubsection{Example of Bootsrapping}\label{example-of-bootsrapping}} +\begin{verbatim} +## +## Bootstrapped Paired t-test (studentized) +## +## data: extra by group +## t-observed = -2.7766, df = 9, p-value = 1 +## alternative hypothesis: equivalence +## null values: +## mean difference mean difference +## -0.5 0.5 +## 90 percent confidence interval: +## -2.853166 -1.071659 +## sample estimates: +## mean of the differences (z = '1' - '2') +## -1.58 +\end{verbatim} -We can use the sleep data to see an example of the bootstrapped results. -If you plot the bootstrap samples, it will show how the resampling via -bootstrapping indicates the instability of Hedges' d(z). Just looking at -the printed results you will notice some differences between confidence -intervals from the bootstrapped result and the t-test. +The output follows the familiar \texttt{htest} format. The reported +p-value is the larger of the two one-sided tests against the equivalence +bounds, and the confidence interval is the studentized bootstrap CI for +the mean difference. Because the p-value is tied to this interval via CI +inversion, a researcher reporting both will find the two tell a +consistent story. The interval will fall within the equivalence bounds +if and only if the equivalence test rejects. For situations where both a +nil hypothesis test and an equivalence test are desired simultaneously +with bootstrapping, \texttt{boot\_t\_TOST} returns a \texttt{TOSTt} +object with bootstrapped SMD estimates \citep{Kirby2013}: \begin{Shaded} \begin{Highlighting}[] @@ -802,8 +1374,6 @@ \subsubsection{Example of Bootsrapping}\label{example-of-bootsrapping}} \AttributeTok{paired =} \ConstantTok{TRUE}\NormalTok{,} \AttributeTok{eqb =}\NormalTok{ .}\DecValTok{5}\NormalTok{,} \AttributeTok{R =} \DecValTok{999}\NormalTok{)} - - \FunctionTok{print}\NormalTok{(test1)} \end{Highlighting} \end{Shaded} @@ -825,128 +1395,103 @@ \subsubsection{Example of Bootsrapping}\label{example-of-bootsrapping}} ## ## Effect Sizes ## Estimate SE C.I. Conf. Level -## Raw -1.580 0.3699 [-2.26, -1.038] 0.9 -## Hedges's g(z) -1.174 0.6491 [-2.7507, -0.9285] 0.9 -## Note: percentile bootstrap method utilized. -\end{verbatim} - -\newpage - -\hypertarget{log-tost}{% -\subsection{Log TOST}\label{log-tost}} - -The natural logarithmic (log) transformation is often utilized to -stabilize the variance of a measure, and it often provides the best -approximation of the normal distribution \citep{logtest}. However, -another, less often reported, advantage of the log transformation is -that the back transformation of the differences of the log-transformed -data is a \emph{ratio} \citep{logtest}. For example, if we had a two -samples (x \& y) with an geometric mean\footnote{The mean of - log-transformed data is the \emph{geometric} not \emph{arithmetic} - mean. I highly recommend reading \citet{logtest} and - \citet{caldwell2019basic} for more details} or 7 and 10.5, x and y -respectively in the code below, we could represent the differences as -ratio of y:x where y is 1.5 times greater than x. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{x }\OtherTok{=} \DecValTok{7}\NormalTok{; y }\OtherTok{=} \FloatTok{10.5} -\FunctionTok{log}\NormalTok{(y) }\SpecialCharTok{{-}} \FunctionTok{log}\NormalTok{(x)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 0.4054651 +## Raw -1.580 0.3509 [-2.9393, -1.0833] 0.9 +## Hedges's g(z) -1.174 0.7990 [-1.3917, 1.1253] 0.9 +## Note: studentized bootstrap ci method utilized. \end{verbatim} -\begin{Shaded} -\begin{Highlighting}[] -\FunctionTok{log}\NormalTok{(y}\SpecialCharTok{/}\NormalTok{x)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 0.4054651 -\end{verbatim} +The bootstrapped SMDs are a useful feature in their own right. Analytic +confidence intervals for standardized effect sizes rely on +approximations, which assume normality and can behave poorly with small +or skewed samples. Bootstrap intervals for SMDs sidestep these +assumptions by resampling the full estimation pipeline, yielding +intervals that inherit the robustness properties of the bootstrap +itself. + +\subsubsection{Permutation t-test}\label{permutation-t-test} + +When \texttt{R\ =\ NULL}, the function enumerates all possible +permutations for an exact test; otherwise, it draws R random +permutations with the ``+1'' correction \citep{phipson2010}. By default +the test is studentized (\texttt{perm\_se\ =\ TRUE}), consistent with +the recommendations discussed in the methods section above. The +\texttt{perm\_se\ =\ FALSE} option is available primarily to match the +behavior of other permutation test packages that do not studentize by +default. \begin{Shaded} \begin{Highlighting}[] -\FunctionTok{exp}\NormalTok{(}\FunctionTok{log}\NormalTok{(y) }\SpecialCharTok{{-}} \FunctionTok{log}\NormalTok{(x))} +\FunctionTok{set.seed}\NormalTok{(}\DecValTok{8812}\NormalTok{)} +\FunctionTok{perm\_t\_test}\NormalTok{(}\AttributeTok{formula =}\NormalTok{ extra }\SpecialCharTok{\textasciitilde{}}\NormalTok{ group,} + \AttributeTok{data =}\NormalTok{ sleep,} + \AttributeTok{paired =} \ConstantTok{TRUE}\NormalTok{,} + \AttributeTok{alternative =} \StringTok{"equivalence"}\NormalTok{,} + \AttributeTok{mu =} \FunctionTok{c}\NormalTok{(}\SpecialCharTok{{-}}\FloatTok{0.5}\NormalTok{, }\FloatTok{0.5}\NormalTok{),} + \AttributeTok{R =} \DecValTok{999}\NormalTok{)} \end{Highlighting} \end{Shaded} \begin{verbatim} -## [1] 1.5 +## +## Randomization Permutation Paired t-test +## +## data: extra by group +## t-observed = -2.7766, df = 9, p-value = 0.999 +## alternative hypothesis: equivalence +## null values: +## difference in means difference in means +## -0.5 0.5 +## 90 percent confidence interval: +## -2.1720 -0.9956 +## sample estimates: +## mean of the differences (z = '1' - '2') +## -1.58 \end{verbatim} -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{y}\SpecialCharTok{/}\NormalTok{x} -\end{Highlighting} -\end{Shaded} +Running the bootstrap and permutation equivalence tests on the same +paired sleep data using matching equivalence bounds produces broadly +similar conclusions, as would be expected. The two approaches can, +however, diverge in small samples, under strong heteroscedasticity, or +in the presence of influential outliers, which is where the choice +between them becomes substantively important. Users interested in +robustness to outliers can additionally supply the \texttt{tr} argument +to either function to use trimmed means, preserving the mean-based +interpretation while downweighting extreme observations. -\begin{verbatim} -## [1] 1.5 -\end{verbatim} +\newpage -The log transformation thereby acts as a useful tool help tame data into -conforming to the normality assumption, and makes the interpretation -fairly simple. In addition, some regulatory agencies, such as the United -States Food and Drug Administration (FDA) \citep{fda}, specifically -require bioequivalence studies to report the geometric means and make -statistical comparisons on the log transformed data \citep{he2022}. In -pharmaceutical reserach, bioequivalence testing involves determining -whether two drugs, a test drug and a reference drug, have the same rate -and extent of absorption in the body. This is typically accomplished by -testing whether the blood concentrations of the drug after -administration of the test drug are sufficiently close to the blood -concentrations after administration of the reference drug. If the two -drugs are bioequivalent, they can be used interchangeably. The area -under the curve (AUC) is the measure of the extent of absorption, and -the peak concentration is the measure of the rate of absorption. In -order to determine bioequivalence, the AUC and peak concentration of the -test drug must be within a certain percentage of the AUC and peak -concentration of the reference drug. - -In my personal experience as a physiologist, it is not uncommon that -biological/physiological phenomenon present have longer right-tailed -distributions, and are often adequately normalized with a natural log -transformation. The additional advantage is the how equivalence bounds -can, almost, be universally applied when making comparisons on the log -scale. The FDA considers to drugs to be bioequivalent when the maximal -concentration and AUC differences between drugs are less than 1.25. To -put it another way, ratio between two means must be between 1.25 and 0.8 -(i.e., 1/1.25) \citep{fda}. - -Therefore, I have implemented two functions to allow for the comparison -of data that is believed to be left skewed (long right tail), and is on -a ratio scale\footnote{Ratio scale means the outcome is measured on a - numerical scale that has equal distances between adjacent values and - true zero.}. The first function is a parametric t-test on the log -transformed scale while the second function is a bootstrapping test -which is more robust than parametric version \citep{he2022}. - -\hypertarget{example-of-log-tost}{% -\subsubsection{Example of Log TOST}\label{example-of-log-tost}} - -The \texttt{log\_TOST} function is almost exactly the same as the -\texttt{t\_TOST} function. First, the primary differences is that it -only accepts paired and two sample comparisons. One sample tests are not -support (i.e., there is no ratio to calculate). Second, standardized -mean differences are not calculated, but a ratio of means is instead -reported \citep{lajeunesse2015bias}\footnote{Also, referred to as a - ``response ratio'' in ecology. Like an SMD, the response ratio can be - utilized in meta-analysis.}. Third, the default equivalence bounds are -by default set to the FDA standards (i.e., \texttt{eqb\ =\ 1.25}), but -can be changed by the user\footnote{Only one value needs to be supplied - to eqb; the reciprocal value of eqb is taken as the other equivalence - bound. For example, if \texttt{eqb\ \ =\ 0.85} then the upper - equivalence bound is 1/0.85 (\textasciitilde1.333)}. - -As an example we can use the \texttt{mtcars} data to compare the type of -transmission (\texttt{am}) effects on the gas mileage (\texttt{mpg}). We -can see from the data below there are significant, non-equivalent, -differences in mpg between transmission types. +\subsection{Log-Transformed Methods}\label{log-transformed-methods} + +The natural log transformation stabilizes variance and approximates +normality for right-skewed data \citep{logtest}. An additional advantage +is that differences on the log scale back-transform to \emph{ratios}, +making interpretation straightforward: if two samples have geometric +means of 7 and 10.5, the ratio is 1.5 (i.e., 10.5/7). This property is +central to pharmaceutical bioequivalence, where the FDA requires +comparisons on the log-transformed scale with equivalence bounds of 0.80 +to 1.25 for the ratio of geometric means \citep{fda, he2022}. + +An alternative to \texttt{log\_TOST} is to log-transform data manually +and analyze with \texttt{t\_TOST}, \texttt{simple\_htest}, or the +resampling methods. Differences on the log scale can then be interpreted +as ``sympercents'' --- symmetric percentage differences on the +\(100 \cdot \ln(x) - ln(y)\) scale \citep{cole2000} --- which are +symmetric in a way that ordinary percentage differences are not because +percentage changes are calculated from different bases, a 10\% loss and +a 10\% gain do not cancel out, whereas +10 and -10 sympercents are +exactly equal in magnitude.\footnote{A concrete illustration that I give + my students: if a stock falls 10\% on Tuesday and rises 10\% from + Tuesday on Wednesday, you still have a loss on of your original + investment. Sympercents do not have this asymmetry.} + +\subsubsection{Log TOST}\label{log-tost} + +The \texttt{log\_TOST} function accepts paired and two-sample +comparisons (one-sample tests are not supported since there is no ratio +to calculate). Instead of an SMD, a ratio of means is reported +\citep{lajeunesse2015bias}. The default equivalence bounds are set to +the FDA standard (\texttt{eqb\ =\ 1.25}; the reciprocal is taken as the +other bound). \begin{Shaded} \begin{Highlighting}[] @@ -977,14 +1522,10 @@ \subsubsection{Example of Log TOST}\label{example-of-log-tost}} \newpage -\hypertarget{example-of-bootstrap-log-tost}{% -\subsubsection{Example of Bootstrap Log -TOST}\label{example-of-bootstrap-log-tost}} +\subsubsection{Bootstrap Log TOST}\label{bootstrap-log-tost} -The bootstrap version of \texttt{log\_TOST}, \texttt{boot\_log\_TOST}, -uses the same bootstrapping method detailed above -(\texttt{boot\_t\_TOST}), but it uses the log-transformed values and -produces the ratio of means as the effect size. +The bootstrap version uses the same bootstrap methods as +\texttt{boot\_t\_test} on the log-transformed values: \begin{Shaded} \begin{Highlighting}[] @@ -996,7 +1537,7 @@ \subsubsection{Example of Bootstrap Log ## ## Bootstrapped Log Welch Two Sample t-test ## -## The equivalence test was non-significant, t(23.96) = -1.363, p = 9.57e-01 +## The equivalence test was non-significant, t(23.96) = -1.363, p = 9.54e-01 ## The null hypothesis test was significant, t(23.96) = -3.826, p = 0e+00 ## NHST: reject null significance hypothesis that the effect is equal to 1 ## TOST: don't reject null equivalence hypothesis @@ -1004,14 +1545,14 @@ \subsubsection{Example of Bootstrap Log ## TOST Results ## t df p.value ## t-test -3.826 23.96 < 0.001 -## TOST Lower -1.363 23.96 0.957 +## TOST Lower -1.363 23.96 0.954 ## TOST Upper -6.288 23.96 < 0.001 ## ## Effect Sizes ## Estimate SE C.I. Conf. Level -## log(Means Ratio) -0.3466 0.08634 [-0.4871, -0.2039] 0.9 -## Means Ratio 0.7071 0.06119 [0.6144, 0.8156] 0.9 -## Note: percentile bootstrap method utilized. +## log(Means Ratio) -0.3466 0.08621 [-0.4932, -0.1893] 0.9 +## Means Ratio 0.7071 0.06116 [0.6106, 0.8275] 0.9 +## Note: studentized bootstrap ci method utilized. \end{verbatim} From this analysis, we would conclude there is a significant effect that @@ -1019,83 +1560,31 @@ \subsubsection{Example of Bootstrap Log \newpage -\hypertarget{equivalence-testing-with-anovas}{% \section{Equivalence Testing with -ANOVAs}\label{equivalence-testing-with-anovas}} - -Many researchers utilize ANOVA as an omnibus test for the -absence/presence of effects before inspecting multiple pairwise -comparisons. This is very useful when implementing factorial designs -wherein multiple experimental factors are tested and/or manipulated. As -\citet{Campbell_2021} suggest, the lack of a significant result at the -ANOVA-level does not necessarily indicate that a factor or interaction -of factors have no effect. However, \citet{Campbell_2021} only suggest -an equivalence test for one-way ANOVAs and therefore exclude -multi-factor or factorial ANOVAs. Therefore, I have extended the work of -\citet{Campbell_2021} to include functions that allow for equivalence -testing of the partial \(\eta^2\) (eta-squared) effect size from ANOVAs. - -\hypertarget{f-test-calculations}{% -\subsection{F-test Calculations}\label{f-test-calculations}} - -Statistical equivalence testing\footnote{Also called ``omnibus - non-inferiority testing'' by \citet{Campbell_2021}} for \emph{F}-tests -are special use case of the cumulative distribution function of the -non-central \emph{F} distribution. As \citet{Campbell_2021} states, this -type of statistical test answers the question: ``Can we reject the -hypothesis that the total proportion of variance in outcome Y -attributable to X is greater than or equal to the equivalence bound -\(\Delta\)?'' - -\hypertarget{hypothesis-tests}{% -\subsubsection{Hypothesis Tests}\label{hypothesis-tests}} - -\[ -H_0 = 1 > \eta^2_p \geq \Delta -\] - -\[ -H_1 = 0 \geq \eta^2_p < \Delta -\] - -In TOSTER, I have gone a tad farther than \citet{Campbell_2021}, and -have included a calculation for a generalization of the non-centrality -parameter that allows the equivalence test for \emph{F}-tests to be -applied to variety of designs. - -\citet{Campbell_2021} calculate the \emph{p}-value as: - -\[ -p = p_f(F; J-1, N-J, \frac{N \cdot \Delta}{1-\Delta}) -\] - -The non-centrality parameter (ncp = \(\lambda\)) can be calculated with -the equivalence bound and the degrees of freedom: - -\[ -\lambda_{eq} = \frac{\Delta}{1-\Delta} \cdot(df_1 + df_2 +1) -\] - -\newpage - -The \emph{p}-value for the equivalence test (\(p_{eq}\)) could then be -calculated from traditional ANOVA results and the distribution function: - -\[ -p_{eq} = p_f(F; df_1, df_2, \lambda_{eq}) -\] - -\hypertarget{example-of-equivalence-anova-testing}{% -\subsection{Example of Equivalence ANOVA -Testing}\label{example-of-equivalence-anova-testing}} - -Using the \texttt{InsectSprays} data set in R and the base R -\texttt{aov} function, I can demonstrate how this omnibus equivalence -testing can be applied with TOSTER. From the initial analysis we an see -a clear ``significant'' effect (very small p-value) of the inspect -spray. However, we \emph{may} be interested in testing if the effect is -practically equivalent. I will arbitrarily set the equivalence bound to -a partial eta-squared of 0.35 (\(H_0: \eta^2_p > 0.35\)). +ANOVAs}\label{equivalence-testing-with-anovas} + +Many researchers use ANOVA as an omnibus test before inspecting pairwise +comparisons, particularly in factorial designs. As \citet{Campbell_2021} +note, the lack of a significant ANOVA result does not indicate that a +factor has no effect. Equivalence testing for ANOVAs addresses this +directly by asking: can we reject the hypothesis that the partial +\(\eta^2\) exceeds a specified bound? + +The equivalence test uses the non-central \emph{F} distribution. The +non-centrality parameter is computed from the equivalence bound and the +degrees of freedom, and the \emph{p}-value is obtained from the +cumulative distribution function of the non-central \emph{F}-value. I +have extended the approach of \citet{Campbell_2021} --- who proposed +equivalence testing for one-way ANOVAs --- to accommodate multi-factor +and factorial designs by generalizing the non-centrality parameter +calculation. The full derivation is described in the package +documentation; here we demonstrate usage. + +\subsection{Example with Summary +Statistics}\label{example-with-summary-statistics} + +Using the \texttt{InsectSprays} data and the \texttt{equ\_ftest} +function, which accepts summary \emph{F}-test statistics directly: \begin{Shaded} \begin{Highlighting}[] @@ -1116,12 +1605,6 @@ \subsection{Example of Equivalence ANOVA ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 \end{verbatim} -We can then use the information in the table above to perform an -equivalence test using the \texttt{equ\_ftest} function. This function -returns an object of the S3 class \texttt{htest} and the output will -look very familiar to that of the t-test. The main difference is the -estimates, and confidence interval, are for partial \(\eta^2_p\). - \begin{Shaded} \begin{Highlighting}[] \FunctionTok{equ\_ftest}\NormalTok{(}\AttributeTok{Fstat =} \FloatTok{34.70228}\NormalTok{, }\AttributeTok{df1 =} \DecValTok{5}\NormalTok{, }\AttributeTok{df2 =} \DecValTok{66}\NormalTok{, }\AttributeTok{eqb =} \FloatTok{0.35}\NormalTok{)} @@ -1140,34 +1623,27 @@ \subsection{Example of Equivalence ANOVA ## [1] 0.724439 \end{verbatim} -Based on the results above we would conclude there is a significant -effect of ``spray'' and the differences due to spray are \emph{not} -statistically equivalent. In essence, we reject the traditional null -hypothesis of ``no effect'' but accept the null hypothesis of the -equivalence test. +Based on these results, there is a significant effect of ``spray'' and +the differences are \emph{not} statistically equivalent --- we reject +the traditional null hypothesis but fail to reject the equivalence null +hypothesis. \newpage -The \texttt{equ\_ftest} function is very useful because all you need is -very basic summary statistics. However, if you are doing all your -analyses in R then you can use the \texttt{equ\_anova} function. This -function accepts objects produced from \texttt{stats::aov}, -\texttt{car::Anova} and \texttt{afex::aov\_car} (or any ANOVA from -derived from \texttt{afex}). +\subsection{Example with R ANOVA +Objects}\label{example-with-r-anova-objects} -As a second example, we can use the afex package's data and ANOVA -\citep{afex}. Again, we will use the equivalence bound of 0.35, which is -a completely arbitrary (and baseless) equivalence bound. Notice that the -output contains 2 p-values: one for the significance (\texttt{p.null}) -and another for the equivalence test (\texttt{p.equ}). +The \texttt{equ\_anova} function accepts objects from +\texttt{stats::aov}, \texttt{car::Anova}, and \texttt{afex::aov\_car}, +avoiding the need to extract summary statistics manually: \begin{Shaded} \begin{Highlighting}[] -\CommentTok{\# Example using a purely within{-}subjects design } +\CommentTok{\# Example using a purely within{-}subjects design} \CommentTok{\# (Maxwell \& Delaney, 2004, Chapter 12, Table 12.5, p. 578):} \FunctionTok{library}\NormalTok{(afex)} \FunctionTok{data}\NormalTok{(md\_12}\FloatTok{.1}\NormalTok{)} -\NormalTok{aovtest2 }\OtherTok{=} \FunctionTok{aov\_ez}\NormalTok{(}\StringTok{"id"}\NormalTok{, }\StringTok{"rt"}\NormalTok{, md\_12}\FloatTok{.1}\NormalTok{, }\AttributeTok{within =} \FunctionTok{c}\NormalTok{(}\StringTok{"angle"}\NormalTok{, }\StringTok{"noise"}\NormalTok{), } +\NormalTok{aovtest2 }\OtherTok{=} \FunctionTok{aov\_ez}\NormalTok{(}\StringTok{"id"}\NormalTok{, }\StringTok{"rt"}\NormalTok{, md\_12}\FloatTok{.1}\NormalTok{, }\AttributeTok{within =} \FunctionTok{c}\NormalTok{(}\StringTok{"angle"}\NormalTok{, }\StringTok{"noise"}\NormalTok{),} \AttributeTok{anova\_table=}\FunctionTok{list}\NormalTok{(}\AttributeTok{correction =} \StringTok{"none"}\NormalTok{, }\AttributeTok{es =} \StringTok{"none"}\NormalTok{))} \FunctionTok{equ\_anova}\NormalTok{(aovtest2,} \AttributeTok{eqb =} \FloatTok{0.35}\NormalTok{)} @@ -1184,213 +1660,173 @@ \subsection{Example of Equivalence ANOVA \newpage -\hypertarget{equivalence-between-replication-studies}{% -\section{Equivalence Between Replication -Studies}\label{equivalence-between-replication-studies}} - -During the development of this TOSTER update, I was helping advise a -team of researchers on a massive replication project for sport and -exercise science \citep{repSES}. How to determine whether a -direct\footnote{Defined as being a as-close-as possible replication to - the original study, in contrast to ``conceptual'' replications.} -replication was a successful replication of the original study was -contentious topic of conversation among the team. Inspired by these -discussions, I created 2 functions that would utilize the basic -principles of SMDs\footnote{The textbook by \citet{borenstein} and the - some of the works of Wolfgang Vietchbauer, metafor R package author, - were a large source of information for developing these functions.} to -test for differences between two studies. - -Overall, the concept is simple: if we have estimates of SMDs from two -very similar studies we can use the large-sample approximation to -compute the sampling variances\footnote{Users can also supply their own - sampling variances using the \texttt{se1} and \texttt{se2} arguments.} -to estimate the degree to which the two studies differ from one another -(i.e., calculate p-values). The users of TOSTER then have the option to -test whether the two SMDs significantly differ, or use TOST to estimate -if they are practically equivalent. Additionally, there are two options -for comparing SMDs: using the summary statistics or using bootstrapping -(assuming original data is available). - -\hypertarget{example-using-summary-statistics}{% -\subsection{Example using Summary -Statistics}\label{example-using-summary-statistics}} - -In this example, let us imagine an ``original'' study that reports an -effect of Cohen's dz = 0.95 in a paired samples design with 25 subjects. -However, a replication doubled the sample size, found a non-significant -effect at an SMD of 0.2. Are these two studies compatible (the lower the -p-value the lower the compatibility)? Or, to put it another way, should -the replication be considered a ``failure'' to replicate the original -study? - -We can use the \texttt{compare\_smd} function to at least measure how -often we would expect a discrepancy between the original and replication -study if the same underlying effect was being measured (also assuming no -publication bias). - -We can see from the results below that, if the null hypothesis were -true, we would only expect to see a discrepancy in SMDs between studies -at least this large \textasciitilde1\% of the time. +\section{Power Analysis and Sample Size +Planning}\label{power-analysis-and-sample-size-planning} -\begin{Shaded} -\begin{Highlighting}[] -\FunctionTok{compare\_smd}\NormalTok{(}\AttributeTok{smd1 =} \FloatTok{0.95}\NormalTok{,} - \AttributeTok{n1 =} \DecValTok{25}\NormalTok{,} - \AttributeTok{smd2 =} \FloatTok{0.23}\NormalTok{,} - \AttributeTok{n2 =} \DecValTok{50}\NormalTok{,} - \AttributeTok{paired =} \ConstantTok{TRUE}\NormalTok{)} -\end{Highlighting} -\end{Shaded} +Researchers planning an equivalence study will need to determine the +required sample size. Power for equivalence tests behaves differently +than for standard tests: larger samples are generally needed, and power +depends on both the true effect size and the width of the equivalence +bounds. -\begin{verbatim} -## -## Difference in Cohen's dz (paired) -## -## data: Summary Statistics -## z = 2.5685, p-value = 0.01021 -## alternative hypothesis: true difference in SMDs is not equal to 0 -## sample estimates: -## difference in SMDs -## 0.72 -\end{verbatim} +\subsection{Power for TOST t-tests}\label{power-for-tost-t-tests} + +The \texttt{power\_t\_TOST} function computes exact power for TOST with +one-sample, two-sample, and paired t-tests. It can solve for any one of +\texttt{n}, \texttt{power}, or \texttt{alpha} given the other parameters +are provided. The \texttt{delta} argument specifies the true mean +difference (set to 0 if the true effect is exactly zero) and +\texttt{eqb} specifies the equivalence bounds. -Let us also imagine a scenario where a replication team considers a -replication successful if the SMDs are within 0.25 units of each other. -We can set the \texttt{TOST} argument to TRUE, and then set the -equivalence bound using \texttt{null} argument. +For example, to determine the sample size needed to achieve 80\% power +to detect equivalence within bounds of \(\pm\) 0.5 (assuming the true +difference is zero and \(SD\) = 1): \begin{Shaded} \begin{Highlighting}[] -\FunctionTok{compare\_smd}\NormalTok{(}\AttributeTok{smd1 =} \FloatTok{0.95}\NormalTok{, }\AttributeTok{n1 =} \DecValTok{25}\NormalTok{, }\AttributeTok{smd2 =} \FloatTok{0.23}\NormalTok{,}\AttributeTok{n2 =} \DecValTok{50}\NormalTok{,} - \AttributeTok{paired =} \ConstantTok{TRUE}\NormalTok{, }\AttributeTok{TOST =} \ConstantTok{TRUE}\NormalTok{, }\AttributeTok{null =}\NormalTok{ .}\DecValTok{25}\NormalTok{)} +\FunctionTok{power\_t\_TOST}\NormalTok{(}\AttributeTok{delta =} \DecValTok{0}\NormalTok{, }\AttributeTok{sd =} \DecValTok{1}\NormalTok{, }\AttributeTok{eqb =} \FloatTok{0.5}\NormalTok{,} + \AttributeTok{alpha =} \FloatTok{0.05}\NormalTok{, }\AttributeTok{power =} \FloatTok{0.8}\NormalTok{,} + \AttributeTok{type =} \StringTok{"two.sample"}\NormalTok{)} \end{Highlighting} \end{Shaded} \begin{verbatim} ## -## Difference in Cohen's dz (paired) +## Two-sample TOST power calculation ## -## data: Summary Statistics -## z = 1.6767, p-value = 0.9532 -## alternative hypothesis: true difference in SMDs is less than 0.25 -## sample estimates: -## difference in SMDs -## 0.72 +## power = 0.8 +## beta = 0.2 +## alpha = 0.05 +## n = 69.19782 +## delta = 0 +## sd = 1 +## bounds = -0.5, 0.5 +## +## NOTE: n is number in *each* group \end{verbatim} -Based on the imaginary studies we outlined above, we would not reject -the null equivalence hypothesis, but reject the null significance -hypothesis. Therefore, we would could conclude that there are -significant differences between the studies that are not practically -equivalent. - -\hypertarget{example-using-bootstrapping}{% -\subsection{Example using -Bootstrapping}\label{example-using-bootstrapping}} +\subsection{Power for Correlations}\label{power-for-correlations} -The above results are only based on an approximating the differences -between the SMDs. If the raw data is available, then the optimal -solution is the bootstrap. This can be accomplished with the -\texttt{boot\_compare\_smd} function. The only drawback to this function -is that TOST is currently not avaiable, and users would instead have to -run 2 one-sided tests manually using the \texttt{null} and -\texttt{alternative} arguments. - -For this example, we will simulate some data. As an alternative approach -to TOST, we can just set the \texttt{alpha} to 0.1, and then check to -see if the confidence interval is within the preset equivalence bounds. +The \texttt{power\_z\_cor} function provides power analysis for +equivalence tests on Pearson correlations using Fisher's +z-transformation: \begin{Shaded} \begin{Highlighting}[] -\FunctionTok{set.seed}\NormalTok{(}\DecValTok{4522}\NormalTok{)} -\NormalTok{boot\_test }\OtherTok{=} \FunctionTok{boot\_compare\_smd}\NormalTok{(}\AttributeTok{x1 =} \FunctionTok{rnorm}\NormalTok{(}\DecValTok{25}\NormalTok{,.}\DecValTok{95}\NormalTok{), }\AttributeTok{x2 =} \FunctionTok{rnorm}\NormalTok{(}\DecValTok{50}\NormalTok{), } - \AttributeTok{paired =} \ConstantTok{TRUE}\NormalTok{, }\AttributeTok{alpha =}\NormalTok{ .}\DecValTok{1}\NormalTok{)} -\NormalTok{boot\_test} +\FunctionTok{power\_z\_cor}\NormalTok{(}\AttributeTok{rho =} \DecValTok{0}\NormalTok{, }\AttributeTok{power =} \FloatTok{0.8}\NormalTok{,} + \AttributeTok{null =} \FloatTok{0.3}\NormalTok{, }\AttributeTok{alpha =} \FloatTok{0.05}\NormalTok{,} + \AttributeTok{alternative =} \StringTok{"equivalence"}\NormalTok{)} \end{Highlighting} \end{Shaded} \begin{verbatim} ## -## Bootstrapped Differences in SMDs (paired) +## Approximate Power for Pearson Product-Moment Correlation (z-test) ## -## data: Bootstrapped -## z (observed) = 2.887, p-value = 0.006003 -## alternative hypothesis: true difference in SMDs is not equal to 0 -## 90 percent confidence interval: -## 0.4070761 1.3508435 -## sample estimates: -## difference in SMDs -## 0.8058872 +## n = 92.38991 +## rho = 0 +## alpha = 0.05 +## beta = 0.2 +## power = 0.8 +## null = 0.3, -0.3 +## alternative = equivalence \end{verbatim} +\subsection{Extensions: PowerTOST and +Superpower}\label{extensions-powertost-and-superpower} + +For more complex designs --- crossover studies, replicate designs, +group-sequential equivalence trials --- the PowerTOST package +\citep{PowerTOST} provides extensive functionality beyond what TOSTER +offers. For factorial designs where equivalence contrasts are tested via +estimated marginal means, simulation-based power analysis via the +Superpower package is recommended \citep{Lakens_Caldwell_2021}. Both +packages complement TOSTER's built-in power tools for common research +designs. + \newpage -\hypertarget{conclusions}{% -\section{Conclusions}\label{conclusions}} - -In this manuscript I have demonstrated most of the new functions and -features within the TOSTER R package. This constitutes a major update to -the package over the past 2 years. I hope that updates to the package -builds upon the original impact of the TOSTER package\footnote{In my - opinion, the impact of the \citet{lakens_ori} cannot be overstated - considering it is cited by over 1000 other papers!}, and has been made -TOST more accessible to the average researcher. In addition, I have -added a number of other functions that offer robust alternatives to the -t-test for performing TOST analyses. I would strongly recommend users of -TOSTER to explore these functions, and, at the very least, compare the -robust results to the t-test results to ensure that the conclusions do -not change due to the chosen analysis\footnote{If they do change, then - it would be prudent to explore what features in the data might explain - this discrepancy.}. Lastly, to my knowledge, this is the first package -to offer equivalence testing options for ANOVAs or for comparing SMDs -between studies. Overall, this package and its functions offer an easily -accessible option for researchers to explore equivalence testing, and -hopefully improve their statistical analyses. +\section{Conclusions}\label{conclusions} + +This paper has described the major updates to the TOSTER R package, +expanding its scope from the original t-test and correlation equivalence +tests to a comprehensive toolkit for frequentist equivalence testing. +The update adds a unified t-test interface with minimal effect testing +support, a simplified hypothesis testing interface that integrates with +the standard R ecosystem, tools for standardized effect sizes and +correlations, robust and resampling-based alternatives, ANOVA +equivalence testing, and power analysis for sample size planning. + +Several limitations should be acknowledged. TOSTER only covers a few +standard research designs --- it does not handle many analysis scenarios +such as those involving multilevel or mixed-effects models. For these +settings, researchers should consider packages like ``emmeans'' or +``marginaleffects'', which can accommodate non-zero null hypotheses in +more complex modeling frameworks. TOSTER also does not provide Bayesian +equivalence testing; researchers interested in Bayesian methods for the +null should consult other R packages such as ``bayestestR''. + +I hope that the updated package continues to build upon the original +impact of \citet{lakens_ori} and makes equivalence testing more +accessible to researchers across disciplines. At the very least, I would +encourage users to compare results across methods (e.g., parametric +vs.~robust) to ensure that conclusions do not depend on the chosen +analysis. If the interpretation substantially changes depending on the +test, this suggests features of the data --- violations of assumptions, +outliers, distributional shape --- that warrant further investigation. \newpage -\hypertarget{additional-information}{% -\section{Additional Information}\label{additional-information}} +\section{Additional Information}\label{additional-information} -All analyses/code in this manuscript are from TOSTER v0.6.0: +The TOSTER package can be installed from CRAN or from GitHub: \begin{verbatim} -# Install the exact release with this code -devtools::install_github("Lakens/TOSTER@v0.6.0") +# Install from CRAN +install.packages("TOSTER") + +# Install the development version from GitHub +devtools::install_github("Lakens/TOSTER") \end{verbatim} -\hypertarget{acknowledgements}{% -\subsection*{Acknowledgement(s)}\label{acknowledgements}} +\subsection*{Acknowledgement(s)}\label{acknowledgements} \addcontentsline{toc}{subsection}{Acknowledgement(s)} -I'd would like to thank everyone from the Lakens' laboratory group for +I would like to thank everyone from the Lakens laboratory group for their input and suggestions. -\hypertarget{disclosure-statement}{% -\subsection*{Disclosure statement}\label{disclosure-statement}} +\subsection*{Disclosure statement}\label{disclosure-statement} \addcontentsline{toc}{subsection}{Disclosure statement} The author of this manuscript is the author of the TOSTER package. Citations of this manuscript will benefit his citation count. -\hypertarget{funding}{% -\subsection*{Funding}\label{funding}} +\section{AI Use Disclosure}\label{ai-use-disclosure} + +During the preparation of the TOSTER package updates and this +manuscript, the author used Claude Opus 4.5-4.7 (Anthropic) to assist +with editing R code and drafting unit tests. For the manuscript text, +Claude Opus was used only for targeted copy-editing --- suggesting +revisions to specific passages, catching typographical and grammatical +errors, and identifying inconsistencies across sections. The manuscript +was written by the author, and all AI-assisted output was reviewed, +revised, and verified by the author, who takes full responsibility for +the content of the package and the manuscript. + +\subsection*{Funding}\label{funding} \addcontentsline{toc}{subsection}{Funding} No funding was provided for this work. -\hypertarget{notes-on-contributors}{% -\subsection*{Notes on contributor(s)}\label{notes-on-contributors}} +\subsection*{Notes on contributor(s)}\label{notes-on-contributors} \addcontentsline{toc}{subsection}{Notes on contributor(s)} Daniel Lakens provided a review of many of the materials that have been incorporated into the update of TOSTER, and was the original author of -this package. Without his help and encouragment, the TOSTER package and +this package. Without his help and encouragement, the TOSTER package and this update would not exist. -\hypertarget{nomenclaturenotation}{% -\subsection*{Nomenclature/Notation}\label{nomenclaturenotation}} +\subsection*{Nomenclature/Notation}\label{nomenclaturenotation} \addcontentsline{toc}{subsection}{Nomenclature/Notation} \begin{itemize} @@ -1411,17 +1847,16 @@ \subsection*{Nomenclature/Notation}\label{nomenclaturenotation}} \item SMD: Standardized Mean Difference (e.g., Cohen's d) \item - TOST: Two-One Sided Tests + TOST: Two One-Sided Tests \item WMW: Wilcoxon-Mann-Whitney \end{itemize} -\hypertarget{notes}{% -\subsection*{Notes}\label{notes}} +\subsection*{Notes}\label{notes} \addcontentsline{toc}{subsection}{Notes} -The R package is also (partially) implemented in jamovi as the TOSTER -module. +The R package is also (partially) implemented in +\href{https://www.jamovi.org/}{jamovi} as the TOSTER module. \newpage diff --git 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b/papers/Avocado_Update/interactcadsample.bib index e46ea717..09a6333c 100644 --- a/papers/Avocado_Update/interactcadsample.bib +++ b/papers/Avocado_Update/interactcadsample.bib @@ -281,7 +281,7 @@ @article{ @article{Arboretti_Pesarin_Salmaso_2020, title={A unified approach to permutation testing for equivalence}, volume={30}, url={http://dx.doi.org/10.1007/s10260-020-00548-0}, DOI={10.1007/s10260-020-00548-0}, -abstractNote={AbstractThe notion of testing for equivalence of two treatments is widely used in clinical trials, pharmaceutical experiments, bioequivalence and quality control. It is traditionally operated within the intersection–union principle (IU). According to this principle the null hypothesis is stated as the set of effects the differences $$delta$$ δ of which lie outside a suitable equivalence interval and the alternative as the set of $$delta$$ δ that lie inside it. In the literature related solutions are essentially based on likelihood techniques, which in turn are rather difficult to deal with. A recently published paper goes beyond most of likelihood limitations by using the IU approach within the permutation theory. One more paper, based on Roy’s union–intersection principle (UI) within the permutation theory, goes beyond some limitations of traditional two-sided tests. Such UI approach, effectively a mirror image of IU, assumes a null hypothesis where $$delta$$ δ lies inside the equivalence interval and an alternative where it lies outside. Since testing for equivalence can rationally be analyzed by both principles but, as the two differ in terms of the mirror-like roles assigned to the hypotheses under study, they are not strictly comparable. The present paper’s main goal is to look into these problems and provide a sort of comparative analysis of both by highlighting the related requirements, properties, limitations, difficulties, and pitfalls so as to get practitioners properly acquainted with their correct use in practical contexts.}, number={3}, journal={Statistical Methods & Applications}, publisher={Springer Science and Business Media LLC}, +abstractNote={AbstractThe notion of testing for equivalence of two treatments is widely used in clinical trials, pharmaceutical experiments, bioequivalence and quality control. It is traditionally operated within the intersection–union principle (IU). According to this principle the null hypothesis is stated as the set of effects the differences $$delta$$ δ of which lie outside a suitable equivalence interval and the alternative as the set of $$delta$$ δ that lie inside it. In the literature related solutions are essentially based on likelihood techniques, which in turn are rather difficult to deal with. A recently published paper goes beyond most of likelihood limitations by using the IU approach within the permutation theory. One more paper, based on Roy’s union–intersection principle (UI) within the permutation theory, goes beyond some limitations of traditional two-sided tests. Such UI approach, effectively a mirror image of IU, assumes a null hypothesis where $$delta$$ δ lies inside the equivalence interval and an alternative where it lies outside. Since testing for equivalence can rationally be analyzed by both principles but, as the two differ in terms of the mirror-like roles assigned to the hypotheses under study, they are not strictly comparable. The present paper’s main goal is to look into these problems and provide a sort of comparative analysis of both by highlighting the related requirements, properties, limitations, difficulties, and pitfalls so as to get practitioners properly acquainted with their correct use in practical contexts.}, number={3}, journal={Statistical Methods \& Applications}, publisher={Springer Science and Business Media LLC}, author={Arboretti, Rosa and Pesarin, Fortunato and Salmaso, Luigi}, year={2020}, month=nov, pages={1033–1052}, language={en} } diff --git a/papers/Avocado_Update/~$ocado_Update.docx b/papers/Avocado_Update/~$ocado_Update.docx new file mode 100644 index 0000000000000000000000000000000000000000..cc171ee6167954c20db9ced727e6ca1187d46d88 GIT binary patch literal 162 zcmd;gPRvOuPtD2EQE*Hw%FkmU3h*;HGbA$PFr+Y)Go%9P90nZ*1qMf;SP?@$1IUo` w6J`KOM_(os^I$xfc&6Ed-#zE Date: Fri, 17 Apr 2026 12:48:02 -0500 Subject: [PATCH 11/40] fixed to ref figure --- papers/Avocado_Update/Avocado_Update.Rmd | 6 +- papers/Avocado_Update/Avocado_Update.log | 767 ------------------ papers/Avocado_Update/Avocado_Update.pdf | Bin 390406 -> 390236 bytes papers/Avocado_Update/Avocado_Update.tex | 34 +- .../figure-latex/cdplot-1.pdf | Bin 18842 -> 18842 bytes .../figure-latex/genericplot-1.pdf | Bin 5027 -> 5027 bytes .../figure-latex/hypplot-1.pdf | Bin 4622 -> 4622 bytes .../figure-latex/shplot1-1.pdf | Bin 4660 -> 4660 bytes .../figure-latex/symmplot-1.pdf | Bin 4534 -> 4534 bytes .../figure-latex/cdplot-1.pdf | Bin 18842 -> 18842 bytes .../figure-latex/genericplot-1.pdf | Bin 5027 -> 5027 bytes .../figure-latex/hypplot-1.pdf | Bin 4622 -> 4622 bytes .../figure-latex/shplot1-1.pdf | Bin 4660 -> 4660 bytes .../figure-latex/symmplot-1.pdf | Bin 4534 -> 4534 bytes 14 files changed, 20 insertions(+), 787 deletions(-) delete mode 100644 papers/Avocado_Update/Avocado_Update.log diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index 7bee11bb..71bba149 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -501,13 +501,13 @@ Based on these results, we would conclude there is a significant difference and #### Checking Assumptions -The WMW tests assume independent observations and exchangeability under the null, the latter meaning that group labels carry no information about outcomes if there is no effect. For the interpretations discussed above to be clean, two additional assumptions matter. The location-shift assumption — that the two group distributions differ only in location, not shape — is what allows WMW results to be interpreted as a test of the median difference via the Hodges-Lehmann estimator. Violations are best diagnosed by overlaying the empirical cumulative density functions of the two groups; under location shift, the curves are horizontal translations of each other, whereas crossing curves or varying horizontal gaps indicate violation. For the paired and one-sample signed-rank variants used in `wilcox_TOST` or `simple_htest`, the signed-rank procedure additionally requires that the difference distribution is symmetric around the null value for the pseudo-median to coincide with the mean or median difference. Asymmetry is readily checked with a symmetry plot of the paired differences (@fig-symmplot). As with the location-shift case, visible departures suggest that the test's output — the pseudo-median and associated p-value — does not map cleanly onto a hypothesis about central tendency. +The WMW tests assume independent observations and exchangeability under the null, the latter meaning that group labels carry no information about outcomes if there is no effect. For the interpretations discussed above to be clean, two additional assumptions matter. The location-shift assumption — that the two group distributions differ only in location, not shape — is what allows WMW results to be interpreted as a test of the median difference via the Hodges-Lehmann estimator. Violations are best diagnosed by overlaying the empirical cumulative density functions of the two groups; under location shift, the curves are horizontal translations of each other, whereas crossing curves or varying horizontal gaps indicate violation. For the paired and one-sample signed-rank variants used in `wilcox_TOST` or `simple_htest`, the signed-rank procedure additionally requires that the difference distribution is symmetric around the null value for the pseudo-median to coincide with the mean or median difference. Asymmetry is readily checked with a symmetry plot of the paired differences (Figure \ref{fig:symmplot}). As with the location-shift case, visible departures suggest that the test's output — the pseudo-median and associated p-value — does not map cleanly onto a hypothesis about central tendency. ```{r } -#| label: fig-symmplot +#| label: symmplot #| echo: false -#| fig.cap: "Symmetry plot of the paired differences from the sleep dataset, pairing each observation's distance above the median with the corresponding distance below. Under symmetry, points fall along the 45-degree reference line (dashed); systematic departures — points consistently above or below the line — indicate asymmetry. The signed-rank variant of WMW requires this symmetry for the pseudo-median to coincide with the mean or median of the differences." +#| fig.cap: "\\label{fig:symmplot}Symmetry plot of the paired differences from the sleep dataset, pairing each observation's distance above the median with the corresponding distance below. Under symmetry, points fall along the 45-degree reference line (dashed); systematic departures — points consistently above or below the line — indicate asymmetry. The signed-rank variant of WMW requires this symmetry for the pseudo-median to coincide with the mean or median of the differences." # symmetry plot for sleep paired differences d_sleep <- sleep$extra[sleep$group == 2] - sleep$extra[sleep$group == 1] m <- median(d_sleep) diff --git a/papers/Avocado_Update/Avocado_Update.log b/papers/Avocado_Update/Avocado_Update.log deleted file mode 100644 index 7128121d..00000000 --- a/papers/Avocado_Update/Avocado_Update.log +++ /dev/null @@ -1,767 +0,0 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.29 (TeX Live 2026) (preloaded format=pdflatex 2026.4.17) 17 APR 2026 12:35 -entering extended mode - restricted \write18 enabled. - %&-line parsing enabled. -**Avocado_Update.tex -(./Avocado_Update.tex -LaTeX2e <2025-11-01> -L3 programming layer <2026-03-20> -(./interact.cls -Document Class: interact 2016/10/17 v1.03 Interact LaTeX document class -(c:/Users/caldwellaaron/AppData/Roaming/TinyTeX/texmf-dist/tex/latex/base/article.cls -Document Class: 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4969022d..5775e300 100644 --- a/papers/Avocado_Update/Avocado_Update.tex +++ b/papers/Avocado_Update/Avocado_Update.tex @@ -790,13 +790,13 @@ \subsection{Standardized Mean Difference ## g[rm]=('1'-'2')/SD_rm) ## ## data: extra by group -## z-observed = -0.78495, p-value = 0.908 +## z-observed = -0.78495, p-value = 0.8874 ## alternative hypothesis: equivalence ## null values: ## lower bound upper bound ## -0.5 0.5 ## 90 percent confidence interval: -## -1.269159 -0.449214 +## -1.3739319 -0.4362495 ## sample estimates: ## SMD (g[rm]) ## -0.7513666 @@ -879,13 +879,13 @@ \subsection{Equivalence Tests for ## Bootstrapped Pearson's product-moment correlation (BCa) ## ## data: bugs$LDLF and bugs$LDHF -## N = 91, p-value = 0.9827 +## N = 91, p-value = 0.9834 ## alternative hypothesis: equivalence ## null values: ## correlation correlation ## 0.4 -0.4 ## 90 percent confidence interval: -## 0.4398575 0.6899351 +## 0.4456997 0.6881875 ## sample estimates: ## r ## 0.5767783 @@ -898,7 +898,7 @@ \subsection{Equivalence Tests for \end{Shaded} \begin{verbatim} -## [1] "The Bootstrapped Pearson's product-moment correlation (BCa) is not statistically significant (p = 0.983, r = 0.577, 90% C.I.[0.44, 0.69]) at a 0.05 alpha-level. The null hypothesis cannot be rejected. At the desired error rate, it cannot be stated that the true correlation is between 0.4 and -0.4." +## [1] "The Bootstrapped Pearson's product-moment correlation (BCa) is not statistically significant (p = 0.983, r = 0.577, 90% C.I.[0.446, 0.688]) at a 0.05 alpha-level. The null hypothesis cannot be rejected. At the desired error rate, it cannot be stated that the true correlation is between 0.4 and -0.4." \end{verbatim} If only summary statistics are available, an approximate method is @@ -1135,21 +1135,21 @@ \subsubsection{Wilcoxon-Mann-Whitney that the difference distribution is symmetric around the null value for the pseudo-median to coincide with the mean or median difference. Asymmetry is readily checked with a symmetry plot of the paired -differences (\citet{fig-symmplot}). As with the location-shift case, -visible departures suggest that the test's output --- the pseudo-median -and associated p-value --- does not map cleanly onto a hypothesis about -central tendency. +differences (Figure \ref{fig:symmplot}). As with the location-shift +case, visible departures suggest that the test's output --- the +pseudo-median and associated p-value --- does not map cleanly onto a +hypothesis about central tendency. \begin{figure} \centering -\pandocbounded{\includegraphics[keepaspectratio]{Avocado_Update_files/Avocado_Update_files/figure-latex/fig-symmplot-1.pdf}} -\caption{Symmetry plot of the paired differences from the sleep dataset, -pairing each observation's distance above the median with the -corresponding distance below. Under symmetry, points fall along the -45-degree reference line (dashed); systematic departures --- points -consistently above or below the line --- indicate asymmetry. The -signed-rank variant of WMW requires this symmetry for the pseudo-median -to coincide with the mean or median of the differences.} +\pandocbounded{\includegraphics[keepaspectratio]{Avocado_Update_files/Avocado_Update_files/figure-latex/symmplot-1.pdf}} +\caption{\label{fig:symmplot}Symmetry plot of the paired differences +from the sleep dataset, pairing each observation's distance above the +median with the corresponding distance below. Under symmetry, points +fall along the 45-degree reference line (dashed); systematic departures +--- points consistently above or below the line --- indicate asymmetry. +The signed-rank variant of WMW requires this symmetry for the +pseudo-median to coincide with the mean or median of the differences.} \end{figure} \subsubsection{Brunner-Munzel Test}\label{brunner-munzel-test} diff --git a/papers/Avocado_Update/Avocado_Update_files/Avocado_Update_files/figure-latex/cdplot-1.pdf b/papers/Avocado_Update/Avocado_Update_files/Avocado_Update_files/figure-latex/cdplot-1.pdf index 29862ce41624d1ff94f871ae9f4fc894a5e845d7..ddf1aa640a27ab68de1855001123e6277b10674e 100644 GIT binary patch delta 25 dcmbO=nQ_)+#tGIeCZ-1F6YV9y^u}m=PXJ}O2TT9} delta 25 dcmbO=nQ_)+#tGIe#-=956YV9y^u}m=PXJ}82TA|{ diff --git a/papers/Avocado_Update/Avocado_Update_files/Avocado_Update_files/figure-latex/genericplot-1.pdf b/papers/Avocado_Update/Avocado_Update_files/Avocado_Update_files/figure-latex/genericplot-1.pdf index aaedeed59e61da9fac6c720298467dbac388e328..f690e2b132857fcfc0093fbec3e3f13bbc2c4481 100644 GIT binary patch delta 23 bcmZ3izF2*NHH(RxW0!(j=-Yf_JPD%!h delta 23 bcmeBE=~J0t&0=h7VmQ%W0!(j=-Yf_JPC5pP diff --git a/papers/Avocado_Update/Avocado_Update_files/Avocado_Update_files/figure-latex/shplot1-1.pdf b/papers/Avocado_Update/Avocado_Update_files/Avocado_Update_files/figure-latex/shplot1-1.pdf index 928e63d5c4ee3e3aa1f135beb1643a546167647f..03e6550bb552a0464abc64985dc5054ebf7e8288 100644 GIT binary patch delta 23 bcmdm@vPETrHH(Re delta 23 bcmdm{yiIw6HH)E?l diff --git a/papers/Avocado_Update/Avocado_Update_files/figure-latex/cdplot-1.pdf b/papers/Avocado_Update/Avocado_Update_files/figure-latex/cdplot-1.pdf index 29862ce41624d1ff94f871ae9f4fc894a5e845d7..ddf1aa640a27ab68de1855001123e6277b10674e 100644 GIT binary patch delta 25 dcmbO=nQ_)+#tGIeCZ-1F6YV9y^u}m=PXJ}O2TT9} delta 25 dcmbO=nQ_)+#tGIe#-=956YV9y^u}m=PXJ}82TA|{ diff --git a/papers/Avocado_Update/Avocado_Update_files/figure-latex/genericplot-1.pdf b/papers/Avocado_Update/Avocado_Update_files/figure-latex/genericplot-1.pdf index aaedeed59e61da9fac6c720298467dbac388e328..f690e2b132857fcfc0093fbec3e3f13bbc2c4481 100644 GIT binary patch delta 23 bcmZ3izF2*NHH(RxW0!(j=-Yf_JPD%!h delta 23 bcmeBE=~J0t&0=h7VmQ%W0!(j=-Yf_JPC5pP diff --git a/papers/Avocado_Update/Avocado_Update_files/figure-latex/shplot1-1.pdf b/papers/Avocado_Update/Avocado_Update_files/figure-latex/shplot1-1.pdf index 928e63d5c4ee3e3aa1f135beb1643a546167647f..03e6550bb552a0464abc64985dc5054ebf7e8288 100644 GIT binary patch delta 23 bcmdm@vPETrHH(Re delta 23 bcmdm{yiIw6HH)E?l From ae5404cb33f72e6f523a21f24d7c6bfc084fc74f Mon Sep 17 00:00:00 2001 From: Aaron Caldwell Date: Fri, 17 Apr 2026 13:22:10 -0500 Subject: [PATCH 12/40] Fixed aspect ratio also added Brunner-Munzel assumption section. --- papers/Avocado_Update/Avocado_Update.R | 7 +++++-- papers/Avocado_Update/Avocado_Update.Rmd | 16 +++++++++++++--- papers/Avocado_Update/Avocado_Update.pdf | Bin 390236 -> 390245 bytes papers/Avocado_Update/Avocado_Update.tex | 10 +++++----- .../figure-latex/cdplot-1.pdf | Bin 18842 -> 18842 bytes .../figure-latex/genericplot-1.pdf | Bin 5027 -> 5027 bytes .../figure-latex/hypplot-1.pdf | Bin 4622 -> 4622 bytes .../figure-latex/shplot1-1.pdf | Bin 4660 -> 4660 bytes .../figure-latex/symmplot-1.pdf | Bin 4534 -> 4548 bytes .../figure-latex/cdplot-1.pdf | Bin 18842 -> 18842 bytes .../figure-latex/genericplot-1.pdf | Bin 5027 -> 5027 bytes .../figure-latex/hypplot-1.pdf | Bin 4622 -> 4622 bytes .../figure-latex/shplot1-1.pdf | Bin 4660 -> 4660 bytes .../figure-latex/symmplot-1.pdf | Bin 4534 -> 4548 bytes 14 files changed, 23 insertions(+), 10 deletions(-) diff --git a/papers/Avocado_Update/Avocado_Update.R b/papers/Avocado_Update/Avocado_Update.R index f1b9053f..8b06df6b 100644 --- a/papers/Avocado_Update/Avocado_Update.R +++ b/papers/Avocado_Update/Avocado_Update.R @@ -3,7 +3,9 @@ knitr::opts_chunk$set( warning = FALSE, message = FALSE, echo = TRUE, - fig.pos = "H" + fig.pos = "H", + fig.width = 6, + fig.height = 4 ) knitr::knit_hooks$set(purl = knitr::hook_purl) library(TOSTER) @@ -266,11 +268,12 @@ upper <- sorted[sorted > m] - m lower <- m - sorted[sorted < m] k <- min(length(upper), length(lower)) sp_df <- data.frame(lower = sort(lower)[1:k], upper = sort(upper)[1:k]) +mx <- max(sp_df$lower, sp_df$upper) ggplot(sp_df, aes(x = lower, y = upper)) + geom_point(size = 2) + geom_abline(intercept = 0, slope = 1, linetype = "dashed") + - coord_equal() + + coord_equal(xlim = c(0, mx), ylim = c(0, mx)) + labs(x = "Distance below median", y = "Distance above median") + theme_minimal() diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index 71bba149..747da6ab 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -56,7 +56,9 @@ knitr::opts_chunk$set( warning = FALSE, message = FALSE, echo = TRUE, - fig.pos = "H" + fig.pos = "H", + fig.width = 6, + fig.height = 4 ) knitr::knit_hooks$set(purl = knitr::hook_purl) library(TOSTER) @@ -507,6 +509,8 @@ The WMW tests assume independent observations and exchangeability under the null ```{r } #| label: symmplot #| echo: false +#| fig.height: 5.5 +#| fig.width: 5.5 #| fig.cap: "\\label{fig:symmplot}Symmetry plot of the paired differences from the sleep dataset, pairing each observation's distance above the median with the corresponding distance below. Under symmetry, points fall along the 45-degree reference line (dashed); systematic departures — points consistently above or below the line — indicate asymmetry. The signed-rank variant of WMW requires this symmetry for the pseudo-median to coincide with the mean or median of the differences." # symmetry plot for sleep paired differences d_sleep <- sleep$extra[sleep$group == 2] - sleep$extra[sleep$group == 1] @@ -516,16 +520,18 @@ upper <- sorted[sorted > m] - m lower <- m - sorted[sorted < m] k <- min(length(upper), length(lower)) sp_df <- data.frame(lower = sort(lower)[1:k], upper = sort(upper)[1:k]) +mx <- max(sp_df$lower, sp_df$upper) ggplot(sp_df, aes(x = lower, y = upper)) + geom_point(size = 2) + geom_abline(intercept = 0, slope = 1, linetype = "dashed") + - coord_equal() + + coord_equal(xlim = c(0, mx), ylim = c(0, mx)) + labs(x = "Distance below median", y = "Distance above median") + theme_minimal() ``` +\newpage ### Brunner-Munzel Test @@ -543,6 +549,10 @@ bm_test = brunner_munzel(formula = extra ~ group, print(bm_test) ``` +#### Checking Assumptions + +The Brunner-Munzel test has unusually light assumption requirements. It requires independent observations within and between groups, and (for the t-approximation used by default) sample sizes sufficient for the asymptotic approximation to be reasonable. That's essentially it. Unlike WMW, there is no location-shift requirement for the test to be interpretable as a hypothesis about location, and unlike the proportional-odds framing of WMW as a score test from an ordinal regression model, no constant-odds-ratio assumption is needed. The relative effect is defined directly from the two marginal distributions and carries the same interpretation --— the probability that a random observation from one group exceeds one from the other --— regardless of whether the distributions are symmetric, skewed, unimodal, or shaped however they happen to be. Visual inspection of the group distributions remains useful for understanding the data and for choosing a reporting scale, but there is no distributional assumption that these checks are verifying. The studentized permutation variant (`method = "perm"`) is worth considering in small samples, where the t-approximation may be less trustworthy. + \newpage @@ -738,7 +748,7 @@ I would like to thank everyone from the Lakens laboratory group for their input The author of this manuscript is the author of the TOSTER package. Citations of this manuscript will benefit his citation count. -# AI Use Disclosure +## AI Use Disclosure {-} During the preparation of the TOSTER package updates and this manuscript, the author used Claude Opus 4.5-4.7 (Anthropic) to assist with editing R code and drafting unit tests. For the manuscript text, Claude Opus was used only for targeted copy-editing — suggesting revisions to specific passages, catching typographical and grammatical errors, and identifying inconsistencies across sections. 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Bootstrapped Pearson's product-moment correlation (BCa) ## ## data: bugs$LDLF and bugs$LDHF -## N = 91, p-value = 0.9834 +## N = 91, p-value = 0.9838 ## alternative hypothesis: equivalence ## null values: ## correlation correlation ## 0.4 -0.4 ## 90 percent confidence interval: -## 0.4456997 0.6881875 +## 0.4443340 0.6923365 ## sample estimates: ## r ## 0.5767783 @@ -898,7 +898,7 @@ \subsection{Equivalence Tests for \end{Shaded} \begin{verbatim} -## [1] "The Bootstrapped Pearson's product-moment correlation (BCa) is not statistically significant (p = 0.983, r = 0.577, 90% C.I.[0.446, 0.688]) at a 0.05 alpha-level. The null hypothesis cannot be rejected. At the desired error rate, it cannot be stated that the true correlation is between 0.4 and -0.4." +## [1] "The Bootstrapped Pearson's product-moment correlation (BCa) is not statistically significant (p = 0.984, r = 0.577, 90% C.I.[0.444, 0.692]) at a 0.05 alpha-level. The null hypothesis cannot be rejected. 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8306fbc5a233cf063ba28ca0529d7bd597501ffb Mon Sep 17 00:00:00 2001 From: Aaron Caldwell Date: Fri, 17 Apr 2026 13:46:33 -0500 Subject: [PATCH 13/40] Add final assumptions --- papers/Avocado_Update/Avocado_Update.Rmd | 14 +++ papers/Avocado_Update/Avocado_Update.pdf | Bin 390245 -> 395447 bytes papers/Avocado_Update/Avocado_Update.tex | 105 +++++++++++++++++- .../figure-latex/cdplot-1.pdf | Bin 18842 -> 18842 bytes .../figure-latex/genericplot-1.pdf | Bin 5027 -> 5027 bytes .../figure-latex/hypplot-1.pdf | Bin 4622 -> 4622 bytes .../figure-latex/shplot1-1.pdf | Bin 4660 -> 4660 bytes .../figure-latex/symmplot-1.pdf | Bin 4548 -> 4548 bytes .../figure-latex/cdplot-1.pdf | Bin 18842 -> 18842 bytes .../figure-latex/genericplot-1.pdf | Bin 5027 -> 5027 bytes .../figure-latex/hypplot-1.pdf | Bin 4622 -> 4622 bytes .../figure-latex/shplot1-1.pdf | Bin 4660 -> 4660 bytes .../figure-latex/symmplot-1.pdf | Bin 4548 -> 4548 bytes 13 files changed, 113 insertions(+), 6 deletions(-) diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index 747da6ab..e7a0196c 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -597,6 +597,10 @@ print(test1) The bootstrapped SMDs are a useful feature in their own right. Analytic confidence intervals for standardized effect sizes rely on approximations, which assume normality and can behave poorly with small or skewed samples. Bootstrap intervals for SMDs sidestep these assumptions by resampling the full estimation pipeline, yielding intervals that inherit the robustness properties of the bootstrap itself. +#### Checking Assumptions + +The bootstrap t-test makes fewer distributional assumptions than its parametric counterpart but is not assumption-free. It requires independent observations and assumes the empirical distribution is a reasonable stand-in for the population distribution. The latter is an assumption that holds better in moderate-to-large samples than in small ones. The studentized bootstrap is reasonably robust to skewness and heavy tails, but remains sensitive to outliers and other high-influence observations, which can be drawn repeatedly across replicates and disproportionately shape the bootstrap distribution. Useful visual checks include the distribution of the data (histograms, boxplots) to identify extreme values, the bootstrap distribution itself to flag problematic skewness or gaps, and the jackknife-after-bootstrap plot to identify individual observations that dominate the bootstrap variability. The `tr` argument for trimmed means provides a straightforward robustness option when outliers are a concern. Diagnostic workflows for all three designs are demonstrated in the online supplement. As with the parametric t-test, these checks reflect assumptions that are relaxed relative to normal-theory inference but not eliminated. + ### Permutation t-test When `R = NULL`, the function enumerates all possible permutations for an exact test; otherwise, it draws R random permutations with the "+1" correction [@phipson2010]. By default the test is studentized (`perm_se = TRUE`), consistent with the recommendations discussed in the methods section above. The `perm_se = FALSE` option is available primarily to match the behavior of other permutation test packages that do not studentize by default. @@ -613,6 +617,10 @@ perm_t_test(formula = extra ~ group, Running the bootstrap and permutation equivalence tests on the same paired sleep data using matching equivalence bounds produces broadly similar conclusions, as would be expected. The two approaches can, however, diverge in small samples, under strong heteroscedasticity, or in the presence of influential outliers, which is where the choice between them becomes substantively important. Users interested in robustness to outliers can additionally supply the `tr` argument to either function to use trimmed means, preserving the mean-based interpretation while downweighting extreme observations. +#### Checking Assumptions + +The permutation t-test carries different assumptions depending on design. For the two-sample case, the studentized version is valid under exchangeability of observations across groups under the null and provides asymptotic validity even when variances differ between groups. For paired and one-sample designs, the sign-flip procedure additionally requires that the distribution of differences (or one-sample observations) is symmetric around the null value; asymmetry makes the test approximate rather than exact, though studentization provides some asymptotic protection. Symmetry is a commonly overlooked assumption for paired permutation tests and is easily checked with a symmetry plot of the differences, as demonstrated for the WMW signed-rank case above. For two-sample designs, overlaid empirical cumulative density functions and separate density plots help assess whether group shapes are comparable enough for exchangeability under the null to be credible. Diagnostic workflows for all three designs are demonstrated in the online supplement. As with the bootstrap, these checks reflect assumptions that are relaxed relative to parametric inference but not eliminated. + \newpage ## Log-Transformed Methods @@ -641,6 +649,10 @@ boot_log_TOST(mpg ~ am, data = mtcars, R=999) From this analysis, we would conclude there is a significant effect that is not practically equivalent. +### Checking Assumptions + +The log-transformed methods inherit the assumptions of the underlying test applied to the transformed data. The `log_TOST` function carries forward the t-test assumptions, and `boot_log_TOST` carries forward the bootstrap t-test assumptions. What changes is where those assumptions need to hold. Normality, symmetry, and homogeneity of variance should be assessed on the log scale rather than the original scale, since that is where inference takes place. The same visual diagnostics apply just computed on log-transformed data. This points to a broader practical use of log transformation. When the symmetry assumption of the sign-flip permutation test is violated because paired differences are right-skewed on the original scale, the differences of logs are often approximately symmetric, and `perm_t_test` on log-transformed values becomes a reasonable option. In such cases the equivalence bounds should be specified on the log scale and interpreted as log ratios (or sympercents). As with the parametric and bootstrap t-tests, assumptions are relaxed by the transformation but not eliminated --— they are simply moved to a different scale. + \newpage # Equivalence Testing with ANOVAs @@ -649,6 +661,8 @@ Many researchers use ANOVA as an omnibus test before inspecting pairwise compari The equivalence test uses the non-central *F* distribution. The non-centrality parameter is computed from the equivalence bound and the degrees of freedom, and the *p*-value is obtained from the cumulative distribution function of the non-central *F*-value. I have extended the approach of @Campbell_2021 --- who proposed equivalence testing for one-way ANOVAs --- to accommodate multi-factor and factorial designs by generalizing the non-centrality parameter calculation. The full derivation is described in the package documentation; here we demonstrate usage. +The equivalence test inherits the standard ANOVA assumptions which should be checked on the fitted model itself rather than on the equivalence output. The `performance::check_model()` function provides a comprehensive visual diagnostic suite for most linear model objects in R on a single call; the online supplement demonstrates its use. + ## Example with Summary Statistics Using the `InsectSprays` data and the `equ_ftest` function, which accepts summary *F*-test statistics directly: diff --git a/papers/Avocado_Update/Avocado_Update.pdf b/papers/Avocado_Update/Avocado_Update.pdf index 8a86c8dc861fc9d4bd20acc7a648b709f0e4ed25..5f4f7b7466b23960317b4827bb2e96aa409e7f29 100644 GIT binary patch delta 88067 zcmZs?V{m3&*S4E0w%M_5+fK($I=0aX`+lCU_WpLgZ`EA0My;CT z{55N>V~%moqf@Mwcx)7MMKK9_W(H1J^4Z0qRaj>3RFpjcS{)fX95^#K7i%i>7=Q+- zW52Xoi$dLe;=EqJCpO||Iwm=-mL~u; zA*Iy0`#xLrE)JD6iHd}3E=WlN&oCfgytU3^ACg9zteQ!*P&G*F4<2O}%G6Ytvk`_T ztx*+;F4_s+UYtnqgE9_l6tpmQsDm!hNJO+$_>vAKu4-0Vq+s5Pj5|pvW5tL_9yF0X zMrzb~5v3-KMj}bbmR%{qL6}?#vZ>bxj_9-!$Ld_-2NME@7+AVLOBEox;0d{jJjC;z 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The null hypothesis cannot be rejected. At the desired error rate, it cannot be stated that the true correlation is between 0.4 and -0.4." +## [1] "The Bootstrapped Pearson's product-moment correlation (BCa) is not statistically significant (p = 0.981, r = 0.577, 90% C.I.[0.43, 0.685]) at a 0.05 alpha-level. The null hypothesis cannot be rejected. At the desired error rate, it cannot be stated that the true correlation is between 0.4 and -0.4." \end{verbatim} If only summary statistics are available, an approximate method is @@ -1152,6 +1152,8 @@ \subsubsection{Wilcoxon-Mann-Whitney pseudo-median to coincide with the mean or median of the differences.} \end{figure} +\newpage + \subsubsection{Brunner-Munzel Test}\label{brunner-munzel-test} The Brunner-Munzel test \citep{brunner2000} estimates the relative @@ -1220,6 +1222,27 @@ \subsubsection{Brunner-Munzel Test}\label{brunner-munzel-test} ## 0.255 \end{verbatim} +\paragraph{Checking Assumptions}\label{checking-assumptions-4} + +The Brunner-Munzel test has unusually light assumption requirements. It +requires independent observations within and between groups, and (for +the t-approximation used by default) sample sizes sufficient for the +asymptotic approximation to be reasonable. That's essentially it. Unlike +WMW, there is no location-shift requirement for the test to be +interpretable as a hypothesis about location, and unlike the +proportional-odds framing of WMW as a score test from an ordinal +regression model, no constant-odds-ratio assumption is needed. The +relative effect is defined directly from the two marginal distributions +and carries the same interpretation ----- the probability that a random +observation from one group exceeds one from the other ----- regardless +of whether the distributions are symmetric, skewed, unimodal, or shaped +however they happen to be. Visual inspection of the group distributions +remains useful for understanding the data and for choosing a reporting +scale, but there is no distributional assumption that these checks are +verifying. The studentized permutation variant +(\texttt{method\ =\ "perm"}) is worth considering in small samples, +where the t-approximation may be less trustworthy. + \newpage \subsection{Resampling Methods}\label{resampling-methods} @@ -1408,6 +1431,28 @@ \subsubsection{Bootstrap t-test}\label{bootstrap-t-test} intervals that inherit the robustness properties of the bootstrap itself. +\paragraph{Checking Assumptions}\label{checking-assumptions-5} + +The bootstrap t-test makes fewer distributional assumptions than its +parametric counterpart but is not assumption-free. It requires +independent observations and assumes the empirical distribution is a +reasonable stand-in for the population distribution. The latter is an +assumption that holds better in moderate-to-large samples than in small +ones. The studentized bootstrap is reasonably robust to skewness and +heavy tails, but remains sensitive to outliers and other high-influence +observations, which can be drawn repeatedly across replicates and +disproportionately shape the bootstrap distribution. Useful visual +checks include the distribution of the data (histograms, boxplots) to +identify extreme values, the bootstrap distribution itself to flag +problematic skewness or gaps, and the jackknife-after-bootstrap plot to +identify individual observations that dominate the bootstrap +variability. The \texttt{tr} argument for trimmed means provides a +straightforward robustness option when outliers are a concern. +Diagnostic workflows for all three designs are demonstrated in the +online supplement. As with the parametric t-test, these checks reflect +assumptions that are relaxed relative to normal-theory inference but not +eliminated. + \subsubsection{Permutation t-test}\label{permutation-t-test} When \texttt{R\ =\ NULL}, the function enumerates all possible @@ -1458,6 +1503,26 @@ \subsubsection{Permutation t-test}\label{permutation-t-test} to either function to use trimmed means, preserving the mean-based interpretation while downweighting extreme observations. +\paragraph{Checking Assumptions}\label{checking-assumptions-6} + +The permutation t-test carries different assumptions depending on +design. For the two-sample case, the studentized version is valid under +exchangeability of observations across groups under the null and +provides asymptotic validity even when variances differ between groups. +For paired and one-sample designs, the sign-flip procedure additionally +requires that the distribution of differences (or one-sample +observations) is symmetric around the null value; asymmetry makes the +test approximate rather than exact, though studentization provides some +asymptotic protection. Symmetry is a commonly overlooked assumption for +paired permutation tests and is easily checked with a symmetry plot of +the differences, as demonstrated for the WMW signed-rank case above. For +two-sample designs, overlaid empirical cumulative density functions and +separate density plots help assess whether group shapes are comparable +enough for exchangeability under the null to be credible. Diagnostic +workflows for all three designs are demonstrated in the online +supplement. As with the bootstrap, these checks reflect assumptions that +are relaxed relative to parametric inference but not eliminated. + \newpage \subsection{Log-Transformed Methods}\label{log-transformed-methods} @@ -1558,6 +1623,26 @@ \subsubsection{Bootstrap Log TOST}\label{bootstrap-log-tost} From this analysis, we would conclude there is a significant effect that is not practically equivalent. +\subsubsection{Checking Assumptions}\label{checking-assumptions-7} + +The log-transformed methods inherit the assumptions of the underlying +test applied to the transformed data. The \texttt{log\_TOST} function +carries forward the t-test assumptions, and \texttt{boot\_log\_TOST} +carries forward the bootstrap t-test assumptions. What changes is where +those assumptions need to hold. Normality, symmetry, and homogeneity of +variance should be assessed on the log scale rather than the original +scale, since that is where inference takes place. The same visual +diagnostics apply just computed on log-transformed data. This points to +a broader practical use of log transformation. When the symmetry +assumption of the sign-flip permutation test is violated because paired +differences are right-skewed on the original scale, the differences of +logs are often approximately symmetric, and \texttt{perm\_t\_test} on +log-transformed values becomes a reasonable option. In such cases the +equivalence bounds should be specified on the log scale and interpreted +as log ratios (or sympercents). As with the parametric and bootstrap +t-tests, assumptions are relaxed by the transformation but not +eliminated ----- they are simply moved to a different scale. + \newpage \section{Equivalence Testing with @@ -1580,6 +1665,13 @@ \section{Equivalence Testing with calculation. The full derivation is described in the package documentation; here we demonstrate usage. +The equivalence test inherits the standard ANOVA assumptions which +should be checked on the fitted model itself rather than on the +equivalence output. The \texttt{performance::check\_model()} function +provides a comprehensive visual diagnostic suite for most linear model +objects in R on a single call; the online supplement demonstrates its +use. + \subsection{Example with Summary Statistics}\label{example-with-summary-statistics} @@ -1801,7 +1893,8 @@ \subsection*{Disclosure statement}\label{disclosure-statement} The author of this manuscript is the author of the TOSTER package. Citations of this manuscript will benefit his citation count. -\section{AI Use Disclosure}\label{ai-use-disclosure} +\subsection*{AI Use Disclosure}\label{ai-use-disclosure} +\addcontentsline{toc}{subsection}{AI Use Disclosure} During the preparation of the TOSTER package updates and this manuscript, the author used Claude Opus 4.5-4.7 (Anthropic) to assist diff --git a/papers/Avocado_Update/Avocado_Update_files/Avocado_Update_files/figure-latex/cdplot-1.pdf b/papers/Avocado_Update/Avocado_Update_files/Avocado_Update_files/figure-latex/cdplot-1.pdf index a30d176c87305b760fa3ad11cbd51ff9479111ce..d01c8541f4bf64c8c6d145acbcad0412f417925a 100644 GIT binary patch delta 25 dcmbO=nQ_)+#tGIeCI%*E6YV9y^u}m=PXJ|z2S@+_ delta 25 dcmbO=nQ_)+#tGIe29`!96YV9y^u}m=PXJ}22TA|{ diff --git a/papers/Avocado_Update/Avocado_Update_files/Avocado_Update_files/figure-latex/genericplot-1.pdf b/papers/Avocado_Update/Avocado_Update_files/Avocado_Update_files/figure-latex/genericplot-1.pdf index 073e874ef52fc572eb393d5184a493c8354d620f..a623fa047c392fa0a802b29e9942c91901d436d5 100644 GIT binary patch delta 23 bcmZ3izF2*NHH(RXiQzF2{64e`lcWNRW=6# delta 23 bcmdm@vPETrHH(3zk;z1R2{64e`lcWNRV)Vp diff --git a/papers/Avocado_Update/Avocado_Update_files/Avocado_Update_files/figure-latex/symmplot-1.pdf b/papers/Avocado_Update/Avocado_Update_files/Avocado_Update_files/figure-latex/symmplot-1.pdf index 9a6a9938bc460630ed84d0e74d69a6cbd1834b4d..823eee9db6e495eb818a2d2c23d6eb1fd4d75d58 100644 GIT binary patch delta 23 bcmX@2d_;MIHH(RXsmVlp2{64eI#Un;R=5VL delta 23 bcmX@2d_;MIHH(3zk=aCh2{64eI#Un;R|W>M diff --git a/papers/Avocado_Update/Avocado_Update_files/figure-latex/cdplot-1.pdf b/papers/Avocado_Update/Avocado_Update_files/figure-latex/cdplot-1.pdf index a30d176c87305b760fa3ad11cbd51ff9479111ce..d01c8541f4bf64c8c6d145acbcad0412f417925a 100644 GIT binary patch delta 25 dcmbO=nQ_)+#tGIeCI%*E6YV9y^u}m=PXJ|z2S@+_ delta 25 dcmbO=nQ_)+#tGIe29`!96YV9y^u}m=PXJ}22TA|{ diff --git a/papers/Avocado_Update/Avocado_Update_files/figure-latex/genericplot-1.pdf b/papers/Avocado_Update/Avocado_Update_files/figure-latex/genericplot-1.pdf index 073e874ef52fc572eb393d5184a493c8354d620f..a623fa047c392fa0a802b29e9942c91901d436d5 100644 GIT binary patch delta 23 bcmZ3izF2*NHH(RXiQzF2{64e`lcWNRW=6# delta 23 bcmdm@vPETrHH(3zk;z1R2{64e`lcWNRV)Vp diff --git a/papers/Avocado_Update/Avocado_Update_files/figure-latex/symmplot-1.pdf b/papers/Avocado_Update/Avocado_Update_files/figure-latex/symmplot-1.pdf index 9a6a9938bc460630ed84d0e74d69a6cbd1834b4d..823eee9db6e495eb818a2d2c23d6eb1fd4d75d58 100644 GIT binary patch delta 23 bcmX@2d_;MIHH(RXsmVlp2{64eI#Un;R=5VL delta 23 bcmX@2d_;MIHH(3zk=aCh2{64eI#Un;R|W>M From 4b00ad3d338c5a90cde9964c1769895760c7557e Mon Sep 17 00:00:00 2001 From: Aaron Caldwell Date: Fri, 17 Apr 2026 16:04:35 -0500 Subject: [PATCH 14/40] Small edits, long ways to go... --- papers/Avocado_Update/Avocado_Update.Rmd | 16 ++++++++++++++-- .../toster_assumption_checks_supplement.qmd | 2 +- papers/Avocado_Update/~$ocado_Update.docx | Bin 162 -> 0 bytes 3 files changed, 15 insertions(+), 3 deletions(-) delete mode 100644 papers/Avocado_Update/~$ocado_Update.docx diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index e7a0196c..9f60401e 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -69,7 +69,7 @@ library(patchwork) # Introduction -Researchers routinely interpret non-significant p-values as evidence that no effect exists --- a logical error that has been recognized for decades [@blandaltman95]. A non-significant result means only that the data are compatible with the null hypothesis; it does not mean the null hypothesis is true. +Researchers routinely interpret non-significant p-values from a nil-hypothesis (i.e., null hypothesis that the mean difference equals zero) test as evidence that no effect exists --- a logical error that has been recognized for decades [@blandaltman95]. A non-significant result means only that the data are compatible with the null hypothesis; it does not mean the null hypothesis is true. Equivalence testing provides a principled framework for evaluating whether an effect is small enough to be considered negligible. The two one-sided tests (TOST) procedure is the most widely used approach [@schuirmann1987]. Researchers specify an upper ($\Delta_U$) and lower ($\Delta_L$) equivalence bound based on the smallest effect size of interest (SESOI). Two one-sided tests then evaluate whether the observed effect falls within those bounds. If both tests reject at a pre-specified alpha level, the effect can be considered practically equivalent to zero [@lakens_ori]. @@ -318,12 +318,24 @@ The `simple_htest` function supports `t.test` and `wilcox.test` as the underlyin ```{r} res_sh1 <- simple_htest(extra ~ group, data = sleep, + paired = TRUE, alternative = "equivalence", mu = 0.5) res_sh1 ``` -Notice how the output is a single clean `htest` object. This is much more concise than the verbose `t_TOST` output shown previously on purpose. If you want to perform additional tests --- such as a complementary nil significance tests --- then that has to be done separately. However, there are some methods (described below) that make this `htest` format easy to deal with. +Additionally, tests other that TOST can be applied with this function. The traditional nil-hypothesis tests, non-inferiority, or even superiority-by-a-margin hypotheses can all be tested with this function. For example, let's say we want to test that a treatment group in the sleep data was just non-inferior to the other treatment rather than equivalent. This can be accomplished by just setting the direction with the `alternative` argument and the margin of non-inferiority by the `mu` argument. + +```{r} +res_sh2 <- simple_htest(extra ~ group, + data = sleep, + paired = TRUE, + alternative = "greater", + mu = -0.5) +res_sh2 +``` + +Notice how the output, in either hypothesis test above, is a single clean `htest` object. This is much more concise than the verbose `t_TOST` output shown previously on purpose. If you want to perform additional tests --- such as a complementary nil significance tests --- then that has to be done separately. However, there are some methods (described below) that make this `htest` format easy to deal with when reporting statistical test results. ## Describing and Tabulating Results diff --git a/papers/Avocado_Update/toster_assumption_checks_supplement.qmd b/papers/Avocado_Update/toster_assumption_checks_supplement.qmd index f02dfb1d..7952aadb 100644 --- a/papers/Avocado_Update/toster_assumption_checks_supplement.qmd +++ b/papers/Avocado_Update/toster_assumption_checks_supplement.qmd @@ -727,7 +727,7 @@ The Wilcoxon signed-rank test assumes that the distribution of the one-sample va - **Location shift plausible, proportional odds plausible, symmetric differences (for paired or one-sample):** WMW and the signed-rank test are interpretable as tests of the location parameter, with a clean odds-ratio interpretation. - **Location shift violated but independence and exchangeability hold:** Brunner-Munzel is the more defensible two-sample test. Report stochastic superiority rather than a difference in medians. -- **Proportional odds violated:** WMW still provides a valid test of $P(X < Y)$ but the odds-ratio interpretation is not clean. Consider whether the mean, estimated via a bootstrap or permutation t-test, is a more interpretable estimand for the question at hand. +- **Proportional odds violated:** WMW still provides a valid test of $P(X < Y)$ but the odds-ratio interpretation is not clean. Consider whether the mean, estimated via a bootstrap or permutation t-test, is a more interpretable estimand for the question at hand. Alternatively, the Brunner-Munzel test should still be a valid test and estimate as well. # ANOVA Diagnostics diff --git a/papers/Avocado_Update/~$ocado_Update.docx b/papers/Avocado_Update/~$ocado_Update.docx deleted file mode 100644 index cc171ee6167954c20db9ced727e6ca1187d46d88..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 162 zcmd;gPRvOuPtD2EQE*Hw%FkmU3h*;HGbA$PFr+Y)Go%9P90nZ*1qMf;SP?@$1IUo` w6J`KOM_(os^I$xfc&6Ed-#zE Date: Thu, 27 Aug 2026 09:26:51 -0500 Subject: [PATCH 15/40] a few fixes --- .Rbuildignore | 1 + .gitignore | 1 + papers/Avocado_Update/Avocado_Update.R | 9 + papers/Avocado_Update/Avocado_Update.Rmd | 20 +- papers/Avocado_Update/Avocado_Update.pdf | Bin 395447 -> 397195 bytes papers/Avocado_Update/Avocado_Update.tex | 190 +++++++++++------- .../figure-latex/cdplot-1.pdf | Bin 18842 -> 18842 bytes .../figure-latex/genericplot-1.pdf | Bin 5027 -> 5027 bytes .../figure-latex/hypplot-1.pdf | Bin 4622 -> 4622 bytes .../figure-latex/shplot1-1.pdf | Bin 4660 -> 4702 bytes .../figure-latex/symmplot-1.pdf | Bin 4548 -> 4548 bytes .../figure-latex/cdplot-1.pdf | Bin 18842 -> 18842 bytes .../figure-latex/genericplot-1.pdf | Bin 5027 -> 5027 bytes .../figure-latex/hypplot-1.pdf | Bin 4622 -> 4622 bytes .../figure-latex/shplot1-1.pdf | Bin 4660 -> 4702 bytes .../figure-latex/symmplot-1.pdf | Bin 4548 -> 4548 bytes 16 files changed, 134 insertions(+), 87 deletions(-) diff --git a/.Rbuildignore b/.Rbuildignore index 93e4be65..c1a111ab 100644 --- a/.Rbuildignore +++ b/.Rbuildignore @@ -29,3 +29,4 @@ NUL ^CLAUDE\.md$ ^references$ ^\.claude$ +^\.positai$ diff --git a/.gitignore b/.gitignore index 8642cbb0..0560ef1b 100644 --- a/.gitignore +++ b/.gitignore @@ -20,3 +20,4 @@ Rplots.pdf references/working/ .claude/ .claude/settings.local.json +.positai diff --git a/papers/Avocado_Update/Avocado_Update.R b/papers/Avocado_Update/Avocado_Update.R index 8b06df6b..130056b1 100644 --- a/papers/Avocado_Update/Avocado_Update.R +++ b/papers/Avocado_Update/Avocado_Update.R @@ -179,10 +179,19 @@ res_tsum ## ----------------------------------------------------------------------------- res_sh1 <- simple_htest(extra ~ group, data = sleep, + paired = TRUE, alternative = "equivalence", mu = 0.5) res_sh1 +## ----------------------------------------------------------------------------- +res_sh2 <- simple_htest(extra ~ group, + data = sleep, + paired = TRUE, + alternative = "greater", + mu = -0.5) +res_sh2 + ## ----------------------------------------------------------------------------- describe_htest(res_sh1) diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index 9f60401e..ccd94614 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -374,7 +374,7 @@ With these caveats in mind, the following subsections describe the tools TOSTER ## Standardized Mean Difference Functions -The `smd_calc` function computes standardized mean differences --- Cohen's $d$, Hedges' $g$ (bias-corrected), or Glass's $\Delta$ --- with confidence intervals and *optional* hypothesis testing (utilizing a z-test: $z = (SMD - \mu)/SE$). It supports one-sample, paired, and two-sample designs, and can return either an `htest` object or a data frame. The SMD is already reported in `t_TOST` output, but `smd_calc` allows standalone computation and supports equivalence testing directly on the effect size. This may be useful as a complementary function when using `simple_htest` as the primary statistical testing function. +The `smd_calc` function computes standardized mean differences --- Cohen's $d$, Hedges' $g$ (bias-corrected), or Glass's $\Delta$ --- with confidence intervals and *optional* hypothesis testing (utilizing a z-test: $z = (d - \delta_0)/SE_d$, where $d$ is the sample SMD estimate, $\delta_0$ is the null value being tested, and $SE_d$ is its standard error.). It supports one-sample, paired, and two-sample designs, and can return either an `htest` object or a data frame. The SMD is already reported in `t_TOST` output, but `smd_calc` allows standalone computation and supports equivalence testing directly on the effect size. This may be useful as a complementary function when using `simple_htest` as the primary statistical testing function. When estimation rather than testing is the goal, `smd_calc` can be called with `alternative = "none"` (the default), which returns only the point estimate and confidence interval without performing a hypothesis test. This is the recommended usage when the primary objective is to report the magnitude of the effect. @@ -400,7 +400,7 @@ smd_calc(formula = extra ~ group, output = "htest") ``` -However, the parametric z-test used by `smd_calc` relies on distributional assumptions about the sampling distribution of the SMD that may not hold in practice. When a hypothesis test on the SMD is the goal, bootstrapping via `boot_smd_calc` is recommended because it provides more robust confidence intervals and p-values without requiring these assumptions. +However, the parametric z-test used by `smd_calc` relies on distributional assumptions about the sampling distribution of the SMD that may not hold in practice. When a hypothesis test on the SMD is the goal, bootstrapping via `boot_smd_calc` is recommended because it provides more robust confidence intervals and p-values. ```{r} boot_smd_calc(formula = extra ~ group, @@ -415,11 +415,11 @@ boot_smd_calc(formula = extra ~ group, ### Checking Assumptions -The SMD functions inherit the assumptions of the underlying t-test: independent observations, approximately normal residuals or differences, and, for two-sample designs, comparable variances across groups. Visual diagnostics from the t-test discussion above apply directly. Two considerations are specific to standardized effect sizes, however. First, outliers and skew affect both the numerator (mean difference) and the denominator (standard deviation) of the SMD, which can amplify their influence on the estimate; the `tr` trimmed-means argument available in the `boot_smd_calc` function is particularly worth considering when the raw data show extreme values. Second, the parametric z-test in `smd_calc` relies on an approximation to the sampling distribution of the SMD that can be very inaccurate in small samples, which is the primary motivation for preferring `boot_smd_calc` when hypothesis testing is the goal. +The SMD functions inherit the assumptions of the underlying t-test: independent observations, approximately normal residuals or differences, and, for two-sample designs, comparable variances across groups. Visual diagnostics from the t-test discussion above apply directly. Two considerations are specific to standardized effect sizes, however. First, outliers and skew affect both the numerator (mean difference) and the denominator (standard deviation) of the SMD, which can amplify their influence on the estimate; the `tr` trimmed-means argument available in the `boot_smd_calc` function is particularly worth considering when the raw data show extreme values. Second, the asymptotic z-test in `smd_calc` relies on an approximation to the sampling distribution of the SMD that can be very inaccurate in small samples, which is the primary motivation for preferring `boot_smd_calc` when hypothesis testing is the goal. ## Equivalence Tests for Correlations -Researchers sometimes need to test that a correlation is practically zero. The `z_cor_test` function provides a parametric approach using the Fisher z-transformation. However, this is only an approximate method and is generally *not recommended* as the primary inferential tool for equivalence testing on correlations. +Researchers sometimes need to test that a correlation is practically zero. The `z_cor_test` function provides a asymptotic approach using the Fisher z-transformation. However, this is only an approximate method and is generally *not recommended* as the primary inferential tool for equivalence testing on correlations. ```{r} z_cor_test( @@ -481,7 +481,7 @@ compare_cor(r1 = 0.45, When raw data are available for both studies, `boot_compare_cor` provides a bootstrap alternative that avoids the distributional approximations underlying the Fisher z and Kraatz methods. -The comparison functions `compare_smd` and `compare_cor` inherit the assumptions of the underlying effect size estimates they combine. Because both studies contribute to the comparison, violations in either study propagate into the test. When raw data are available, the bootstrap alternatives (`boot_compare_smd` and `boot_compare_cor`) are preferred because they avoid the Fisher z and related parametric approximations, which can be inaccurate particularly when effect sizes are large or sample sizes are modest. When only summary statistics are available, readers should report the diagnostic context of the original and replication studies where possible. +The comparison functions `compare_smd` and `compare_cor` inherit the assumptions of the underlying effect size estimates they combine. Because both studies contribute to the comparison, violations in either study propagate into the test. When raw data are available, the bootstrap alternatives (`boot_compare_smd` and `boot_compare_cor`) are preferred because they avoid the Fisher z and related asymptotic approximations, which can be inaccurate particularly when effect sizes are large or sample sizes are modest. When only summary statistics are available, readers should report the diagnostic context of the original and replication studies where possible. \newpage @@ -572,7 +572,7 @@ The Brunner-Munzel test has unusually light assumption requirements. It requires When the goal is to test hypotheses about means while relaxing distributional assumptions, resampling methods provide powerful tools. TOSTER implements both bootstrap and permutation approaches through `boot_t_test` and `perm_t_test`, each with distinct inferential logic but a shared motivation to retain the mean as the estimand while obtaining reference distributions empirically --- through resampling --- rather than relying on the normal-theory derivations (i.e., Student's t-test). -Both approaches are natural extensions of the traditional t-test rather than alternatives to it. The test statistic itself is unchanged; what changes is how the null distribution is constructed. In the parametric t-test, the null distribution of the standardized mean difference is derived analytically under the assumption that the data are drawn from a population with a normal distribution (and, for Student's version, with equal variances). The bootstrap approximates the sampling distribution of the same statistic by resampling from the observed data with replacement, whereas the permutation approach constructs an exact or near-exact null distribution by shuffling group labels (two-sample) or flipping signs (one-sample and paired). In both cases, the researcher continues to ask the same question about means; only the calibration of the reference distribution differs. This is important for researchers who may view resampling methods as a departure from familiar testing logic: the estimand, the null hypothesis, and the test statistic remain the same. +Both approaches are natural extensions of the classic t-test rather than alternatives to it. The test statistic itself is unchanged; what changes is how the null distribution is constructed. In the conventional t-test, the null distribution of the standardized mean difference is derived analytically under the assumption that the data are drawn from a normal distribution, with equal variances assumed in Student's t-test and unequal variances accommodated in Welch's t-test. The bootstrap approximates the sampling distribution of the same statistic by resampling from the observed data with replacement, whereas the permutation approach constructs an exact or near-exact null distribution by shuffling group labels (two-sample) or flipping signs (one-sample and paired). In both cases, the researcher continues to ask the same question about means; only the calibration of the reference distribution differs. This is important for researchers who may view resampling methods as a departure from familiar testing logic: the estimand, the null hypothesis, and the test statistic remain the same. Because the reference distribution is generated from the data rather than assumed, distributional requirements are relaxed, though not eliminated. The bootstrap relies on asymptotic arguments and assumes that the empirical distribution is a reasonable stand-in for the population distribution; it performs well with moderate-to-large samples and skewed or heavy-tailed data, but can struggle with very small samples or extreme outliers. The permutation approach is exact under the null hypothesis of exchangeability, meaning group labels carry no information about outcomes. In its basic form this exactness requires identical distributions under the null, which implicitly assumes equal variances. The studentized permutation test [@janssen1997; @chung2013] addresses this limitation by permuting a scale-invariant statistic (essentially the Welch t-statistic), yielding asymptotically valid inference even under heteroscedasticity. @Arboretti_Pesarin_Salmaso_2020 extended this logic to the equivalence testing setting, showing that studentized permutation TOST maintains nominal Type I error rates when variances differ across groups. In short, these methods are assumption-lighter rather than assumption-free, and users should still consider whether independence and exchangeability are reasonable for their design. @@ -611,7 +611,7 @@ The bootstrapped SMDs are a useful feature in their own right. Analytic confiden #### Checking Assumptions -The bootstrap t-test makes fewer distributional assumptions than its parametric counterpart but is not assumption-free. It requires independent observations and assumes the empirical distribution is a reasonable stand-in for the population distribution. The latter is an assumption that holds better in moderate-to-large samples than in small ones. The studentized bootstrap is reasonably robust to skewness and heavy tails, but remains sensitive to outliers and other high-influence observations, which can be drawn repeatedly across replicates and disproportionately shape the bootstrap distribution. Useful visual checks include the distribution of the data (histograms, boxplots) to identify extreme values, the bootstrap distribution itself to flag problematic skewness or gaps, and the jackknife-after-bootstrap plot to identify individual observations that dominate the bootstrap variability. The `tr` argument for trimmed means provides a straightforward robustness option when outliers are a concern. Diagnostic workflows for all three designs are demonstrated in the online supplement. As with the parametric t-test, these checks reflect assumptions that are relaxed relative to normal-theory inference but not eliminated. +The bootstrap t-test makes fewer distributional assumptions than its conventional, analytic, counterpart but is not assumption-free. It requires independent observations and assumes the empirical distribution is a reasonable stand-in for the population distribution. The latter is an assumption that holds better in moderate-to-large samples than in small ones. The studentized bootstrap is reasonably robust to skewness and heavy tails, but remains sensitive to outliers and other high-influence observations, which can be drawn repeatedly across replicates and disproportionately shape the bootstrap distribution. Useful visual checks include the distribution of the data (histograms, boxplots) to identify extreme values, the bootstrap distribution itself to flag problematic skewness or gaps, and the jackknife-after-bootstrap plot to identify individual observations that dominate the bootstrap variability. The `tr` argument for trimmed means provides a straightforward robustness option when outliers are a concern. Diagnostic workflows for all three designs are demonstrated in the online supplement. As with the t-test, these checks reflect assumptions that are relaxed relative to normal-theory inference but not eliminated. ### Permutation t-test @@ -631,7 +631,7 @@ Running the bootstrap and permutation equivalence tests on the same paired sleep #### Checking Assumptions -The permutation t-test carries different assumptions depending on design. For the two-sample case, the studentized version is valid under exchangeability of observations across groups under the null and provides asymptotic validity even when variances differ between groups. For paired and one-sample designs, the sign-flip procedure additionally requires that the distribution of differences (or one-sample observations) is symmetric around the null value; asymmetry makes the test approximate rather than exact, though studentization provides some asymptotic protection. Symmetry is a commonly overlooked assumption for paired permutation tests and is easily checked with a symmetry plot of the differences, as demonstrated for the WMW signed-rank case above. For two-sample designs, overlaid empirical cumulative density functions and separate density plots help assess whether group shapes are comparable enough for exchangeability under the null to be credible. Diagnostic workflows for all three designs are demonstrated in the online supplement. As with the bootstrap, these checks reflect assumptions that are relaxed relative to parametric inference but not eliminated. +The permutation t-test carries different assumptions depending on design. For the two-sample case, the studentized version is valid under exchangeability of observations across groups under the null and provides asymptotic validity even when variances differ between groups. For paired and one-sample designs, the sign-flip procedure additionally requires that the distribution of differences (or one-sample observations) is symmetric around the null value; asymmetry makes the test approximate rather than exact, though studentization provides some asymptotic protection. Symmetry is a commonly overlooked assumption for paired permutation tests and is easily checked with a symmetry plot of the differences, as demonstrated for the WMW signed-rank case above. For two-sample designs, overlaid empirical cumulative density functions and separate density plots help assess whether group shapes are comparable enough for exchangeability under the null to be credible. Diagnostic workflows for all three designs are demonstrated in the online supplement. As with the bootstrap, these checks reflect assumptions that are relaxed relative to Student's t-test but not eliminated. \newpage @@ -663,7 +663,7 @@ From this analysis, we would conclude there is a significant effect that is not ### Checking Assumptions -The log-transformed methods inherit the assumptions of the underlying test applied to the transformed data. The `log_TOST` function carries forward the t-test assumptions, and `boot_log_TOST` carries forward the bootstrap t-test assumptions. What changes is where those assumptions need to hold. Normality, symmetry, and homogeneity of variance should be assessed on the log scale rather than the original scale, since that is where inference takes place. The same visual diagnostics apply just computed on log-transformed data. This points to a broader practical use of log transformation. When the symmetry assumption of the sign-flip permutation test is violated because paired differences are right-skewed on the original scale, the differences of logs are often approximately symmetric, and `perm_t_test` on log-transformed values becomes a reasonable option. In such cases the equivalence bounds should be specified on the log scale and interpreted as log ratios (or sympercents). As with the parametric and bootstrap t-tests, assumptions are relaxed by the transformation but not eliminated --— they are simply moved to a different scale. +The log-transformed methods inherit the assumptions of the underlying test applied to the transformed data. The `log_TOST` function carries forward the t-test assumptions, and `boot_log_TOST` carries forward the bootstrap t-test assumptions. What changes is where those assumptions need to hold. Normality, symmetry, and homogeneity of variance should be assessed on the log scale rather than the original scale, since that is where inference takes place. The same visual diagnostics apply just computed on log-transformed data. This points to a broader practical use of log transformation. When the symmetry assumption of the sign-flip permutation test is violated because paired differences are right-skewed on the original scale, the differences of logs are often approximately symmetric, and `perm_t_test` on log-transformed values becomes a reasonable option. In such cases the equivalence bounds should be specified on the log scale and interpreted as log ratios (or sympercents). As with Student's t-test or the bootstrap t-test, assumptions are relaxed by the transformation but not eliminated --— they are simply moved to a different scale. \newpage @@ -749,7 +749,7 @@ This paper has described the major updates to the TOSTER R package, expanding it Several limitations should be acknowledged. TOSTER only covers a few standard research designs --- it does not handle many analysis scenarios such as those involving multilevel or mixed-effects models. For these settings, researchers should consider packages like "emmeans" or "marginaleffects", which can accommodate non-zero null hypotheses in more complex modeling frameworks. TOSTER also does not provide Bayesian equivalence testing; researchers interested in Bayesian methods for the null should consult other R packages such as "bayestestR". -I hope that the updated package continues to build upon the original impact of @lakens_ori and makes equivalence testing more accessible to researchers across disciplines. At the very least, I would encourage users to compare results across methods (e.g., parametric vs. robust) to ensure that conclusions do not depend on the chosen analysis. If the interpretation substantially changes depending on the test, this suggests features of the data --- violations of assumptions, outliers, distributional shape --- that warrant further investigation. +I hope that the updated package continues to build upon the original impact of @lakens_ori and makes equivalence testing more accessible to researchers across disciplines. At the very least, I would encourage users to compare results across methods (e.g., Student's t-test vs. permutation t-test vs. bootstrap t-test) to ensure that conclusions do not entirely depend on the chosen analysis. If the interpretation substantially changes depending on the test, this suggests features of the data --- violations of assumptions, outliers, distributional shape --- that warrant further investigation. \newpage diff --git a/papers/Avocado_Update/Avocado_Update.pdf b/papers/Avocado_Update/Avocado_Update.pdf index 5f4f7b7466b23960317b4827bb2e96aa409e7f29..f89192861277ff13e46efc26f2f8305397b195ce 100644 GIT binary patch delta 90437 zcmV(>K-j;xkr<1g7?39cG?&r40V#hO+jiSH?mb_jx0TO{RoySUPiZGJ)17YHb=;jZ zr=15{Vr(_Cq>@t8^z6qkK!CEGDBH-U@2Z?~&IxBmp>XpZbV>oE87A|TZ3Z+(B$sk+^- z)=`$Y*I&l^>naUh(a5%x1l+EQ*p+&iueTVnBL-RVILHOQ{imjwHHl}A7skRJ%}JKy7Fw*u9; z-S?y58@}qAM(!m|h-U1*A9qc?3fz*Wze>>z2d4d@1?{0El&*(WnAv|LU@t0QP0W)u zusqe>DoI_@YN3hXZKr6{zhqUHqM^x6S))NP&f89b1;9ZhcRgGZ{H9mPTDz~nei(LT zln?gnJAtFnh;gv_1&2_4Up6odNzqst1cWk$A-6 ztM1s>jWPE)bg$s*D62Sf!}yX1ehAV?nN)sGDh$0MU|+a;d9&SItx)CL<%3a9ku^Bo z*8^BZoD^;)OHrX(aDpa+-=iE3WK31pbbHbwF6_s;5}?LQnks*(rBLN6EZhT_3<1lI 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A non-significant result means only that -the data are compatible with the null hypothesis; it does not mean the -null hypothesis is true. +Researchers routinely interpret non-significant p-values from a +nil-hypothesis (i.e., null hypothesis that the mean difference equals +zero) test as evidence that no effect exists --- a logical error that +has been recognized for decades \citep{blandaltman95}. A non-significant +result means only that the data are compatible with the null hypothesis; +it does not mean the null hypothesis is true. Equivalence testing provides a principled framework for evaluating whether an effect is small enough to be considered negligible. The two @@ -564,6 +565,7 @@ \subsection{The simplified function}\label{the-simplified-function} \begin{Highlighting}[] \NormalTok{res\_sh1 }\OtherTok{\textless{}{-}} \FunctionTok{simple\_htest}\NormalTok{(extra }\SpecialCharTok{\textasciitilde{}}\NormalTok{ group, } \AttributeTok{data =}\NormalTok{ sleep,} + \AttributeTok{paired =} \ConstantTok{TRUE}\NormalTok{,} \AttributeTok{alternative =} \StringTok{"equivalence"}\NormalTok{, } \AttributeTok{mu =} \FloatTok{0.5}\NormalTok{)} \NormalTok{res\_sh1} @@ -572,29 +574,62 @@ \subsection{The simplified function}\label{the-simplified-function} \begin{verbatim} ## -## Welch Two Sample t-test +## Paired t-test ## ## data: extra by group -## t = -1.2719, df = 17.776, p-value = 0.8901 +## t = -2.7766, df = 9, p-value = 0.9892 ## alternative hypothesis: equivalence ## null values: -## difference in means difference in means -## -0.5 0.5 +## mean difference mean difference +## -0.5 0.5 ## 90 percent confidence interval: -## -3.0533815 -0.1066185 +## -2.2930053 -0.8669947 ## sample estimates: -## mean of group '1' mean of group '2' -## 0.75 2.33 -## mean difference ('1' - '2') -## -1.58 +## mean of the differences (z = '1' - '2') +## -1.58 +\end{verbatim} + +Additionally, tests other that TOST can be applied with this function. +The traditional nil-hypothesis tests, non-inferiority, or even +superiority-by-a-margin hypotheses can all be tested with this function. +For example, let's say we want to test that a treatment group in the +sleep data was just non-inferior to the other treatment rather than +equivalent. This can be accomplished by just setting the direction with +the \texttt{alternative} argument and the margin of non-inferiority by +the \texttt{mu} argument. + +\begin{Shaded} +\begin{Highlighting}[] +\NormalTok{res\_sh2 }\OtherTok{\textless{}{-}} \FunctionTok{simple\_htest}\NormalTok{(extra }\SpecialCharTok{\textasciitilde{}}\NormalTok{ group, } + \AttributeTok{data =}\NormalTok{ sleep,} + \AttributeTok{paired =} \ConstantTok{TRUE}\NormalTok{,} + \AttributeTok{alternative =} \StringTok{"greater"}\NormalTok{, } + \AttributeTok{mu =} \SpecialCharTok{{-}}\FloatTok{0.5}\NormalTok{)} +\NormalTok{res\_sh2} +\end{Highlighting} +\end{Shaded} + +\begin{verbatim} +## +## Paired t-test +## +## data: extra by group +## t = -2.7766, df = 9, p-value = 0.9892 +## alternative hypothesis: true mean difference is greater than -0.5 +## 95 percent confidence interval: +## -2.293005 Inf +## sample estimates: +## mean of the differences (z = '1' - '2') +## -1.58 \end{verbatim} -Notice how the output is a single clean \texttt{htest} object. This is -much more concise than the verbose \texttt{t\_TOST} output shown -previously on purpose. If you want to perform additional tests --- such -as a complementary nil significance tests --- then that has to be done -separately. However, there are some methods (described below) that make -this \texttt{htest} format easy to deal with. +Notice how the output, in either hypothesis test above, is a single +clean \texttt{htest} object. This is much more concise than the verbose +\texttt{t\_TOST} output shown previously on purpose. If you want to +perform additional tests --- such as a complementary nil significance +tests --- then that has to be done separately. However, there are some +methods (described below) that make this \texttt{htest} format easy to +deal with when reporting statistical test results. \subsection{Describing and Tabulating Results}\label{describing-and-tabulating-results} @@ -609,7 +644,7 @@ \subsection{Describing and Tabulating \end{Shaded} \begin{verbatim} -## [1] "The Welch Two Sample t-test is not statistically significant (t(17.776) = -1.27, p = 0.89, mean of group '1' = 0.75, mean of group '2' = 2.33, mean difference ('1' - '2') = -1.58, 90% C.I.[-3.05, -0.107]) at a 0.05 alpha-level. The null hypothesis cannot be rejected. At the desired error rate, it cannot be stated that the true difference in means is between -0.5 and 0.5." +## [1] "The Paired t-test is not statistically significant (t(9) = -2.78, p = 0.989, mean of the differences (z = '1' - '2') = -1.58, 90% C.I.[-2.29, -0.867]) at a 0.05 alpha-level. The null hypothesis cannot be rejected. At the desired error rate, it cannot be stated that the true mean difference is between -0.5 and 0.5." \end{verbatim} \subsection{Plotting Results}\label{plotting-results} @@ -691,13 +726,15 @@ \subsection{Standardized Mean Difference The \texttt{smd\_calc} function computes standardized mean differences --- Cohen's \(d\), Hedges' \(g\) (bias-corrected), or Glass's \(\Delta\) --- with confidence intervals and \emph{optional} hypothesis testing -(utilizing a z-test: \(z = (SMD - \mu)/SE\)). It supports one-sample, -paired, and two-sample designs, and can return either an \texttt{htest} -object or a data frame. The SMD is already reported in \texttt{t\_TOST} -output, but \texttt{smd\_calc} allows standalone computation and -supports equivalence testing directly on the effect size. This may be -useful as a complementary function when using \texttt{simple\_htest} as -the primary statistical testing function. +(utilizing a z-test: \(z = (d - \delta_0)/SE_d\), where \(d\) is the +sample SMD estimate, \(\delta_0\) is the null value being tested, and +\(SE_d\) is its standard error.). It supports one-sample, paired, and +two-sample designs, and can return either an \texttt{htest} object or a +data frame. The SMD is already reported in \texttt{t\_TOST} output, but +\texttt{smd\_calc} allows standalone computation and supports +equivalence testing directly on the effect size. This may be useful as a +complementary function when using \texttt{simple\_htest} as the primary +statistical testing function. When estimation rather than testing is the goal, \texttt{smd\_calc} can be called with \texttt{alternative\ =\ "none"} (the default), which @@ -769,8 +806,7 @@ \subsection{Standardized Mean Difference distributional assumptions about the sampling distribution of the SMD that may not hold in practice. When a hypothesis test on the SMD is the goal, bootstrapping via \texttt{boot\_smd\_calc} is recommended because -it provides more robust confidence intervals and p-values without -requiring these assumptions. +it provides more robust confidence intervals and p-values. \begin{Shaded} \begin{Highlighting}[] @@ -790,13 +826,13 @@ \subsection{Standardized Mean Difference ## g[rm]=('1'-'2')/SD_rm) ## ## data: extra by group -## z-observed = -0.78495, p-value = 0.906 +## z-observed = -0.78495, p-value = 0.8934 ## alternative hypothesis: equivalence ## null values: ## lower bound upper bound ## -0.5 0.5 ## 90 percent confidence interval: -## -1.3479445 -0.4457752 +## -1.2978837 -0.4361528 ## sample estimates: ## SMD (g[rm]) ## -0.7513666 @@ -814,7 +850,7 @@ \subsubsection{Checking Assumptions}\label{checking-assumptions-1} influence on the estimate; the \texttt{tr} trimmed-means argument available in the \texttt{boot\_smd\_calc} function is particularly worth considering when the raw data show extreme values. Second, the -parametric z-test in \texttt{smd\_calc} relies on an approximation to +asymptotic z-test in \texttt{smd\_calc} relies on an approximation to the sampling distribution of the SMD that can be very inaccurate in small samples, which is the primary motivation for preferring \texttt{boot\_smd\_calc} when hypothesis testing is the goal. @@ -823,7 +859,7 @@ \subsection{Equivalence Tests for Correlations}\label{equivalence-tests-for-correlations} Researchers sometimes need to test that a correlation is practically -zero. The \texttt{z\_cor\_test} function provides a parametric approach +zero. The \texttt{z\_cor\_test} function provides a asymptotic approach using the Fisher z-transformation. However, this is only an approximate method and is generally \emph{not recommended} as the primary inferential tool for equivalence testing on correlations. @@ -879,13 +915,13 @@ \subsection{Equivalence Tests for ## Bootstrapped Pearson's product-moment correlation (BCa) ## ## data: bugs$LDLF and bugs$LDHF -## N = 91, p-value = 0.9808 +## N = 91, p-value = 0.9825 ## alternative hypothesis: equivalence ## null values: ## correlation correlation ## 0.4 -0.4 ## 90 percent confidence interval: -## 0.4299380 0.6853442 +## 0.4419166 0.6857747 ## sample estimates: ## r ## 0.5767783 @@ -898,7 +934,7 @@ \subsection{Equivalence Tests for \end{Shaded} \begin{verbatim} -## [1] "The Bootstrapped Pearson's product-moment correlation (BCa) is not statistically significant (p = 0.981, r = 0.577, 90% C.I.[0.43, 0.685]) at a 0.05 alpha-level. The null hypothesis cannot be rejected. At the desired error rate, it cannot be stated that the true correlation is between 0.4 and -0.4." +## [1] "The Bootstrapped Pearson's product-moment correlation (BCa) is not statistically significant (p = 0.982, r = 0.577, 90% C.I.[0.442, 0.686]) at a 0.05 alpha-level. The null hypothesis cannot be rejected. At the desired error rate, it cannot be stated that the true correlation is between 0.4 and -0.4." \end{verbatim} If only summary statistics are available, an approximate method is @@ -1011,7 +1047,7 @@ \subsection{Comparing Standardized Effect Sizes Between in either study propagate into the test. When raw data are available, the bootstrap alternatives (\texttt{boot\_compare\_smd} and \texttt{boot\_compare\_cor}) are preferred because they avoid the Fisher -z and related parametric approximations, which can be inaccurate +z and related asymptotic approximations, which can be inaccurate particularly when effect sizes are large or sample sizes are modest. When only summary statistics are available, readers should report the diagnostic context of the original and replication studies where @@ -1256,22 +1292,22 @@ \subsection{Resampling Methods}\label{resampling-methods} resampling --- rather than relying on the normal-theory derivations (i.e., Student's t-test). -Both approaches are natural extensions of the traditional t-test rather -than alternatives to it. The test statistic itself is unchanged; what -changes is how the null distribution is constructed. In the parametric -t-test, the null distribution of the standardized mean difference is -derived analytically under the assumption that the data are drawn from a -population with a normal distribution (and, for Student's version, with -equal variances). The bootstrap approximates the sampling distribution -of the same statistic by resampling from the observed data with -replacement, whereas the permutation approach constructs an exact or -near-exact null distribution by shuffling group labels (two-sample) or -flipping signs (one-sample and paired). In both cases, the researcher -continues to ask the same question about means; only the calibration of -the reference distribution differs. This is important for researchers -who may view resampling methods as a departure from familiar testing -logic: the estimand, the null hypothesis, and the test statistic remain -the same. +Both approaches are natural extensions of the classic t-test rather than +alternatives to it. The test statistic itself is unchanged; what changes +is how the null distribution is constructed. In the conventional t-test, +the null distribution of the standardized mean difference is derived +analytically under the assumption that the data are drawn from a normal +distribution, with equal variances assumed in Student's t-test and +unequal variances accommodated in Welch's t-test. The bootstrap +approximates the sampling distribution of the same statistic by +resampling from the observed data with replacement, whereas the +permutation approach constructs an exact or near-exact null distribution +by shuffling group labels (two-sample) or flipping signs (one-sample and +paired). In both cases, the researcher continues to ask the same +question about means; only the calibration of the reference distribution +differs. This is important for researchers who may view resampling +methods as a departure from familiar testing logic: the estimand, the +null hypothesis, and the test statistic remain the same. Because the reference distribution is generated from the data rather than assumed, distributional requirements are relaxed, though not @@ -1434,24 +1470,23 @@ \subsubsection{Bootstrap t-test}\label{bootstrap-t-test} \paragraph{Checking Assumptions}\label{checking-assumptions-5} The bootstrap t-test makes fewer distributional assumptions than its -parametric counterpart but is not assumption-free. It requires -independent observations and assumes the empirical distribution is a -reasonable stand-in for the population distribution. The latter is an -assumption that holds better in moderate-to-large samples than in small -ones. The studentized bootstrap is reasonably robust to skewness and -heavy tails, but remains sensitive to outliers and other high-influence -observations, which can be drawn repeatedly across replicates and -disproportionately shape the bootstrap distribution. Useful visual -checks include the distribution of the data (histograms, boxplots) to -identify extreme values, the bootstrap distribution itself to flag -problematic skewness or gaps, and the jackknife-after-bootstrap plot to -identify individual observations that dominate the bootstrap +conventional, analytic, counterpart but is not assumption-free. It +requires independent observations and assumes the empirical distribution +is a reasonable stand-in for the population distribution. The latter is +an assumption that holds better in moderate-to-large samples than in +small ones. The studentized bootstrap is reasonably robust to skewness +and heavy tails, but remains sensitive to outliers and other +high-influence observations, which can be drawn repeatedly across +replicates and disproportionately shape the bootstrap distribution. +Useful visual checks include the distribution of the data (histograms, +boxplots) to identify extreme values, the bootstrap distribution itself +to flag problematic skewness or gaps, and the jackknife-after-bootstrap +plot to identify individual observations that dominate the bootstrap variability. The \texttt{tr} argument for trimmed means provides a straightforward robustness option when outliers are a concern. Diagnostic workflows for all three designs are demonstrated in the -online supplement. As with the parametric t-test, these checks reflect -assumptions that are relaxed relative to normal-theory inference but not -eliminated. +online supplement. As with the t-test, these checks reflect assumptions +that are relaxed relative to normal-theory inference but not eliminated. \subsubsection{Permutation t-test}\label{permutation-t-test} @@ -1521,7 +1556,7 @@ \subsubsection{Permutation t-test}\label{permutation-t-test} enough for exchangeability under the null to be credible. Diagnostic workflows for all three designs are demonstrated in the online supplement. As with the bootstrap, these checks reflect assumptions that -are relaxed relative to parametric inference but not eliminated. +are relaxed relative to Student's t-test but not eliminated. \newpage @@ -1639,8 +1674,8 @@ \subsubsection{Checking Assumptions}\label{checking-assumptions-7} logs are often approximately symmetric, and \texttt{perm\_t\_test} on log-transformed values becomes a reasonable option. In such cases the equivalence bounds should be specified on the log scale and interpreted -as log ratios (or sympercents). As with the parametric and bootstrap -t-tests, assumptions are relaxed by the transformation but not +as log ratios (or sympercents). As with Student's t-test or the +bootstrap t-test, assumptions are relaxed by the transformation but not eliminated ----- they are simply moved to a different scale. \newpage @@ -1861,11 +1896,12 @@ \section{Conclusions}\label{conclusions} I hope that the updated package continues to build upon the original impact of \citet{lakens_ori} and makes equivalence testing more accessible to researchers across disciplines. At the very least, I would -encourage users to compare results across methods (e.g., parametric -vs.~robust) to ensure that conclusions do not depend on the chosen -analysis. If the interpretation substantially changes depending on the -test, this suggests features of the data --- violations of assumptions, -outliers, distributional shape --- that warrant further investigation. +encourage users to compare results across methods (e.g., Student's +t-test vs.~permutation t-test vs.~bootstrap t-test) to ensure that +conclusions do not entirely depend on the chosen analysis. 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The top panel displays the raw mean difference with its 90% confidence interval, and the bottom panel displays the standardized effect size (Hedges's g~av~) with its 90% confidence interval. In both panels, the dashed vertical lines indicate the equivalence bounds (raw: -0.5 to 0.5; standardized: approximately -0.3 to 0.3), and the point estimate is marked by a filled circle. A 90% confidence interval is used because it corresponds to the two one-sided tests (TOST) procedure at an alpha level of 0.05."---- plot(res1) @@ -159,113 +168,191 @@ plot(res1) plot(res1, type = "cd") ## ----------------------------------------------------------------------------- -res4 = t_TOST(x = bugs$LDHF, # simple sample, vector from which to estimate mean - mu = 7.5, # set nil significance test to 7.5 - eqb = c(5.5,8.5) # set TOST bounds to 5.5 and 8.5 - ) +res4 = t_TOST( + # single sample, vector from which to estimate mean + x = bugs$LDHF, + # set nil significance test to 7.5 + mu = 7.5, + # set TOST bounds to 5.5 and 8.5 + eqb = c(5.5, 8.5) +) res4 ## ----------------------------------------------------------------------------- res_tsum = tsum_TOST( - m1 = mean(bugs$LDHF, na.rm=TRUE), - sd1 = sd(bugs$LDHF, na.rm=TRUE), + # sample mean + m1 = mean(bugs$LDHF, na.rm = TRUE), + # sample standard deviation + sd1 = sd(bugs$LDHF, na.rm = TRUE), + # sample size n1 = length(na.omit(bugs$LDHF)), - hypothesis = "EQU", smd_ci = "t", + # equivalence rather than minimal effects test + hypothesis = "EQU", + # sets type of SMD confidence interval + smd_ci = "t", + # equivalence bounds eqb = c(5.5, 8.5) ) res_tsum ## ----------------------------------------------------------------------------- -res_sh1 <- simple_htest(extra ~ group, - data = sleep, - paired = TRUE, - alternative = "equivalence", - mu = 0.5) +res_sh1 = simple_htest( + extra ~ group, + data = sleep, + paired = TRUE, + # equivalence hypothesis + alternative = "equivalence", + # symmetric bounds of -0.5 and 0.5 + mu = 0.5 +) res_sh1 ## ----------------------------------------------------------------------------- -res_sh2 <- simple_htest(extra ~ group, - data = sleep, - paired = TRUE, - alternative = "greater", - mu = -0.5) +res_sh2 = simple_htest( + extra ~ group, + data = sleep, + paired = TRUE, + # non-inferiority rather than equivalence + alternative = "greater", + # margin of non-inferiority + mu = -0.5 +) res_sh2 -## ----------------------------------------------------------------------------- -describe_htest(res_sh1) +## ----results = "asis"--------------------------------------------------------- +cat(describe_htest(res_sh1)) ## ----shplot1, fig.width=6, fig.height=3.5, fig.cap = "Estimate plot for an equivalence test via simple_htest."---- plot_htest_est(res_sh1) ## ----------------------------------------------------------------------------- -smd_calc(formula = extra ~ group, - data = sleep, - paired = TRUE, - bias_correction = TRUE, - rm_correction = TRUE, - output = "htest") +smd_calc( + formula = extra ~ group, + data = sleep, + paired = TRUE, + # apply Hedges' correction + bias_correction = TRUE, + # apply the repeated measures correction + rm_correction = TRUE, + # return an htest object rather than a data frame + output = "htest" +) ## ----------------------------------------------------------------------------- -smd_calc(formula = extra ~ group, - data = sleep, - paired = TRUE, - bias_correction = TRUE, - rm_correction = TRUE, - alternative = "equivalence", - null.value = c(-0.5,0.5), - output = "htest") +smd_calc( + formula = extra ~ group, + data = sleep, + paired = TRUE, + # apply Hedges' correction + bias_correction = TRUE, + # apply the repeated measures correction + rm_correction = TRUE, + # equivalence hypothesis + alternative = "equivalence", + # equivalence bounds on the SMD + null.value = c(-0.5, 0.5), + # return an htest object rather than a data frame + output = "htest" +) ## ----------------------------------------------------------------------------- -boot_smd_calc(formula = extra ~ group, - data = sleep, - paired = TRUE, - bias_correction = TRUE, - rm_correction = TRUE, - alternative = "equivalence", - null.value = c(-0.5,0.5)) +boot_smd_calc( + formula = extra ~ group, + data = sleep, + paired = TRUE, + # apply Hedges' correction + bias_correction = TRUE, + # apply the repeated measures correction + rm_correction = TRUE, + # equivalence hypothesis + alternative = "equivalence", + # equivalence bounds on the SMD + null.value = c(-0.5, 0.5) +) ## ----------------------------------------------------------------------------- z_cor_test( x = bugs$LDLF, y = bugs$LDHF, + # type of correlation coefficient method = "pearson", - alternative = "t") + # two-sided nil hypothesis test + alternative = "t" +) ## ----------------------------------------------------------------------------- res_boot_cor = boot_cor_test( x = bugs$LDLF, y = bugs$LDHF, + # type of correlation coefficient method = "pearson", + # equivalence hypothesis alternative = "equivalence", - null = 0.4) + # equivalence bounds of -0.4 and 0.4 + null = 0.4 +) print(res_boot_cor) -describe_htest(res_boot_cor) +## ----results = "asis"--------------------------------------------------------- +cat(describe_htest(res_boot_cor)) ## ----------------------------------------------------------------------------- -compare_smd(smd1 = 0.95, - n1 = 25, - smd2 = 0.23, - n2 = 50, - paired = TRUE, - null = .25, - alternative = "equivalence") +# Significance test +compare_smd( + # SMD and sample size of the original study + smd1 = 0.95, + n1 = 25, + # SMD and sample size of the replication + smd2 = 0.23, + n2 = 50, + # both estimates come from paired designs + paired = TRUE, + # nil hypothesis test + null = 0, + # significance hypothesis + alternative = "two.sided" +) + +# Equivalence test +compare_smd( + # SMD and sample size of the original study + smd1 = 0.95, + n1 = 25, + # SMD and sample size of the replication + smd2 = 0.23, + n2 = 50, + # both estimates come from paired designs + paired = TRUE, + # equivalence bounds on the difference in SMDs + null = .25, + # equivalence hypothesis + alternative = "equivalence" +) ## ----------------------------------------------------------------------------- -compare_cor(r1 = 0.45, - df1 = 48, - r2 = 0.25, - df2 = 78, - alternative = "equivalence", - null = 0.25) +compare_cor( + # correlation and degrees of freedom of the first study + r1 = 0.45, + df1 = 48, + # correlation and degrees of freedom of the second study + r2 = 0.25, + df2 = 78, + # equivalence hypothesis + alternative = "equivalence", + # equivalence bounds on the difference in correlations + null = 0.25 +) ## ----------------------------------------------------------------------------- -test1 = wilcox_TOST(formula = extra ~ group, - data = sleep, - paired = TRUE, - eqb = .5) +test1 = wilcox_TOST( + formula = extra ~ group, + data = sleep, + paired = TRUE, + # equivalence bounds on the pseudo-median + eqb = .5 +) print(test1) ## ----------------------------------------------------------------------------- @@ -288,38 +375,56 @@ ggplot(sp_df, aes(x = lower, y = upper)) + theme_minimal() ## ----------------------------------------------------------------------------- -bm_test = brunner_munzel(formula = extra ~ group, - data = sleep, - alternative = "equivalence", - mu = c(0.3, 0.7)) +bm_test = brunner_munzel( + formula = extra ~ group, + data = sleep, + # equivalence hypothesis + alternative = "equivalence", + # equivalence bounds on the probability scale + mu = c(0.3, 0.7) +) print(bm_test) ## ----------------------------------------------------------------------------- set.seed(4522) -boot_t_test(formula = extra ~ group, - data = sleep, - paired = TRUE, - alternative = "equivalence", - mu = c(-0.5, 0.5), - R = 999) +boot_t_test( + formula = extra ~ group, + data = sleep, + paired = TRUE, + # equivalence hypothesis + alternative = "equivalence", + # equivalence bounds + mu = c(-0.5, 0.5), + # number of bootstrap resamples + R = 999 +) ## ----------------------------------------------------------------------------- set.seed(891111) -test1 = boot_t_TOST(formula = extra ~ group, - data = sleep, - paired = TRUE, - eqb = .5, - R = 999) +test1 = boot_t_TOST( + formula = extra ~ group, + data = sleep, + paired = TRUE, + # equivalence bounds + eqb = .5, + # number of bootstrap resamples + R = 999 +) print(test1) ## ----------------------------------------------------------------------------- set.seed(8812) -perm_t_test(formula = extra ~ group, - data = sleep, - paired = TRUE, - alternative = "equivalence", - mu = c(-0.5, 0.5), - R = 999) +perm_t_test( + formula = extra ~ group, + data = sleep, + paired = TRUE, + # equivalence hypothesis + alternative = "equivalence", + # equivalence bounds + mu = c(-0.5, 0.5), + # number of random permutations + R = 999 +) ## ----error=FALSE-------------------------------------------------------------- log_TOST(mpg ~ am, data = mtcars) @@ -334,25 +439,57 @@ anova(aovtest) ## ----------------------------------------------------------------------------- -equ_ftest(Fstat = 34.70228, df1 = 5, df2 = 66, eqb = 0.35) +equ_ftest( + # F statistic and degrees of freedom from the ANOVA table + Fstat = 34.70228, + df1 = 5, + df2 = 66, + # equivalence bound on partial eta-squared + eqb = 0.35 +) ## ----------------------------------------------------------------------------- # Example using a purely within-subjects design # (Maxwell & Delaney, 2004, Chapter 12, Table 12.5, p. 578): library(afex) data(md_12.1) -aovtest2 = aov_ez("id", "rt", md_12.1, within = c("angle", "noise"), - anova_table=list(correction = "none", es = "none")) -equ_anova(aovtest2, - eqb = 0.35) +aovtest2 = aov_ez( + # participant identifier + "id", + # dependent variable + "rt", + # data + md_12.1, + # within-subjects factors + within = c("angle", "noise"), + anova_table = list(correction = "none", es = "none") +) +equ_anova(aovtest2, eqb = 0.35) ## ----------------------------------------------------------------------------- -power_t_TOST(delta = 0, sd = 1, eqb = 0.5, - alpha = 0.05, power = 0.8, - type = "two.sample") +power_t_TOST( + # true mean difference + delta = 0, + # assumed standard deviation + sd = 1, + # equivalence bounds + eqb = 0.5, + alpha = 0.05, + # leaving n unspecified solves for the n giving 80% power + power = 0.8, + type = "two.sample" +) ## ----------------------------------------------------------------------------- -power_z_cor(rho = 0, power = 0.8, - null = 0.3, alpha = 0.05, - alternative = "equivalence") +power_z_cor( + # true correlation + rho = 0, + # leaving n unspecified solves for the n giving 80% power + power = 0.8, + # equivalence bounds of -0.3 and 0.3 + null = 0.3, + alpha = 0.05, + # equivalence hypothesis + alternative = "equivalence" +) diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index ccd94614..8ff48a58 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -79,7 +79,7 @@ However, the original package had a number of limitations. It offered a function This paper provides a tutorial-style introduction to the updated TOSTER package, organized around its major capabilities: (1) a unified t-test interface (`t_TOST`) that pairs a standard significance test with an equivalence test and supports minimal effect testing, (2) a simplified hypothesis testing interface (`simple_htest`) that returns standard `htest` objects, (3) tools for computing and testing standardized effect sizes --- including standardized mean differences, rank-based effect sizes, and correlations, (4) robust and resampling-based alternatives including the Brunner-Munzel test, bootstrap and permutation t-tests, and log-transformed methods, (5) equivalence testing for ANOVAs, and (6) power analysis and sample size planning. Each section motivates when and why a researcher would use the tool before demonstrating how. -TOSTER occupies a specific niche in the R ecosystem in that it is a frequentist toolkit for equivalence testing of standard designs. Researchers interested in Bayesian approaches to evidence for the null have excellent options (e.g., `bayestestR`). For equivalence-style contrasts in more complex models --- mixed models, marginal means --- the `emmeans` and `marginaleffects` packages can accommodate non-zero null hypotheses. TOSTER fills the space of accessible, self-contained equivalence testing for common, albeit simple, research designs. For a broader introduction to the methodology, see @lakens_ori, @lakens2018equivalence, @lakens2020improving, and @mazzolari2022myths. TOSTER is also available as a [jamovi](https://www.jamovi.org/) module for researchers who prefer a graphical interface although it has limited capabilities compared to the R package. +TOSTER occupies a specific niche in the R ecosystem in that it is a frequentist toolkit for equivalence testing of standard designs. Researchers interested in Bayesian approaches to evidence for the null have excellent options (e.g., `bayestestR`). For equivalence-style contrasts in more complex models --- mixed models, marginal means --- the `emmeans` and `marginaleffects` packages can accommodate non-zero null hypotheses. TOSTER fills the space of accessible, self-contained equivalence testing for common, albeit simple, research designs. For a broader introduction to the methodology, see @lakens_ori, @lakens2018equivalence, @lakens2020improving, @mazzolari2022myths, and @Tomek_Caldwell_Eisner_2026. TOSTER is also available as a [jamovi](https://www.jamovi.org/) module for researchers who prefer a graphical interface although it has limited capabilities compared to the R package. # Setting Equivalence Bounds @@ -89,11 +89,17 @@ If a researcher has an equivalence hypothesis, they should be able to articulate Using "default" bounds on the standardized mean difference (SMD; also known as Cohen's d) based on Cohen's benchmarks (e.g., SMD = 0.2, 0.5, 0.8 to imply small, medium, or large effects, respectively) without domain-specific justification is strongly discouraged. Cohen himself stated that these conventions were "recommended for use only when no better basis for estimating the index is available" (p. 25, as cited in @panzarella2021) and reportedly came to regret having proposed them at all [@funder2019]. These benchmarks were never intended as universal thresholds; applied uncritically, they can produce equivalence bounds that are too wide, too narrow, and likely entirely disconnected from the substantive research question. -If a researcher cannot specify meaningful bounds, this itself is informative --- it suggests the equivalence hypothesis is not well enough developed to test such a hypothesis. In such cases, it may be more productive to focus on effect estimation and description rather than leaping to hypothesis tests [@scheel_hypothesis]. Throughout the examples in this paper, equivalence bounds, and the data sets themselves, are chosen for illustrative purposes; readers should not treat them as direct recommendations for their own research. +If a researcher cannot specify meaningful bounds, this itself is informative as it suggests the equivalence hypothesis is not well enough developed to test such a hypothesis. In such cases, it may be more productive to focus on effect estimation and description rather than leaping to hypothesis tests [@scheel_hypothesis]. Throughout the examples in this paper, equivalence bounds, and the data sets themselves, are chosen for illustrative purposes; readers should not treat them as direct recommendations for their own research. # TOST with t-tests -The `t_TOST` function provides a unified interface for equivalence and minimal effect testing with t-tests. It runs a standard nil significance test ($H_0: $) alongside the equivalence (or minimal effect) test, because in practice researchers may want both: "Is there a significant effect?" and "Can I rule out a meaningful effect?" The function handles independent, paired, and one-sample designs through the `paired` argument and accepts data via either a formula interface or raw vectors, making it a direct analog to base R's `t.test`. Additionally, the output includes an SMD calculation alongside the unstandarized mean difference with appropriate, albeit approximate, confidence intervals. For researchers who only want to test a single hypothesis at a time, the `simple_htest` function (described in the next section) provides a lighter alternative. +The t-test is the standard tool for comparing means. Its construction is simple: we estimate a mean (one-sample), the mean of the paired differences (paired samples), or the difference in means between two groups, and then divide that estimate by its standard error. The resulting t-statistic is compared against a t-distribution to obtain p-values and confidence intervals. Written generally, the statistic takes the form $t = (\hat{\mu} - \mu_0)/SE_{\hat{\mu}}$, where $\hat{\mu}$ is the estimate and $\mu_0$ is the value being tested against. + +What matters for equivalence testing is that $\mu_0$ is a choice rather than a fixed feature of the test. A traditional nil hypothesis test fixes $\mu_0 = 0$ and asks whether the mean, or mean difference, is exactly zero. TOST changes none of the underlying machinery; it changes only what is substituted for $\mu_0$. Two one-sided tests are performed --- one against the lower bound ($\mu_0 = \Delta_L$) and one against the upper bound ($\mu_0 = \Delta_U$) --- and equivalence is concluded only if both reject. Because each test is one-sided at $\alpha$, the interval estimate corresponding to the procedure is a $1-2\alpha$ (typically 90%) confidence interval, which is what TOSTER reports and plots. + +Throughout, the estimand for a t-test --- the quantity being estimated --- is the mean, or mean difference, on the original scale of measurement.^[This describes the default analysis, not an inviolable property of the test. Several choices substitute a different estimand: trimming (the `tr` argument in the resampling functions, `tr = 0` by default) targets the trimmed mean, which coincides with the mean only under symmetry; the log-transformed methods target a ratio of geometric means; and in applied settings, decisions about how intercurrent events --- post-randomization events such as dropout or treatment discontinuation --- and missing data are handled determine which population summary is being estimated at all. The estimand is worth choosing deliberately rather than inheriting from whichever test happens to be run [@Lundberg_Johnson_Stewart_2021].] This is worth stating plainly because several of the alternatives described later in this paper --- the rank-based tests in particular --- estimate different quantities entirely, and are not interchangeable substitutes when the hypothesis concerns means. + +The `t_TOST` function in TOSTER provides a unified interface for equivalence and minimal effect testing with t-tests. It runs a standard nil significance test alongside the equivalence (or minimal effect) test, because in practice researchers may want both: "Is there a significant effect?" and "Can I rule out a meaningful effect?" The function handles independent, paired, and one-sample designs through the `paired` argument and accepts data via either a formula interface or raw vectors, making it a direct analog to base R's `t.test`. Additionally, the output includes an SMD calculation alongside the unstandarized mean difference with appropriate, albeit approximate, confidence intervals. For researchers who only want to test a single hypothesis at a time, the `simple_htest` function (described in the next section) provides a lighter alternative. Minimal effect testing (MET) is also supported. MET is useful when the hypothesis is that an effect *exceeds* a meaningful threshold --- the null hypothesis is equivalence, and rejection implies a practically meaningful effect (see Figure 1) [@mazzolari2022myths]. @@ -238,10 +244,16 @@ data('bugs') For the first example, we use the `sleep` data with equivalence bounds (set by the `eqb` argument) of $\pm$ 0.5 units. The `smd_ci` argument can also be set to use one of four possible calculations for the SMD confidence interval. In the example below, I have opted to go with the normal approximation for computational ease; such approximations tend provide adequate coverage [@Viechtbauer_2007]. The formula interface works almost identically to `t.test`: ```{r} -res1 = t_TOST(formula = extra ~ group, # difference in "group" of outcome "extra" - data = sleep, # data - eqb = .5, # equivalence bound - smd_ci = "z") # sets type of SMD confidence interval calculation +res1 = t_TOST( + # difference in "group" of outcome "extra" + formula = extra ~ group, + # data + data = sleep, + # equivalence bound + eqb = .5, + # sets type of SMD confidence interval + smd_ci = "z" +) ``` The `print` method provides a verbose summary including the two-tailed test, both one-sided equivalence tests, and the standardized mean difference (Hedges' $g$ by default): @@ -250,6 +262,12 @@ The `print` method provides a verbose summary including the two-tailed test, bot print(res1) ``` +A greater description of the results can be accomplished with `describe` method. + +```{r, results = "asis"} +cat(describe(res1)) +``` + \newpage The default visualization provided by the generic `plot` method for `t_TOST` @@ -274,10 +292,14 @@ Other plot types are available: `type = "c"` produces a consonance plot showing When the equivalence bounds are not symmetric around the null, two values can be supplied to `eqb`. This is common when the research question implies asymmetric consequences. Similarly, we may want to test the mean itself rather than a mean difference. This harkens back to the original use of the t-test by Gosset at Guinness Brewery, where the concern was whether a single sample mean fell within acceptable limits. Here we test whether the mean of LDHF falls within the bounds of 5.5 and 8.5 around a hypothesized value of 7.5: ```{r} -res4 = t_TOST(x = bugs$LDHF, # simple sample, vector from which to estimate mean - mu = 7.5, # set nil significance test to 7.5 - eqb = c(5.5,8.5) # set TOST bounds to 5.5 and 8.5 - ) +res4 = t_TOST( + # single sample, vector from which to estimate mean + x = bugs$LDHF, + # set nil significance test to 7.5 + mu = 7.5, + # set TOST bounds to 5.5 and 8.5 + eqb = c(5.5, 8.5) +) res4 ``` @@ -295,10 +317,17 @@ When only summary statistics are available (e.g., from a published article), the ```{r} res_tsum = tsum_TOST( - m1 = mean(bugs$LDHF, na.rm=TRUE), - sd1 = sd(bugs$LDHF, na.rm=TRUE), + # sample mean + m1 = mean(bugs$LDHF, na.rm = TRUE), + # sample standard deviation + sd1 = sd(bugs$LDHF, na.rm = TRUE), + # sample size n1 = length(na.omit(bugs$LDHF)), - hypothesis = "EQU", smd_ci = "t", + # equivalence rather than minimal effects test + hypothesis = "EQU", + # sets type of SMD confidence interval + smd_ci = "t", + # equivalence bounds eqb = c(5.5, 8.5) ) @@ -316,22 +345,30 @@ Alongside `simple_htest`, a set of helper functions makes it easy to work with a The `simple_htest` function supports `t.test` and `wilcox.test` as the underlying test (via the `test` argument) and all five alternative hypotheses: `"two.sided"`, `"less"`, `"greater"`, `"equivalence"`, and `"minimal.effect"`. The `mu` parameter specifies equivalence bounds --- a single value creates symmetric bounds (e.g., if 1 is provided at `mu` with `alternative = "equivalence"` then the equivalence region will be [-1,1]), or two values can be given for asymmetric bounds. ```{r} -res_sh1 <- simple_htest(extra ~ group, - data = sleep, - paired = TRUE, - alternative = "equivalence", - mu = 0.5) +res_sh1 = simple_htest( + extra ~ group, + data = sleep, + paired = TRUE, + # equivalence hypothesis + alternative = "equivalence", + # symmetric bounds of -0.5 and 0.5 + mu = 0.5 +) res_sh1 ``` Additionally, tests other that TOST can be applied with this function. The traditional nil-hypothesis tests, non-inferiority, or even superiority-by-a-margin hypotheses can all be tested with this function. For example, let's say we want to test that a treatment group in the sleep data was just non-inferior to the other treatment rather than equivalent. This can be accomplished by just setting the direction with the `alternative` argument and the margin of non-inferiority by the `mu` argument. ```{r} -res_sh2 <- simple_htest(extra ~ group, - data = sleep, - paired = TRUE, - alternative = "greater", - mu = -0.5) +res_sh2 = simple_htest( + extra ~ group, + data = sleep, + paired = TRUE, + # non-inferiority rather than equivalence + alternative = "greater", + # margin of non-inferiority + mu = -0.5 +) res_sh2 ``` @@ -341,8 +378,8 @@ Notice how the output, in either hypothesis test above, is a single clean `htest For writing up results, `describe_htest` generates a formatted text string following typical scientific reporting conventions: -```{r} -describe_htest(res_sh1) +```{r, results = "asis"} +cat(describe_htest(res_sh1)) ``` @@ -358,9 +395,9 @@ plot_htest_est(res_sh1) # Standardized Effect Sizes -This section provides a unified treatment of the standardized effect size tools in TOSTER --- standardized mean differences (SMDs), correlation coefficients, and tools for comparing both between studies. Before demonstrating these tools, it is worth discussing the limitations of standardized effect sizes, because these limitations are especially important for equivalence testing. +This section provides a unified treatment of the standardized effect size tools in TOSTER and I cover standardized mean differences (SMDs), correlation coefficients, and tools for comparing both between studies. Before demonstrating these tools, it is worth discussing the limitations of standardized effect sizes, because these limitations are especially important for equivalence testing. -## Problems with Standardized Effect Sizes +## A Brief Warning on Standardized Effect Sizes Standardized effect sizes --- Cohen's $d$, correlation coefficients, standardized regression coefficients --- are appealing because they allow comparison across studies and scales, are required by many journals, and serve as the primary currency of meta-analysis. However, they have well-documented problems that are especially relevant to equivalence testing. @@ -379,37 +416,55 @@ The `smd_calc` function computes standardized mean differences --- Cohen's $d$, When estimation rather than testing is the goal, `smd_calc` can be called with `alternative = "none"` (the default), which returns only the point estimate and confidence interval without performing a hypothesis test. This is the recommended usage when the primary objective is to report the magnitude of the effect. ```{r} -smd_calc(formula = extra ~ group, - data = sleep, - paired = TRUE, - bias_correction = TRUE, - rm_correction = TRUE, - output = "htest") +smd_calc( + formula = extra ~ group, + data = sleep, + paired = TRUE, + # apply Hedges' correction + bias_correction = TRUE, + # apply the repeated measures correction + rm_correction = TRUE, + # return an htest object rather than a data frame + output = "htest" +) ``` When hypothesis testing on the SMD is desired, `smd_calc` supports equivalence and minimal-effect alternatives directly: ```{r} -smd_calc(formula = extra ~ group, - data = sleep, - paired = TRUE, - bias_correction = TRUE, - rm_correction = TRUE, - alternative = "equivalence", - null.value = c(-0.5,0.5), - output = "htest") +smd_calc( + formula = extra ~ group, + data = sleep, + paired = TRUE, + # apply Hedges' correction + bias_correction = TRUE, + # apply the repeated measures correction + rm_correction = TRUE, + # equivalence hypothesis + alternative = "equivalence", + # equivalence bounds on the SMD + null.value = c(-0.5, 0.5), + # return an htest object rather than a data frame + output = "htest" +) ``` However, the parametric z-test used by `smd_calc` relies on distributional assumptions about the sampling distribution of the SMD that may not hold in practice. When a hypothesis test on the SMD is the goal, bootstrapping via `boot_smd_calc` is recommended because it provides more robust confidence intervals and p-values. ```{r} -boot_smd_calc(formula = extra ~ group, - data = sleep, - paired = TRUE, - bias_correction = TRUE, - rm_correction = TRUE, - alternative = "equivalence", - null.value = c(-0.5,0.5)) +boot_smd_calc( + formula = extra ~ group, + data = sleep, + paired = TRUE, + # apply Hedges' correction + bias_correction = TRUE, + # apply the repeated measures correction + rm_correction = TRUE, + # equivalence hypothesis + alternative = "equivalence", + # equivalence bounds on the SMD + null.value = c(-0.5, 0.5) +) ``` @@ -425,8 +480,11 @@ Researchers sometimes need to test that a correlation is practically zero. The ` z_cor_test( x = bugs$LDLF, y = bugs$LDHF, + # type of correlation coefficient method = "pearson", - alternative = "t") + # two-sided nil hypothesis test + alternative = "t" +) ``` Because the z-test relies on distributional approximations that can be inaccurate --- particularly with small samples --- bootstrapping via `boot_cor_test` is recommended for more robust inference. The `boot_cor_test` function supports all three major correlation methods (Pearson, Kendall, and Spearman) as well as the Winsorized [@Wilcox_1993] and percentage bend [@Wilcox_1994] correlation coefficients as robust alternatives, and returns standard `htest` objects that integrate with the helper functions described above. @@ -435,13 +493,19 @@ Because the z-test relies on distributional approximations that can be inaccurat res_boot_cor = boot_cor_test( x = bugs$LDLF, y = bugs$LDHF, + # type of correlation coefficient method = "pearson", + # equivalence hypothesis alternative = "equivalence", - null = 0.4) + # equivalence bounds of -0.4 and 0.4 + null = 0.4 +) print(res_boot_cor) +``` -describe_htest(res_boot_cor) +```{r, results = "asis"} +cat(describe_htest(res_boot_cor)) ``` If only summary statistics are available, an approximate method is provided through `corsum_test`, which uses the Fisher z-transformation to compute confidence intervals and p-values. @@ -454,29 +518,59 @@ Correlation-based inference in `z_cor_test` and `boot_cor_test` assumes independ When evaluating whether a replication study produced results consistent with the original, researchers need a way to formally compare effect sizes. The `compare_smd` function tests whether two SMD estimates --- from an original study and a replication, for example --- differ more than would be expected by sampling variability alone. -Consider an original study reporting Cohen's $d_z$ = 0.95 with 25 participants and a replication with $d$ = 0.23 and 50 participants. Are these estimates compatible? +Consider an original study reporting Cohen's $d_z$ = 0.95 with 25 participants and a replication with $d$ = 0.23 and 50 participants. Are these estimates compatible? Let's say that we deter ```{r} -compare_smd(smd1 = 0.95, - n1 = 25, - smd2 = 0.23, - n2 = 50, - paired = TRUE, - null = .25, - alternative = "equivalence") +# Significance test +compare_smd( + # SMD and sample size of the original study + smd1 = 0.95, + n1 = 25, + # SMD and sample size of the replication + smd2 = 0.23, + n2 = 50, + # both estimates come from paired designs + paired = TRUE, + # nil hypothesis test + null = 0, + # significance hypothesis + alternative = "two.sided" +) + +# Equivalence test +compare_smd( + # SMD and sample size of the original study + smd1 = 0.95, + n1 = 25, + # SMD and sample size of the replication + smd2 = 0.23, + n2 = 50, + # both estimates come from paired designs + paired = TRUE, + # equivalence bounds on the difference in SMDs + null = .25, + # equivalence hypothesis + alternative = "equivalence" +) ``` -Based on these results, we would reject the null significance hypothesis (the SMDs differ significantly) but fail to reject the equivalence null hypothesis (we cannot conclude the SMDs are practically equivalent). When raw data are available, `boot_compare_smd` provides a bootstrap alternative (see the package vignettes for details). +Based on these results, we would reject the null significance hypothesis (conclude the SMDs differ) but fail to reject the equivalence null hypothesis (we cannot conclude the SMDs are practically equivalent). When raw data are available, `boot_compare_smd` provides a bootstrap alternative (see the package vignettes for details). For comparing correlation coefficients between independent studies, the `compare_cor` function provides an analogous test. It uses either the Fisher z-transformation (default) or the Kraatz (Anderson-Hauck) method to compare two correlation coefficients. ```{r} -compare_cor(r1 = 0.45, - df1 = 48, - r2 = 0.25, - df2 = 78, - alternative = "equivalence", - null = 0.25) +compare_cor( + # correlation and degrees of freedom of the first study + r1 = 0.45, + df1 = 48, + # correlation and degrees of freedom of the second study + r2 = 0.25, + df2 = 78, + # equivalence hypothesis + alternative = "equivalence", + # equivalence bounds on the difference in correlations + null = 0.25 +) ``` When raw data are available for both studies, `boot_compare_cor` provides a bootstrap alternative that avoids the distributional approximations underlying the Fisher z and Kraatz methods. @@ -485,8 +579,6 @@ The comparison functions `compare_smd` and `compare_cor` inherit the assumptions \newpage -**A note on argument names for equivalence bounds.** The equivalence bound argument is named differently across TOSTER functions for historical and compatibility reasons: `eqb` in `t_TOST`, `wilcox_TOST`, `equ_ftest`, and `equ_anova`; `mu` in `simple_htest`, `brunner_munzel`, `boot_t_test`, and `perm_t_test` (for compatibility with base R's `t.test`); and `null` in `compare_smd` and `z_cor_test`. Despite the different names, the interpretation is the same: the bound(s) defining the region of practical equivalence. - # Robust Methods for Equivalence Testing When t-test assumptions (normality, homoscedasticity) are seriously violated, or when the mean difference is a poor summary of the effect (e.g., with heavy skew or ordinal psychometric scales) alternative approaches are worth considering. TOSTER organizes these into three categories: rank-based tests, resampling methods, and log-transformed methods. @@ -504,10 +596,13 @@ However, there are important caveats. Many researchers mistakenly believe these For equivalence testing, bounds are applied to the pseudo-median, which corresponds to the mean or median difference only under the location-shift assumption. For testing hypotheses about mean differences, `perm_t_test` or `boot_t_test` are more appropriate. For stochastic superiority, `brunner_munzel` provides a clearer framework. ```{r} -test1 = wilcox_TOST(formula = extra ~ group, - data = sleep, - paired = TRUE, - eqb = .5) +test1 = wilcox_TOST( + formula = extra ~ group, + data = sleep, + paired = TRUE, + # equivalence bounds on the pseudo-median + eqb = .5 +) print(test1) ``` @@ -554,10 +649,14 @@ Stochastic superiority tests may be particularly well-suited to Likert-type scal The `brunner_munzel` function supports equivalence and minimal effect testing via `alternative`, with bounds on the probability scale via `mu`. Results can be reported on different scales (`"probability"`, `"difference"`, `"logodds"`, or `"odds"`), and three test methods are available: `"t"` (default), `"logit"` (range-preserving CIs), and `"perm"` (studentized permutation; @neubert2007). ```{r} -bm_test = brunner_munzel(formula = extra ~ group, - data = sleep, - alternative = "equivalence", - mu = c(0.3, 0.7)) +bm_test = brunner_munzel( + formula = extra ~ group, + data = sleep, + # equivalence hypothesis + alternative = "equivalence", + # equivalence bounds on the probability scale + mu = c(0.3, 0.7) +) print(bm_test) ``` @@ -572,26 +671,31 @@ The Brunner-Munzel test has unusually light assumption requirements. It requires When the goal is to test hypotheses about means while relaxing distributional assumptions, resampling methods provide powerful tools. TOSTER implements both bootstrap and permutation approaches through `boot_t_test` and `perm_t_test`, each with distinct inferential logic but a shared motivation to retain the mean as the estimand while obtaining reference distributions empirically --- through resampling --- rather than relying on the normal-theory derivations (i.e., Student's t-test). -Both approaches are natural extensions of the classic t-test rather than alternatives to it. The test statistic itself is unchanged; what changes is how the null distribution is constructed. In the conventional t-test, the null distribution of the standardized mean difference is derived analytically under the assumption that the data are drawn from a normal distribution, with equal variances assumed in Student's t-test and unequal variances accommodated in Welch's t-test. The bootstrap approximates the sampling distribution of the same statistic by resampling from the observed data with replacement, whereas the permutation approach constructs an exact or near-exact null distribution by shuffling group labels (two-sample) or flipping signs (one-sample and paired). In both cases, the researcher continues to ask the same question about means; only the calibration of the reference distribution differs. This is important for researchers who may view resampling methods as a departure from familiar testing logic: the estimand, the null hypothesis, and the test statistic remain the same. +Both approaches are natural extensions of the classic t-test. The test statistic itself is unchanged; what changes is how the null distribution is constructed. In the conventional t-test, the null distribution of the standardized mean difference is derived analytically under the assumption that the data are drawn from a normal distribution, with equal variances assumed in Student's t-test and unequal variances accommodated in Welch's t-test. The bootstrap approximates the sampling distribution of the same statistic by resampling from the observed data with replacement, whereas the permutation approach constructs an exact or near-exact null distribution by shuffling group labels (two-sample) or flipping signs (one-sample and paired). In both cases, the researcher continues to ask the same question about means; only the calibration of the reference distribution differs. This is important for researchers who may view resampling methods as a departure from familiar testing logic because the estimand, the null hypothesis, and the test statistic remain the same. Because the reference distribution is generated from the data rather than assumed, distributional requirements are relaxed, though not eliminated. The bootstrap relies on asymptotic arguments and assumes that the empirical distribution is a reasonable stand-in for the population distribution; it performs well with moderate-to-large samples and skewed or heavy-tailed data, but can struggle with very small samples or extreme outliers. The permutation approach is exact under the null hypothesis of exchangeability, meaning group labels carry no information about outcomes. In its basic form this exactness requires identical distributions under the null, which implicitly assumes equal variances. The studentized permutation test [@janssen1997; @chung2013] addresses this limitation by permuting a scale-invariant statistic (essentially the Welch t-statistic), yielding asymptotically valid inference even under heteroscedasticity. @Arboretti_Pesarin_Salmaso_2020 extended this logic to the equivalence testing setting, showing that studentized permutation TOST maintains nominal Type I error rates when variances differ across groups. In short, these methods are assumption-lighter rather than assumption-free, and users should still consider whether independence and exchangeability are reasonable for their design. A common reflex when normality seems implausible is to switch to the WMW or other rank-based tests. However, these tests answer a different question. As discussed earlier in this manuscript, rank-based procedures test hypotheses about stochastic ordering or pseudo-medians rather than means, and under heteroscedasticity they can reject the null even when the groups have identical means. When the scientific question is genuinely about means --— as it often is in applied research where effect sizes and interventions are naturally expressed on the original scale —-- bootstrap and permutation t-tests should generally be preferred over rank-based methods for handling such violations of assumptions. The rank-based tests remain useful, but as answers to different questions, not as robust substitutes for tests of means. -Choosing between the bootstrap and permutation approaches involves tradeoffs rather than a clear winner in most situations. The studentized permutation test has the strongest theoretical footing for two-sample mean comparisons under heteroscedasticity, and is particularly attractive for equivalence testing given the @Arboretti_Pesarin_Salmaso_2020 results. The bootstrap offers greater flexibility because it extends naturally to statistics beyond the mean difference, provides readily interpretable confidence intervals via CI inversion [@Thulin_2024], and ensures consistency between reported intervals and p-values. In small samples, permutation approaches tend to have better finite-sample calibration because the null distribution is constructed exactly (or nearly so) from the data at hand, whereas the bootstrap's asymptotic justification can produce liberal tests with very small sample sizes. When outliers are a concern, both methods accept the `tr` argument for trimmed means, which reduces the influence of extreme observations while preserving the mean-based interpretation. As a rough heuristic, researchers comparing two independent groups with suspected unequal variances may find the studentized permutation test most defensible, while those wanting a single framework that yields coherent intervals and p-values across a range of designs may prefer the bootstrap. Both are substantial improvements over ignoring assumption violations, and both return standard `htest` objects that integrate with the TOST framework for equivalence and minimal effect testing. +Choosing between the bootstrap and permutation approaches involves tradeoffs rather than a clear winner in most situations. The studentized permutation test has the strongest theoretical footing for two-sample mean comparisons under heteroscedasticity, and is particularly attractive for equivalence testing given the @Arboretti_Pesarin_Salmaso_2020 results. The bootstrap offers greater flexibility because it extends naturally to statistics beyond the mean difference, provides readily interpretable confidence intervals via CI inversion [@Thulin_2024], and ensures consistency between reported intervals and p-values. In small samples, permutation approaches tend to have better finite-sample calibration because the null distribution is constructed exactly (or nearly so) from the data at hand, whereas the bootstrap's asymptotic justification can produce liberal tests with very small sample sizes. When outliers are a concern, both methods accept the `tr` argument for trimmed means, which reduces the influence of extreme observations while preserving a location-based interpretation on the original scale. As a rough heuristic, researchers comparing two independent groups with suspected unequal variances may find the studentized permutation test most defensible, while those wanting a single framework that yields coherent intervals and p-values across a range of designs may prefer the bootstrap. Both are substantial improvements over ignoring assumption violations of the t-test, and both return standard `htest` objects that integrate with the TOST framework for equivalence and minimal effect testing. ### Bootstrap t-test -The bootstrap t-test in TOSTER defaults to the studentized bootstrap (`boot_ci = "stud"`) because this matches the original bootstrap t-test proposed by @efron93 (Chapter 16), which is the procedure `boot_t_TOST` implements. In this approach, the t-statistic itself is resampled rather than the raw mean difference, and confidence intervals are constructed from the bootstrap distribution of these pivotal quantities. Other CI methods (percentile, basic, and BCa) remain available via the `boot_ci` argument, but the studentized default preserves consistency with the canonical Efron and Tibshirani formulation. It also ensures that the reported p-value and confidence interval are derived from the same bootstrap distribution, so the two will always agree; previous versions of the package used the studentized p-value regardless of the selected CI method, which could produce disagreement between reported intervals and p-values. +The bootstrap t-test in TOSTER defaults to the studentized bootstrap (`boot_ci = "stud"`) because this matches the original bootstrap t-test proposed by @efron93 (Chapter 16), which is the procedure `boot_t_TOST` implements. In this approach, the t-statistic itself is resampled rather than the raw mean difference, and confidence intervals are constructed from the bootstrap distribution of these pivotal quantities. Other CI methods (percentile, basic, and BCa) remain available via the `boot_ci` argument, but the studentized default preserves consistency with the canonical @efron93 formulation. It also ensures that the reported p-value and confidence interval are derived from the same bootstrap distribution, so the two will always agree; previous versions of the package used the studentized p-value regardless of the selected CI method, which could produce disagreement between reported intervals and p-values. ```{r} set.seed(4522) -boot_t_test(formula = extra ~ group, - data = sleep, - paired = TRUE, - alternative = "equivalence", - mu = c(-0.5, 0.5), - R = 999) +boot_t_test( + formula = extra ~ group, + data = sleep, + paired = TRUE, + # equivalence hypothesis + alternative = "equivalence", + # equivalence bounds + mu = c(-0.5, 0.5), + # number of bootstrap resamples + R = 999 +) ``` The output follows the familiar `htest` format. The reported p-value is the larger of the two one-sided tests against the equivalence bounds, and the confidence interval is the studentized bootstrap CI for the mean difference. Because the p-value is tied to this interval via CI inversion, a researcher reporting both will find the two tell a consistent story. The interval will fall within the equivalence bounds if and only if the equivalence test rejects. @@ -599,11 +703,15 @@ For situations where both a nil hypothesis test and an equivalence test are desi ```{r} set.seed(891111) -test1 = boot_t_TOST(formula = extra ~ group, - data = sleep, - paired = TRUE, - eqb = .5, - R = 999) +test1 = boot_t_TOST( + formula = extra ~ group, + data = sleep, + paired = TRUE, + # equivalence bounds + eqb = .5, + # number of bootstrap resamples + R = 999 +) print(test1) ``` @@ -615,19 +723,24 @@ The bootstrap t-test makes fewer distributional assumptions than its conventiona ### Permutation t-test -When `R = NULL`, the function enumerates all possible permutations for an exact test; otherwise, it draws R random permutations with the "+1" correction [@phipson2010]. By default the test is studentized (`perm_se = TRUE`), consistent with the recommendations discussed in the methods section above. The `perm_se = FALSE` option is available primarily to match the behavior of other permutation test packages that do not studentize by default. +When `R = NULL`, the function enumerates all possible permutations for an exact test; otherwise, it draws R random permutations with the "+1" correction [@phipson2010]. By default the test is studentized (`perm_se = TRUE`), consistent with the recommendations discussed in the methods section above. The `perm_se = FALSE` option is available primarily to match the behavior of other R packages that provide permutation tests that do not studentize by default. ```{r} set.seed(8812) -perm_t_test(formula = extra ~ group, - data = sleep, - paired = TRUE, - alternative = "equivalence", - mu = c(-0.5, 0.5), - R = 999) +perm_t_test( + formula = extra ~ group, + data = sleep, + paired = TRUE, + # equivalence hypothesis + alternative = "equivalence", + # equivalence bounds + mu = c(-0.5, 0.5), + # number of random permutations + R = 999 +) ``` -Running the bootstrap and permutation equivalence tests on the same paired sleep data using matching equivalence bounds produces broadly similar conclusions, as would be expected. The two approaches can, however, diverge in small samples, under strong heteroscedasticity, or in the presence of influential outliers, which is where the choice between them becomes substantively important. Users interested in robustness to outliers can additionally supply the `tr` argument to either function to use trimmed means, preserving the mean-based interpretation while downweighting extreme observations. +Running the bootstrap and permutation equivalence tests on the same paired sleep data using matching equivalence bounds produces broadly similar conclusions, as would be expected. The two approaches can, however, diverge in small samples, under strong heteroscedasticity, or in the presence of influential outliers, which is where the choice between them becomes substantively important. Users interested in robustness to outliers can additionally supply the `tr` argument to either function to use trimmed means, which downweight extreme observations while preserving a location-based interpretation on the original scale. #### Checking Assumptions @@ -663,7 +776,7 @@ From this analysis, we would conclude there is a significant effect that is not ### Checking Assumptions -The log-transformed methods inherit the assumptions of the underlying test applied to the transformed data. The `log_TOST` function carries forward the t-test assumptions, and `boot_log_TOST` carries forward the bootstrap t-test assumptions. What changes is where those assumptions need to hold. Normality, symmetry, and homogeneity of variance should be assessed on the log scale rather than the original scale, since that is where inference takes place. The same visual diagnostics apply just computed on log-transformed data. This points to a broader practical use of log transformation. When the symmetry assumption of the sign-flip permutation test is violated because paired differences are right-skewed on the original scale, the differences of logs are often approximately symmetric, and `perm_t_test` on log-transformed values becomes a reasonable option. In such cases the equivalence bounds should be specified on the log scale and interpreted as log ratios (or sympercents). As with Student's t-test or the bootstrap t-test, assumptions are relaxed by the transformation but not eliminated --— they are simply moved to a different scale. +The log-transformed methods inherit the assumptions of the underlying test applied to the transformed data. The `log_TOST` function carries forward the t-test assumptions, and `boot_log_TOST` carries forward the bootstrap t-test assumptions. What changes is where those assumptions need to hold. Normality, symmetry, and homogeneity of variance should be assessed on the log scale rather than the original scale, since that is where inference takes place. The same visual diagnostics apply just computed on log-transformed data. This points to a broader practical use of log transformation. When the symmetry assumption of the sign-flip permutation test is violated because paired differences are right-skewed on the original scale, the differences of logs are often approximately symmetric, and `perm_t_test` on log-transformed values becomes a reasonable option. In such cases the equivalence bounds should be specified on the log scale and interpreted as log ratios (or sympercents). \newpage @@ -687,7 +800,14 @@ anova(aovtest) ``` ```{r} -equ_ftest(Fstat = 34.70228, df1 = 5, df2 = 66, eqb = 0.35) +equ_ftest( + # F statistic and degrees of freedom from the ANOVA table + Fstat = 34.70228, + df1 = 5, + df2 = 66, + # equivalence bound on partial eta-squared + eqb = 0.35 +) ``` Based on these results, there is a significant effect of "spray" and the differences are *not* statistically equivalent --- we reject the traditional null hypothesis but fail to reject the equivalence null hypothesis. @@ -703,10 +823,18 @@ The `equ_anova` function accepts objects from `stats::aov`, `car::Anova`, and `a # (Maxwell & Delaney, 2004, Chapter 12, Table 12.5, p. 578): library(afex) data(md_12.1) -aovtest2 = aov_ez("id", "rt", md_12.1, within = c("angle", "noise"), - anova_table=list(correction = "none", es = "none")) -equ_anova(aovtest2, - eqb = 0.35) +aovtest2 = aov_ez( + # participant identifier + "id", + # dependent variable + "rt", + # data + md_12.1, + # within-subjects factors + within = c("angle", "noise"), + anova_table = list(correction = "none", es = "none") +) +equ_anova(aovtest2, eqb = 0.35) ``` \newpage @@ -722,9 +850,18 @@ The `power_t_TOST` function computes exact power for TOST with one-sample, two-s For example, to determine the sample size needed to achieve 80% power to detect equivalence within bounds of $\pm$ 0.5 (assuming the true difference is zero and $SD$ = 1): ```{r} -power_t_TOST(delta = 0, sd = 1, eqb = 0.5, - alpha = 0.05, power = 0.8, - type = "two.sample") +power_t_TOST( + # true mean difference + delta = 0, + # assumed standard deviation + sd = 1, + # equivalence bounds + eqb = 0.5, + alpha = 0.05, + # leaving n unspecified solves for the n giving 80% power + power = 0.8, + type = "two.sample" +) ``` ## Power for Correlations @@ -732,14 +869,22 @@ power_t_TOST(delta = 0, sd = 1, eqb = 0.5, The `power_z_cor` function provides power analysis for equivalence tests on Pearson correlations using Fisher's z-transformation: ```{r} -power_z_cor(rho = 0, power = 0.8, - null = 0.3, alpha = 0.05, - alternative = "equivalence") +power_z_cor( + # true correlation + rho = 0, + # leaving n unspecified solves for the n giving 80% power + power = 0.8, + # equivalence bounds of -0.3 and 0.3 + null = 0.3, + alpha = 0.05, + # equivalence hypothesis + alternative = "equivalence" +) ``` ## Extensions: PowerTOST and Superpower -For more complex designs --- crossover studies, replicate designs, group-sequential equivalence trials --- the PowerTOST package [@PowerTOST] provides extensive functionality beyond what TOSTER offers. For factorial designs where equivalence contrasts are tested via estimated marginal means, simulation-based power analysis via the Superpower package is recommended [@Lakens_Caldwell_2021]. Both packages complement TOSTER's built-in power tools for common research designs. +For more complex designs --- crossover studies, replicate designs, group-sequential equivalence trials --- the PowerTOST package [@PowerTOST] provides extensive functionality beyond what TOSTER offers. For factorial designs where equivalence contrasts are tested via estimated marginal means, simulation-based power analysis via the Superpower package is recommended [@Lakens_Caldwell_2021]. Both packages complement TOSTER's built-in power tools for common research designs. In some situations, customized simulations may be required to accurately determine power for an equivalence test. \newpage @@ -776,7 +921,7 @@ Citations of this manuscript will benefit his citation count. ## AI Use Disclosure {-} -During the preparation of the TOSTER package updates and this manuscript, the author used Claude Opus 4.5-4.7 (Anthropic) to assist with editing R code and drafting unit tests. For the manuscript text, Claude Opus was used only for targeted copy-editing — suggesting revisions to specific passages, catching typographical and grammatical errors, and identifying inconsistencies across sections. The manuscript was written by the author, and all AI-assisted output was reviewed, revised, and verified by the author, who takes full responsibility for the content of the package and the manuscript. +During the preparation of the TOSTER package updates and this manuscript, the author used Claude Opus 4.5-4.8 (Anthropic) to assist with editing R code and drafting unit tests. For the manuscript text, Claude Opus was used only for targeted copy-editing — suggesting revisions to specific passages, catching typographical and grammatical errors, and identifying inconsistencies across sections. The manuscript was written by the author, and all AI-assisted output was reviewed, revised, and verified by the author, who takes full responsibility for the content of the package and the manuscript. ## Funding {-} diff --git a/papers/Avocado_Update/Avocado_Update.pdf b/papers/Avocado_Update/Avocado_Update.pdf index f89192861277ff13e46efc26f2f8305397b195ce..2533a9d45d92d3af4d003f47ecf5ddd9b2cae754 100644 GIT binary patch delta 199451 zcmZU)QMFAN6Y5%R zkagG~srVT#+a!KRuHkXgtC<)ODu&LitXE6WMjBBApzA{e%#lV4wq zN?J)d`1is`lM@RU3tL24mzmobZa>a~%JK^V1_4ain{AsQJrn*`%0%>EE_4+p)}J$n z1L1=K+s0bmc{W70S~4zZ@#>Px*m)f8{so7q#QIN(E$nQ7#v)FX;8O_P@NB{fRGd)s z7+&mEF|L6ytAyS={9lm$%8gJ5CeC@fbg5dxG5#3Bpa*8HKh$IS5)uPo4))aZc*%wp zK9ZR8q4ZYgP=-v?v-pK^O{j$OAXqmdpH%}Av?`|azR~VTp!9?EO8!BUr%OiJisrSG zaoS*>cd!9~p2I#)5xB8HXJI5avRQF@A~&tV<;8`gkJbS$%)qC$q?J8+CWJ7djsVNq z1jQ~CMbW{*Wm2p)XXTHTc4f;TX92GLM2iuiwRaL1_P4W-HBQt|C~l%rL^2>FDVxGJ 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+@article{Tomek_Caldwell_Eisner_2026, +title={Beyond “non-significant” results: Why and how to test for practical equivalence}, volume={123}, +url={http://dx.doi.org/10.1073/pnas.2611548123}, +DOI={10.1073/pnas.2611548123}, +abstractNote={ Reporting comparisons with P ≥ 0.05 as showing “no effect” or “no difference” remains one of the most widespread and problematic misinterpretations in the scientific literature. A statistically nonsignificant result shows only that the data do not provide strong evidence for a difference. This distinction matters because such findings can arise for two very different reasons: Either there is no meaningful difference, or a meaningful difference is present but cannot be detected reliably because of limited sample size or high variability. We highlight equivalence testing as a practical framework for distinguishing between these possibilities. Using the two one-sided tests (TOST) approach, investigators can formally test whether effects large enough to be scientifically, clinically, or practically meaningful can be ruled out. We explain the logic of TOST, show how it distinguishes practical equivalence from inconclusive evidence, and illustrate its use in paired and unpaired comparisons. To support broader adoption, we also introduce a freely available online calculator that enables researchers to perform common equivalence tests without coding. }, number={33}, journal={Proceedings of the National Academy of Sciences}, publisher={National Academy of Sciences}, author={Tomek, Jakub and Caldwell, Aaron and Eisner, David A.}, year={2026}, +month=aug, language={en} } + +@article{Lundberg_Johnson_Stewart_2021, + title={What Is Your Estimand? Defining the Target Quantity Connects Statistical Evidence to Theory}, + author={Lundberg, Ian and Johnson, Rebecca and Stewart, Brandon M.}, + journal={American Sociological Review}, + volume={86}, + number={3}, + pages={532--565}, + year={2021}, + month=jun, + publisher={SAGE Publications}, + DOI={10.1177/00031224211004187}, + url={http://dx.doi.org/10.1177/00031224211004187} +} \ No newline at end of file diff --git a/papers/Avocado_Update/toster_assumption_checks_supplement.pdf b/papers/Avocado_Update/toster_assumption_checks_supplement.pdf index 4ce84a4aaf7509af6ba44bd7c8e66e2755cac629..c668f2e6e4a83da812a085b6b69f25be4a05ba78 100644 GIT binary patch delta 105430 zcmag_Q*b717d8rZoIFv-wr$(CZQITh+qP}nNyoO+v6GIS?DzZk?3$UXshYEOAKhn{ z*0ruPS!YsN6ZpZnI9L;f(11Bp_9h%~f-i4?A3?Q9Z!n=>Jdv=jTBaJVTUu^QKL#n@ zGChbNoAOzvF+#t`Uc{6oNX70D_P*Bdb{r5tr}x6!!wur2ZUb0M1cz^~Z=nJuUgfVs z6sK)khrwh4%|0I%FQ+7@|0X*@XaBIA-Us9xaIOWp1n*(x#|`eEV*sseIep?S@a`!5 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zyPaD0K!v_LwMl@(g`J>RXsIbxiGk5p#Y7pT4dz9DO*tfWcx+v-H}@arT0rkVm;+CppO9XUIGfbPd=iFfY$P~rU&oPtl%3rzAsdPu zPC?W+kHIhZGpD<@cx?K9Q2wWpbSp~wcI4R - group_by(supp) |> - summarise(n = n(), variance = var(len)) - -# Levene's test from lawstat for robust variance comparison -lawstat::levene.test(tg$len, tg$supp, location = "median") -``` - -The median-centered Levene test (Brown–Forsythe) is preferred over the classical version because it is itself robust to non-normality. ## Paired-Sample Permutation @@ -453,7 +438,7 @@ lawstat::symmetry.test(d_paired, option = "MGG") lawstat::symmetry.test(d_paired, option = "CM") ``` -The Miao–Gel–Gastwirth ("MGG") and Cabilio–Masaro ("CM") tests provide formal assessment. With small samples these tests have limited power, so a non-significant result should not be taken as strong evidence of symmetry — the visual checks remain more informative. Conversely, a significant result with small `n` is worth taking seriously. +The Miao–Gel–Gastwirth ("MGG") and Cabilio–Masaro ("CM") tests provide formal assessment. With small samples these tests have limited power, so a non-significant result should not be taken as strong evidence of symmetry — the visual checks remain more informative. Conversely, a significant result with small `n` is worth taking very seriously. ### Check 5: Skewness with bootstrap confidence interval @@ -551,7 +536,7 @@ plot(cd_pearson, type = "h", abline(h = 4 / length(cd_pearson), col = "red", lty = 2) ``` -Unlike the balanced two-sample t-test case, leverage varies meaningfully in regression, so Cook's D is genuinely informative here: observations with extreme `wt` values (high leverage) combined with large residuals will show up as candidates for driving the estimated correlation. A Pearson correlation that rests on one or two high-leverage points is fragile, and the bootstrap jackknife in the next subsection provides a complementary check. +Unlike the balanced two-sample t-test case, leverage varies meaningfully in regression, so observations with extreme `wt` values (high leverage) combined with large residuals will show up as candidates for driving the estimated correlation. A Pearson correlation that rests on one or two high-leverage points is fragile, and the bootstrap jackknife in the next subsection provides a complementary check. Fisher's z-transformation, $z = \tfrac{1}{2}\log\left((1+r)/(1-r)\right)$, is the standard route to parametric confidence intervals on Pearson `r`. It is sometimes treated as assumption-free but in fact still relies on bivariate normality; its coverage can degrade substantially when the bivariate distribution has heavy tails or strong skew. @@ -623,7 +608,7 @@ The bootstrap distribution should be reasonably symmetric. Near $r = \pm 1$, bou The Wilcoxon-Mann-Whitney (WMW) test is commonly presented as a nonparametric test of medians, but that interpretation requires the **location shift assumption**: the two group distributions differ only in location, not in shape. When shapes differ (for example when one group has greater variance, or the groups have different skewness), WMW tests a hypothesis about stochastic ordering, $P(X < Y)$, rather than a difference in medians. Users who report WMW as a median test when shapes are clearly different are over-interpreting the result. -**Brunner-Munzel** drops the location-shift requirement and directly tests the stochastic-superiority hypothesis $P(X < Y) + 0.5 \cdot P(X = Y) = 0.5$. The tradeoff is analogous to Welch's correction of Student's t-test: Brunner-Munzel gives up a cleaner location-difference interpretation in exchange for validity under unequal shapes and variances. +**Brunner-Munzel** drops the location-shift requirement and directly tests the stochastic-superiority hypothesis $P(X < Y) + 0.5 \cdot P(X = Y) = 0.5$. The tradeoff is analogous to Welch's correction of Student's t-test: Brunner-Munzel gives up a location-difference interpretation in exchange for validity under unequal shapes and variances. The **proportional odds** assumption is a stronger condition used in ordinal regression: the log odds ratio between groups is constant across all cutpoints of the outcome. WMW can be derived as the score test from a proportional odds model, so when proportional odds holds, WMW has a clean odds-ratio interpretation in Harrell's framework. When it fails, WMW still provides a valid test of $P(X < Y)$ but its interpretation as an odds ratio breaks down. From 4115bf6f353b81d7d05eef5b72bb5b2334ff27a4 Mon Sep 17 00:00:00 2001 From: Aaron Caldwell Date: Fri, 28 Aug 2026 08:13:02 -0500 Subject: [PATCH 17/40] small edits --- papers/Avocado_Update/Avocado_Update.Rmd | 4 +- papers/Avocado_Update/Avocado_Update.pdf | Bin 401738 -> 407028 bytes papers/Avocado_Update/Avocado_Update.tex | 89 +++++++++++++----- .../figure-latex/cdplot-1.pdf | Bin 18842 -> 18842 bytes .../figure-latex/genericplot-1.pdf | Bin 5027 -> 5027 bytes .../figure-latex/hypplot-1.pdf | Bin 4622 -> 4622 bytes .../figure-latex/shplot1-1.pdf | Bin 4702 -> 4702 bytes .../figure-latex/symmplot-1.pdf | Bin 4548 -> 4548 bytes .../figure-latex/cdplot-1.pdf | Bin 18842 -> 18842 bytes .../figure-latex/genericplot-1.pdf | Bin 5027 -> 5027 bytes .../figure-latex/hypplot-1.pdf | Bin 4622 -> 4622 bytes .../figure-latex/shplot1-1.pdf | Bin 4702 -> 4702 bytes .../figure-latex/symmplot-1.pdf | Bin 4548 -> 4548 bytes 13 files changed, 70 insertions(+), 23 deletions(-) diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index 8ff48a58..25a09a79 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -673,7 +673,7 @@ When the goal is to test hypotheses about means while relaxing distributional as Both approaches are natural extensions of the classic t-test. The test statistic itself is unchanged; what changes is how the null distribution is constructed. In the conventional t-test, the null distribution of the standardized mean difference is derived analytically under the assumption that the data are drawn from a normal distribution, with equal variances assumed in Student's t-test and unequal variances accommodated in Welch's t-test. The bootstrap approximates the sampling distribution of the same statistic by resampling from the observed data with replacement, whereas the permutation approach constructs an exact or near-exact null distribution by shuffling group labels (two-sample) or flipping signs (one-sample and paired). In both cases, the researcher continues to ask the same question about means; only the calibration of the reference distribution differs. This is important for researchers who may view resampling methods as a departure from familiar testing logic because the estimand, the null hypothesis, and the test statistic remain the same. -Because the reference distribution is generated from the data rather than assumed, distributional requirements are relaxed, though not eliminated. The bootstrap relies on asymptotic arguments and assumes that the empirical distribution is a reasonable stand-in for the population distribution; it performs well with moderate-to-large samples and skewed or heavy-tailed data, but can struggle with very small samples or extreme outliers. The permutation approach is exact under the null hypothesis of exchangeability, meaning group labels carry no information about outcomes. In its basic form this exactness requires identical distributions under the null, which implicitly assumes equal variances. The studentized permutation test [@janssen1997; @chung2013] addresses this limitation by permuting a scale-invariant statistic (essentially the Welch t-statistic), yielding asymptotically valid inference even under heteroscedasticity. @Arboretti_Pesarin_Salmaso_2020 extended this logic to the equivalence testing setting, showing that studentized permutation TOST maintains nominal Type I error rates when variances differ across groups. In short, these methods are assumption-lighter rather than assumption-free, and users should still consider whether independence and exchangeability are reasonable for their design. +Because the reference distribution is generated from the data rather than assumed, distributional requirements are relaxed, though not eliminated. The bootstrap relies on asymptotic arguments and assumes that the empirical distribution is a reasonable stand-in for the population distribution; it performs well with moderate-to-large samples and skewed or heavy-tailed data, but can struggle with very small samples or extreme outliers. The permutation approach is exact under the null hypothesis of exchangeability, meaning group labels carry no information about outcomes. In its basic form this exactness requires identical distributions under the null, which implicitly assumes equal variances. The studentized permutation test [@janssen1997; @chung2013] addresses this limitation by permuting a scale-invariant statistic (essentially the Welch t-statistic), yielding asymptotically valid inference even under heteroscedasticity. @Arboretti_Pesarin_Salmaso_2020 extended this logic to the equivalence testing setting, showing that studentized permutation TOST maintains nominal Type I error rates when variances differ across groups. In short, these methods are assumption-lighter rather than assumption-free, and users should still consider whether these assumptions are reasonable for their design. A common reflex when normality seems implausible is to switch to the WMW or other rank-based tests. However, these tests answer a different question. As discussed earlier in this manuscript, rank-based procedures test hypotheses about stochastic ordering or pseudo-medians rather than means, and under heteroscedasticity they can reject the null even when the groups have identical means. When the scientific question is genuinely about means --— as it often is in applied research where effect sizes and interventions are naturally expressed on the original scale —-- bootstrap and permutation t-tests should generally be preferred over rank-based methods for handling such violations of assumptions. The rank-based tests remain useful, but as answers to different questions, not as robust substitutes for tests of means. @@ -816,7 +816,7 @@ Based on these results, there is a significant effect of "spray" and the differe ## Example with R ANOVA Objects -The `equ_anova` function accepts objects from `stats::aov`, `car::Anova`, and `afex::aov_car`, avoiding the need to extract summary statistics manually: +The `equ_anova` function accepts objects from `stats::aov`, `car::Anova`, and `afex::aov_car`, avoiding the need to extract summary statistics manually. In the example below, the traditional significance test p-value is located in the "p.null" column while to right of that value is the estimated standardized effect size (pes which stands for partial eta-squared), equivalence bound (eqb), and equivalence test p-value (p.equ). ```{r} # Example using a purely within-subjects design diff --git a/papers/Avocado_Update/Avocado_Update.pdf b/papers/Avocado_Update/Avocado_Update.pdf index 2533a9d45d92d3af4d003f47ecf5ddd9b2cae754..70807433855a9a871b071603506f81e76660b5e8 100644 GIT binary patch delta 107881 zcmZsBV{9%;7jA9awr$(Cc5Q3dZ*AMQZQHwUcWvA5J?DJMO>T1kO=ebRGBe3q57y)$ z*5d_My&EwYD;EoM8uBDCH9$u;?tlZO_fH)Y2;B;NGkOmpwDX#26476}#;h}_;vNzyZS=rAEO6*96SQ zeosN+w?R+tK!;k_xV+IRXUrB{$vn>Iwj02&{c7vi>=5rxeVkwUsFSlmo9VuGFVDRk z1QLGSbN78)zKXc(Ugb5 zIlp;C$mZMo{xVgWr~(QAbl8d&ZpSm0yd$04$u4m*LF>Pe*wc*?ceO-`;(WG4Fm$f zhjjM@{;hkAQTr5zq3v6U|6Hi*l2;Y@!g3O#W^%HZC$ZD`=bZ9;3h@2q5}oF@Q}~pL z5VPXtuVa^6U#Uxz5kaB{SWbNYQs)`yBnbrGeKBLiLsEO~7W1tIl8N_Z+1~CFPg@%b zW}JKJxy1n50m$0p(@6#$O^f<$#>v8vQw~?E)f>|A5|oJYxdW}3wQ&M*)1RXEvGO-Z z8=fPbOWLM~5nei4jsd$=J!`F-v&tJzjOj*y2|R2e!TB2%V}j;E0^0Vx=2&`!T{a>m`+JwNTIAyf zAfF*iRn;`)?=i3LEHfj+8K^#EkFYWRdZE3G%g7LI5Y2&h(FQcw7>1vNY(PLrd9Foz z+7^oC&Lw_@2QbBQ0TI*>59``JW6?ICKWDZwOsM*Sa9ylSfZC6*55=fZ+Fdy|VDp31 zn%|NT0l0n)>hvxM88zxSoKEY0MlMojc<<-k^cyuo2i|2D|9Wrq`E^emv~5vVyKdj>nF4 zJJy!fsz(}c84UzXva6b-Pm5*54~MMRz1IMhIoKaiD>Bf6o7SiRGLOXuSZ_XwSfG|UZ2 zXPv6UoA2$u)g?K7BqU?rfNS$;arcT557+~OM3d`)6s6P>?TzM7m!$|PumX8cez2>B z)R%`G;!>_qKIkAoa%8672Y#_L&2y@*)u&7KDzzH>0;Wni_G`)H8)u=IXH{%|scwwKwx#oZ2JDvKM_VX zm&_iz9o&)I!3=EXE$B1|;jyH$z9Qg{Lbg>}%M0^$QkfFlU0u5(K`*asA+jk};szMQ z!}0QVYO+Pfx6E|$I+*L~XLiHbQ?2LQ*_?E!Jp|HaF+;}o}tepb5snyonF~6qt@!zZ;G55B zW2csnJi~Fy5*0c{yensNbwmIL=CLtHUe#^+r6kpSOC|S)2JaOi%*ipy^I~3>ooae$ zQpLt-XK>cQYe^Pd4R%hD)Rd{Vo0)*MNiM1j-ZpcPl%wGyMs6f-aX{l_h7Q>!e`o7K z5Cv<9HMuQ0f69bXXfS(B6}~L2U4Uf@+SzPHFw$TA%L=grBno`a6qb|#=Lh8!3S@ab zmWGdlW}UTMj`J~zB5}H5Bk3?2)7#9X&QXyjgGZE_L&rdD6$l51op% zbFI;!8e~(MD_8jU`Nua4C|1t#Xw@_=%P)wBIoQu^_x%#($e35z>QMs>@ha@pC2`rR znm!NDCgP`8FC%tmBe_tGsYW>0P5VXc^(r>0W4K?AD#6VktN?Ir><<;%5>WUf42%m8iJy4(kJW=QGlk%_PoJr66aKO=Ak&SSC-xwar%-eLeP z>Ib=QYjJS~%Dco%5(~`_J*ji`IKaGjw9w5mSquErsDD0eh7;A)K0*LfJUpaoM00+; zlZU*Dkq{&v0HCFkfmt$9QtYvef=5i&hJgp6ia@=YSSp1xk%Mw@DYZ?4jkv-qLpC~F zj#abw$qo8gzZdJ??oBCNkc<8$sFo==1ztiE7 z@I5oi^dcq27kE<=a$~%t&Unms3(JcmQ)W188J*Y`KOn4AHwA;d!zxuUE?fCD?Ij zp`)J{0br%aWde+=lx$KXmQoXRm|AR z;xDNrLm^z9G*(#A~1Q+WqL9ap`0pu3N%H-WL}lS>crM{>x9$HQ$(!1cF~;o_90 zQQ_?m3-yTjtQ;ht?Cu-=A846-uJ!cuQF;kN1a}_E$^m3%l9Skq5$64iyk-$jFZq>- z#i&hI%7UD8_tUx^oAW0*UWEJpT^3>Me8B#R>X3<00l8~9O{y~ITy47X%tzE;2Z`T^ zyyNTy`8s&}-9u-o)2ARe!m1)s zPevGsl3iD;=DBJR97EE|Omll7!fm>3E2j0qRmpl5AC#)~iHx6v>D^f^N#!F+-&MoG zZ|Yk;cpc4RCT$PH4N{u1;5fGj5-{S^EPb!8~aa%)EW>&N%_<^H6rnum{FZL>^=49e9is zXaMy(gWlJdMZrANZyI9pA|nLfVoKn*e!3*%mxxOFNcPp~HCUa6Cdt){%4y%Z}&~3U-9%ehKUDWJglbA-&JK{3tdk+a5%*PxlUaB(+4P~D$Ua+ zSq-tCf%>%KC>CLYaYtm`{72>I*@+64(&^l|bXIdM@Z}0>vzrjjc3Rn@?`ziD7^u1? 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42ab3feaa4e81bcb1d22d99920700038a444195a..20cf1b05ecbbb36b2c8efcf9fe9084e9118211b1 100644 GIT binary patch delta 25 ccmX@2d_;MI6|1R Date: Mon, 31 Aug 2026 11:12:34 -0500 Subject: [PATCH 18/40] Update Avocado_Update.Rmd --- papers/Avocado_Update/Avocado_Update.Rmd | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index 25a09a79..3731bbce 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -518,7 +518,7 @@ Correlation-based inference in `z_cor_test` and `boot_cor_test` assumes independ When evaluating whether a replication study produced results consistent with the original, researchers need a way to formally compare effect sizes. The `compare_smd` function tests whether two SMD estimates --- from an original study and a replication, for example --- differ more than would be expected by sampling variability alone. -Consider an original study reporting Cohen's $d_z$ = 0.95 with 25 participants and a replication with $d$ = 0.23 and 50 participants. Are these estimates compatible? Let's say that we deter +Consider an original study reporting Cohen's $d_z$ = 0.95 with 25 participants and a replication with $d$ = 0.23 and 50 participants. Are these estimates compatible? Let's say we determine that equivalence between studies would be SMDs within 0.25 of each other. We can then perform a significance and equivalence test between the two study estimates. ```{r} # Significance test From 4d24bbc758f829ca285fac3c7e50f9aceff0fe43 Mon Sep 17 00:00:00 2001 From: Aaron Caldwell Date: Tue, 1 Sep 2026 15:17:18 -0500 Subject: [PATCH 19/40] Update Avocado_Update.Rmd --- papers/Avocado_Update/Avocado_Update.Rmd | 15 ++++++++------- 1 file changed, 8 insertions(+), 7 deletions(-) diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index 3731bbce..6a01d9b5 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -518,7 +518,7 @@ Correlation-based inference in `z_cor_test` and `boot_cor_test` assumes independ When evaluating whether a replication study produced results consistent with the original, researchers need a way to formally compare effect sizes. The `compare_smd` function tests whether two SMD estimates --- from an original study and a replication, for example --- differ more than would be expected by sampling variability alone. -Consider an original study reporting Cohen's $d_z$ = 0.95 with 25 participants and a replication with $d$ = 0.23 and 50 participants. Are these estimates compatible? Let's say we determine that equivalence between studies would be SMDs within 0.25 of each other. We can then perform a significance and equivalence test between the two study estimates. +Consider an original study reporting Cohen's $d_z$ = 0.95 with 25 participants and a replication with $d$ = 0.23 and 50 participants. A researcher might wonder if these these estimates compatible, and it would be prudent to formally test this. Let's say these researchers determine that equivalence between studies would be SMDs within 0.25 of each other. We can then perform a significance and equivalence test between the two original and replication study estimates. ```{r} # Significance test @@ -554,7 +554,7 @@ compare_smd( ) ``` -Based on these results, we would reject the null significance hypothesis (conclude the SMDs differ) but fail to reject the equivalence null hypothesis (we cannot conclude the SMDs are practically equivalent). When raw data are available, `boot_compare_smd` provides a bootstrap alternative (see the package vignettes for details). +Based on these results, we would reject the null significance hypothesis (conclude the SMDs differ) but fail to reject the equivalence null hypothesis (we cannot conclude the SMDs are practically equivalent). When raw data are available, `boot_compare_smd` provides a bootstrap alternative (see the [package vignettes](https://aaroncaldwell.us/TOSTERpkg/articles/SMD_calcs.html#comparing-smds) for details). For comparing correlation coefficients between independent studies, the `compare_cor` function provides an analogous test. It uses either the Fisher z-transformation (default) or the Kraatz (Anderson-Hauck) method to compare two correlation coefficients. @@ -581,7 +581,7 @@ The comparison functions `compare_smd` and `compare_cor` inherit the assumptions # Robust Methods for Equivalence Testing -When t-test assumptions (normality, homoscedasticity) are seriously violated, or when the mean difference is a poor summary of the effect (e.g., with heavy skew or ordinal psychometric scales) alternative approaches are worth considering. TOSTER organizes these into three categories: rank-based tests, resampling methods, and log-transformed methods. +When t-test assumptions (normality, homoscedasticity) are seriously violated, or when the mean difference is a poor summary of the effect (e.g., with heavy skew or ordinal psychometric scales) alternative approaches are worth considering. I have organized TOSTER into three robust alternative categories: rank-based tests, resampling methods, and log-transformed methods. ## Rank-Based Tests @@ -654,7 +654,8 @@ bm_test = brunner_munzel( data = sleep, # equivalence hypothesis alternative = "equivalence", - # equivalence bounds on the probability scale + # equivalence bounds + # on the stochastic superiority scale mu = c(0.3, 0.7) ) print(bm_test) @@ -671,13 +672,13 @@ The Brunner-Munzel test has unusually light assumption requirements. It requires When the goal is to test hypotheses about means while relaxing distributional assumptions, resampling methods provide powerful tools. TOSTER implements both bootstrap and permutation approaches through `boot_t_test` and `perm_t_test`, each with distinct inferential logic but a shared motivation to retain the mean as the estimand while obtaining reference distributions empirically --- through resampling --- rather than relying on the normal-theory derivations (i.e., Student's t-test). -Both approaches are natural extensions of the classic t-test. The test statistic itself is unchanged; what changes is how the null distribution is constructed. In the conventional t-test, the null distribution of the standardized mean difference is derived analytically under the assumption that the data are drawn from a normal distribution, with equal variances assumed in Student's t-test and unequal variances accommodated in Welch's t-test. The bootstrap approximates the sampling distribution of the same statistic by resampling from the observed data with replacement, whereas the permutation approach constructs an exact or near-exact null distribution by shuffling group labels (two-sample) or flipping signs (one-sample and paired). In both cases, the researcher continues to ask the same question about means; only the calibration of the reference distribution differs. This is important for researchers who may view resampling methods as a departure from familiar testing logic because the estimand, the null hypothesis, and the test statistic remain the same. +Both approaches are natural extensions of the classic t-test. The test statistic itself is unchanged; what changes is how the null distribution is constructed. In the conventional t-test, the null distribution of the standardized mean difference is derived analytically under the assumption that the data are drawn from a normal distribution, with equal variances assumed in Student's t-test and unequal variances accommodated in Welch's t-test. The bootstrap approximates the sampling distribution of the same statistic by resampling from the observed data with replacement, whereas the permutation approach constructs an exact or near-exact null distribution by shuffling group labels (two-sample) or flipping signs (one-sample and paired). In both cases, the researcher continues to ask the same question about means; only the calibration of the reference distribution differs. Researchers may find the resampling methods appealing because it follows a familiar testing logic because the estimand, the null hypothesis, and the test statistic remain the same. Because the reference distribution is generated from the data rather than assumed, distributional requirements are relaxed, though not eliminated. The bootstrap relies on asymptotic arguments and assumes that the empirical distribution is a reasonable stand-in for the population distribution; it performs well with moderate-to-large samples and skewed or heavy-tailed data, but can struggle with very small samples or extreme outliers. The permutation approach is exact under the null hypothesis of exchangeability, meaning group labels carry no information about outcomes. In its basic form this exactness requires identical distributions under the null, which implicitly assumes equal variances. The studentized permutation test [@janssen1997; @chung2013] addresses this limitation by permuting a scale-invariant statistic (essentially the Welch t-statistic), yielding asymptotically valid inference even under heteroscedasticity. @Arboretti_Pesarin_Salmaso_2020 extended this logic to the equivalence testing setting, showing that studentized permutation TOST maintains nominal Type I error rates when variances differ across groups. In short, these methods are assumption-lighter rather than assumption-free, and users should still consider whether these assumptions are reasonable for their design. A common reflex when normality seems implausible is to switch to the WMW or other rank-based tests. However, these tests answer a different question. As discussed earlier in this manuscript, rank-based procedures test hypotheses about stochastic ordering or pseudo-medians rather than means, and under heteroscedasticity they can reject the null even when the groups have identical means. When the scientific question is genuinely about means --— as it often is in applied research where effect sizes and interventions are naturally expressed on the original scale —-- bootstrap and permutation t-tests should generally be preferred over rank-based methods for handling such violations of assumptions. The rank-based tests remain useful, but as answers to different questions, not as robust substitutes for tests of means. -Choosing between the bootstrap and permutation approaches involves tradeoffs rather than a clear winner in most situations. The studentized permutation test has the strongest theoretical footing for two-sample mean comparisons under heteroscedasticity, and is particularly attractive for equivalence testing given the @Arboretti_Pesarin_Salmaso_2020 results. The bootstrap offers greater flexibility because it extends naturally to statistics beyond the mean difference, provides readily interpretable confidence intervals via CI inversion [@Thulin_2024], and ensures consistency between reported intervals and p-values. In small samples, permutation approaches tend to have better finite-sample calibration because the null distribution is constructed exactly (or nearly so) from the data at hand, whereas the bootstrap's asymptotic justification can produce liberal tests with very small sample sizes. When outliers are a concern, both methods accept the `tr` argument for trimmed means, which reduces the influence of extreme observations while preserving a location-based interpretation on the original scale. As a rough heuristic, researchers comparing two independent groups with suspected unequal variances may find the studentized permutation test most defensible, while those wanting a single framework that yields coherent intervals and p-values across a range of designs may prefer the bootstrap. Both are substantial improvements over ignoring assumption violations of the t-test, and both return standard `htest` objects that integrate with the TOST framework for equivalence and minimal effect testing. +Choosing between the bootstrap and permutation approaches involves tradeoffs rather than a clear winner in most situations. The studentized permutation test has the strongest theoretical footing for two-sample mean comparisons under heteroscedasticity, and is particularly attractive for equivalence testing [@Arboretti_Pesarin_Salmaso_2020]. The bootstrap offers greater flexibility because it extends naturally to statistics beyond the mean difference, provides readily interpretable confidence intervals via CI inversion [@Thulin_2024], and ensures consistency between reported intervals and p-values. In small samples, permutation approaches tend to have better finite-sample calibration because the null distribution is constructed exactly (or nearly so) from the data at hand, whereas the bootstrap's asymptotic justification can produce liberal tests with very small sample sizes. When outliers are a concern, both methods accept the `tr` argument for trimmed means, which reduces the influence of extreme observations while preserving a location-based interpretation on the original scale. As a rough heuristic, researchers comparing two independent groups with suspected unequal variances may find the studentized permutation test most defensible, while those wanting a single framework that yields coherent intervals and p-values across a range of designs may prefer the bootstrap. Both are substantial improvements over ignoring assumption violations of the t-test, and both return standard `htest` objects that integrate with the TOST framework for equivalence testing. ### Bootstrap t-test @@ -784,7 +785,7 @@ The log-transformed methods inherit the assumptions of the underlying test appli Many researchers use ANOVA as an omnibus test before inspecting pairwise comparisons, particularly in factorial designs. As @Campbell_2021 note, the lack of a significant ANOVA result does not indicate that a factor has no effect. Equivalence testing for ANOVAs addresses this directly by asking: can we reject the hypothesis that the partial $\eta^2$ exceeds a specified bound? -The equivalence test uses the non-central *F* distribution. The non-centrality parameter is computed from the equivalence bound and the degrees of freedom, and the *p*-value is obtained from the cumulative distribution function of the non-central *F*-value. I have extended the approach of @Campbell_2021 --- who proposed equivalence testing for one-way ANOVAs --- to accommodate multi-factor and factorial designs by generalizing the non-centrality parameter calculation. The full derivation is described in the package documentation; here we demonstrate usage. +The equivalence test uses the non-central *F* distribution. The non-centrality parameter is computed from the equivalence bound and the degrees of freedom, and the *p*-value is obtained from the cumulative distribution function of the non-central *F*-value. I have extended the approach of @Campbell_2021 --- who proposed equivalence testing for one-way ANOVAs --- to accommodate multi-factor and factorial designs by generalizing the non-centrality parameter calculation. The full derivation is described in the [package documentation](https://aaroncaldwell.us/TOSTERpkg/articles/the_ftestTOSTER.html); here we demonstrate usage. The equivalence test inherits the standard ANOVA assumptions which should be checked on the fitted model itself rather than on the equivalence output. The `performance::check_model()` function provides a comprehensive visual diagnostic suite for most linear model objects in R on a single call; the online supplement demonstrates its use. From 4562b141f3cef298e878dd30fe9c717ce97ad29b Mon Sep 17 00:00:00 2001 From: Aaron Caldwell Date: Fri, 4 Sep 2026 08:49:08 -0500 Subject: [PATCH 20/40] Small edits on WMW --- papers/Avocado_Update/Avocado_Update.Rmd | 9 +- .../revision-plan_jakub-comments.md | 157 ++++++++++++++++++ 2 files changed, 163 insertions(+), 3 deletions(-) create mode 100644 papers/Avocado_Update/revision-plan_jakub-comments.md diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index 6a01d9b5..33d88bc9 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -589,11 +589,14 @@ Rank-based tests operate on the ranks of the data rather than the raw values, wh ### Wilcoxon-Mann-Whitney Tests -The WMW group of tests provide non-parametric tests based on ranks. Within the TOST framework, two directional location-shift tests determine if the shift is within (equivalence) or outside (minimal effect) the equivalence bounds. +The WMW group of tests provide non-parametric tests based on ranks. Within the TOST framework, the WMW tests are comprised of two directional location-shift tests determine if the shift is within (equivalence) or outside (minimal effect) the equivalence bounds. There are many common misconceptions about the WMW tests that I will try to briefly address. -However, there are important caveats. Many researchers mistakenly believe these are tests of medians, but without a location-shift assumption, the WMW test is a test of stochastic equality. As @Fay_Malinovsky_2018 demonstrate, the effect size can even be presented as stochastic superiority/dominance. Only under a location-shift assumption does it become a test of central tendency through the Hodges-Lehmann estimator. @median_test demonstrate striking counterexamples: groups can have equal medians yet produce a significant WMW test, very different medians yet a non-significant WMW test, or even significance in the opposite direction from the median comparison. -For equivalence testing, bounds are applied to the pseudo-median, which corresponds to the mean or median difference only under the location-shift assumption. For testing hypotheses about mean differences, `perm_t_test` or `boot_t_test` are more appropriate. For stochastic superiority, `brunner_munzel` provides a clearer framework. +First, many researchers mistakenly believe these are tests of medians, but without a location-shift assumption, the WMW test is a test of stochastic equality. As @Fay_Malinovsky_2018 demonstrate, the effect size can even be presented as stochastic superiority/dominance. + +Second, only under a location-shift assumption does it become a test of central tendency through the Hodges-Lehmann estimator. @median_test demonstrate striking counterexamples: groups can have equal medians yet produce a significant WMW test, very different medians yet a non-significant WMW test, or even significance in the opposite direction from the median comparison. + +For equivalence testing in the TOSTER package, the function `wilcox_TOST` applies the bounds to the pseudo-median, which corresponds to the mean or median difference only under the location-shift assumption. For testing hypotheses about mean differences, `perm_t_test` or `boot_t_test` are more appropriate. For stochastic superiority, `brunner_munzel` provides a clearer framework. ```{r} test1 = wilcox_TOST( diff --git a/papers/Avocado_Update/revision-plan_jakub-comments.md b/papers/Avocado_Update/revision-plan_jakub-comments.md new file mode 100644 index 00000000..84ebc9b4 --- /dev/null +++ b/papers/Avocado_Update/revision-plan_jakub-comments.md @@ -0,0 +1,157 @@ +# Revision plan — Jakub's comments on *Exploring Equivalence Testing with the Updated TOSTER R Package* + +Working notes, drafted 2026-09-04. Line numbers refer to `Avocado_Update.Rmd` at commit `4d24bbc` and will shift once editing starts. + +**Summary of my read:** six comments, four of which I think are clearly worth acting on (2, 3, 4, 5), one that deserves a partial/cheap response rather than the restructure it implies (1), and one that is a one-sentence addition with a caveat (6). The heaviest lift by far is #5 (a validation simulation for the ANOVA extension); #4 is the highest value-per-hour. + +| # | Topic | Verdict | Effort | +|---|---|---|---| +| 4 | ANOVA section underdeveloped | Accept in full | ~2–3 h writing | +| 3 | Heteroscedasticity "under the null" | Accept, one clause | ~15 min | +| 2 | WMW location-shift vs. no-assumption | Accept, restructure section | ~1–2 h | +| 6 | McNabb & Murayama citation | Accept with caveat | ~30 min | +| 1 | Tempo/genre shifts | Partial — signposting, not restructure | ~1 h | +| 5 | Validate the ANOVA method | Accept — targeted simulation in supplement | ~1 day | + +--- + +## 1. Genre shifts: general stats resource / tool tutorial / new methods + +> *"the paper seems to be changing tempo/style between general statistical learning resource, specific tool's tutorial, and presentation of new methods (for ANOVA)"* + +**Verdict: real observation, but the fix is not restructuring. Partial accept.** + +Pushing back on the implied remedy: this is a software/tutorial paper, and the mixed register is largely intrinsic to the format. The conceptual material — what WMW actually tests, the warning on standardized effect sizes, the estimand footnote in "TOST with t-tests" — is arguably the most valuable content in the paper, and it is what separates this from package documentation. Jakub says as much himself about the effect-size warning. Stripping it out to make a cleaner tutorial would make the paper worse. Splitting the ANOVA method into a separate methods paper would leave the tutorial with a hole in it. + +What *is* a genuine problem is that the reader currently has no map. Fixes: + +- **Roadmap paragraph at the end of the Introduction** (after the "TOSTER occupies a specific niche" paragraph, ~line 82) that names the three registers explicitly and tells the reader they can skip between them: sections X–Y are conceptual background on choosing an estimand and bounds; sections Y–Z are worked demonstrations organised by design; the ANOVA section describes a generalisation of Campbell & Lakens (2021) that is new to this package. +- **Function-selection table**, early — probably right after "Setting Equivalence Bounds", or as the first thing in "TOST with t-tests". Columns: *research question / estimand*, *design*, *function*, *section*. This is the single highest-return navigational change and doubles as a reference the reader keeps. Rows: mean or mean difference → `t_TOST` / `tsum_TOST` / `simple_htest`; mean with relaxed assumptions → `boot_t_TOST`, `perm_t_TOST`; ratio of geometric means → `log_TOST`; pseudo-median → `wilcox_TOST`; stochastic superiority → `brunner_munzel`; correlation → `z_cor_test`, `boot_cor_test`; SMD → `smd_calc`, `boot_smd_calc`; variance explained / omnibus → `equ_anova`, `equ_ftest`; planning → `power_t_TOST`, `power_z_cor`. +- **Flag the ANOVA novelty in the abstract.** Item (5) currently reads "equivalence testing for ANOVAs", which sounds like a wrapper around existing work. If the generalisation of the non-centrality parameter to factorial and within-subjects designs is new — it is — say so. This also pre-empts comment #4's "most novel but least described" complaint. +- **Optional, larger: promote the estimand framing to a first-class section.** The footnote at ~line 100 is doing a lot of load-bearing conceptual work for a footnote, and it is the thread tying together the rank-based, resampling, log-transform, and ANOVA sections. A short "Choosing an estimand" section sitting beside "Setting Equivalence Bounds" would make the paper read as one argument rather than three. + +Explicitly **not** doing: splitting into two papers, or cutting conceptual material. + +--- + +## 2. WMW section jumps between location-shift and no-assumption versions + +> *"a more explicit separation via section structure might help people navigate this — in my experience, the awareness of what WMW tests is abysmal"* + +**Verdict: accept. He is right, and his read on reader awareness is right.** + +The current text (~lines 590–610) interleaves three things in consecutive sentences: what the test does by default, what it does under location shift, and what `wilcox_TOST` puts bounds on. Restructure `### Wilcoxon-Mann-Whitney Tests` into labelled sub-parts: + +1. **What WMW tests without extra assumptions.** Stochastic equality; H₀: P(X > Y) + ½P(X = Y) = 0.5. Keep the Fay & Malinovsky (2018) citation here. +2. **What WMW tests under a location-shift assumption.** Only here does the Hodges–Lehmann estimator become a median difference. Keep the counterexamples (`@median_test`) here — their force is precisely that the assumption fails silently. +3. **What `wilcox_TOST` puts bounds on, and what that means in practice.** Bounds go on the pseudo-median. State plainly: *if you are not willing to assert location shift, the pseudo-median is your estimand and your equivalence bound has to be justified on that scale.* Then the existing redirect — `perm_t_test`/`boot_t_test` for means, `brunner_munzel` for stochastic superiority. +4. Worked example, then `#### Checking Assumptions`. + +Two supporting changes: + +- **One-line estimand statement at the top of each robust-method subsection** (WMW, Brunner–Munzel, bootstrap/permutation, log-TOST), so the parallel structure is visible at a glance. Cheap, and it makes the whole "Robust Methods" section navigable. +- **Consider an ECDF figure for location shift.** Symmetry currently gets a figure (`fig:symmplot`) while location shift — the more consequential assumption in the two-sample case, and the one Jakub is worried readers do not understand — gets a sentence of prose. A two-panel figure (ECDFs as horizontal translations vs. crossing curves) would carry it better. Check `toster_assumption_checks_supplement.qmd` first; there may already be something usable. + +--- + +## 3. "Valid under heteroscedasticity" — specify *under the null* + +> *"is it worth specifying the heteroskedasticity is under null? Because outside the null, heteroskedasticity is not an issue?"* + +**Verdict: accept. Cheap and correct.** + +He is right on the substance. The claim at ~line 645 (`@karch2021`, "should be preferred as the default non-parametric procedure because it is valid under heteroscedasticity") is a Type I error statement: this is the nonparametric Behrens–Fisher problem. WMW's exactness rests on exchangeability, i.e. identical distributions under H₀; when the distributions differ in scale or shape but stochastic equality holds, WMW's error rate is not nominal. Brunner–Munzel tests H₀: p = 0.5 directly and does not require matched distributions. + +I would go slightly further than the clause Jakub suggests and name the problem — "nonparametric Behrens–Fisher" is the searchable term, and readers who know the parametric version will get it immediately. Draft replacement: + +> `@karch2021` argues it should be preferred as the default non-parametric procedure. Its main advantage over WMW is that the null hypothesis of stochastic equality does not require the two distributions to be identical — the nonparametric Behrens–Fisher problem. WMW's Type I error rate can depart from nominal when the groups are stochastically equal but differ in variance or shape; the Brunner–Munzel test remains valid in that setting and provides a directly interpretable effect size. + +Also worth a sentence in the Brunner–Munzel `#### Checking Assumptions` (~line 664). The text correctly says the assumptions are essentially independence plus adequate n for the t-approximation, but the *reason* heteroscedasticity is a non-issue — the estimand is defined from the two marginal distributions, so nothing about it presumes matched shapes — sits a paragraph away from the claim. Tighten. + +*(Struck: an earlier version of this note flagged the p̂ direction convention as needing a check, on the mistaken grounds that `mu = c(0.3, 0.7)` was asymmetric. It is symmetric about 0.5, and since swapping X and Y maps p̂ → 1 − p̂, symmetric bounds are invariant to the convention anyway. The convention is also already fixed and documented: the formula method takes X as the first factor level and Y as the second (`R/brunner_munzel.R:1213`), and the print method labels the estimate with the actual level names. No action. The only residual thought — very low priority — is that a reader using genuinely asymmetric bounds, e.g. `mu = c(0.4, 0.8)`, has to know which level is X to place them correctly; if the paper ever shows such an example, name the direction in the surrounding prose. The package's own reporting convention follows `t.test`'s x/y argument style rather than the notation used in the methodological literature, which is a deliberate choice and does not need defending in the paper.)* + +--- + +## 4. ANOVA section: most novel, least described; η² undefined + +> *"it's not specified what η^2 is, and overall given that equivalence interpretation for ANOVA is maybe a bit less obvious than some other tests"* + +**Verdict: accept in full. Most actionable comment in the email.** + +The section (~lines 784–840) is about five paragraphs for the paper's genuinely new methodological contribution, and it assumes the reader already knows partial η², already knows why an omnibus equivalence test is coherent, and already knows how to pick a bound on a variance-explained scale. None of those are safe assumptions. + +- **Define partial eta-squared.** η²ₚ = SS_effect / (SS_effect + SS_error) — the proportion of variance attributable to the factor after removing variance attributable to other factors. Say explicitly that this differs from η² (which divides by total SS) and that the two coincide only in a one-way design. Readers pulling values out of `afex` or SPSS output need this. +- **Explain why the test is one-sided, and that it is not a "TOST".** Because η²ₚ ≥ 0 there is no lower bound to test against; the hypotheses are H₀: η²ₚ ≥ Δ vs. H₁: η²ₚ < Δ. This is a non-inferiority / omnibus test, which is exactly why Campbell & Lakens frame it that way. The manuscript never says the two-one-sided-tests logic does not apply here — in a paper built entirely around TOST, that is a real trip hazard. +- **Say what a rejection licenses.** Rejecting means no effect in the design accounts for more than Δ of the (residual-adjusted) variance. That is a statement about the factor as a whole, not about any particular pairwise contrast, and it does not license "the groups are equivalent" in the mean-difference sense. Conversely, failing to reject the equivalence null is not evidence of a large effect. Two or three sentences. +- **Bound selection on the η²ₚ scale, which is harder than on the raw scale.** Two problems worth naming: (a) partial η² is not comparable across designs, because the denominator depends on which other factors are in the model and, for within-subjects designs, on which error term the effect is tested against — so a bound borrowed from a published study in a different design is not portable; (b) the Δ = 0.35 used in *both* worked examples is very large and is there only to make the demo produce a non-degenerate result. The paper has a global "these are illustrative" disclaimer at the end of "Setting Equivalence Bounds", but 0.35 appearing in two consecutive examples reads as a default and needs a local repeat. Consider giving the conversion to Cohen's *f* (f² = η²ₚ/(1−η²ₚ)) as a bridge for readers with *f*-scale intuition, and pointing at the bound-selection section of the `the_ftestTOSTER` vignette. +- **State the generalisation in the manuscript, not only on the website.** Put both formulas inline: Campbell & Lakens' one-way λ = NΔ/(1−Δ), and the generalised λ_eq = [Δ/(1−Δ)]·(df₁ + df₂ + 1). Then note that for a one-way design df₁ + df₂ + 1 = (J−1) + (N−J) + 1 = N, so the generalisation reduces *exactly* to the published result. That reduction is free, it is a real (if partial) validation, and it answers both #4 and part of #5. It is currently nowhere in the manuscript. +- **Walk the `equ_anova` output columns.** The existing sentence lists `p.null`, `pes`, `eqb`, `p.equ` in running prose; a short definition list would be easier to use. The within-subjects example is a good place to note that each effect gets its own error term and therefore its own denominator for η²ₚ. + +--- + +## 5. Was the original ANOVA method validated? Add validation here. + +> *"I'm not sure how much the original work on equivalence testing for ANOVA validated the method, but it feels like something that might work nicely if added here."* + +**Verdict: accept, with a correction to the premise. This is the big one.** + +Correcting the premise: Campbell & Lakens (2021) *did* validate their method — the paper includes a simulation study of Type I error and power for the one-way ANOVA and multivariable regression R² cases, plus a comparison against a Bayesian alternative. So the published method is not unvalidated, and the manuscript should say so (one clause, currently absent, worth adding regardless). + +What is **not** validated is my generalisation. The λ_eq = [Δ/(1−Δ)]·(df₁ + df₂ + 1) substitution is what extends the method to factorial and within-subjects designs, and it reduces to the published formula only in the one-way case. Honest framing: the base method is validated, the extension is not. A reviewer is likely to find this too, so better to close it now. + +Proposed scope — a simulation supplement (probably its own file rather than folded into `toster_assumption_checks_supplement.qmd`, since it is a different kind of content): + +- **Designs:** (a) one-way between-subjects, as a check that the implementation reproduces Campbell & Lakens; (b) 2×2 between-subjects factorial — main effects and interaction; (c) 2×2 fully within-subjects, which is the case the generalisation is least obviously right for, since the ncp for a repeated-measures F depends on the correlation among measures and it is not self-evident that df₁ + df₂ + 1 carries the right information there; (d) mixed design, if (c) behaves. +- **Conditions:** true η²ₚ below, at, and above the bound — the boundary is the least favourable case under H₀ and gives the Type I error; a few n per cell (small / medium / large); for the within-subjects cases, a range of correlations among repeated measures; Δ spanning a plausible range (say 0.02 / 0.06 / 0.14) rather than the 0.35 used for demos. +- **Outcomes:** empirical rejection rate at the boundary (should be ≤ α, and ideally close to α rather than badly conservative), power curves, and a direct comparison of the generalised p-value against the Campbell & Lakens formula in the one-way case — should be numerically *identical*, not merely close. +- **Deliverable:** one figure (rejection rate vs. true η²ₚ, panelled by design) plus a short table, referenced from the ANOVA section with a single summary sentence in the manuscript. Detail stays in the supplement. + +Add as regression tests in `tests/testthat/` once the simulation exists: exact equality of `equ_ftest` against the closed-form one-way result, and a fixed-seed boundary Type I error check for a factorial case. + +**Risk to flag:** if the within-subjects case turns out to be miscalibrated, that is a package bug rather than a paper problem, and it changes the schedule. Run condition (c) first as a smoke test before committing to the full grid. + +--- + +## 6. Cite the "multilevel modelling is often unnecessary" paper + +> *"it might be worth mentioning this paper showing that quite often multilevel-handling-statistics is unnecessary and simple averaging can do a decent job"* + +**Verdict: accept, but with a caveat and in a specific place.** + +The reference is McNabb & Murayama (2021), *Unnecessary reliance on multilevel modelling to analyse nested data in neuroscience: When a traditional summary-statistics approach suffices*, **Current Research in Neurobiology** 2, 100024, doi:10.1016/j.crneur.2021.100024. Note for the .bib: the ScienceDirect PII Jakub sent resolves to *Current Research in Neurobiology*, not *NeuroImage: Reports*. + +It fits naturally in the **Conclusions** limitations paragraph (~line 894), which currently notes that TOSTER does not handle multilevel/mixed-effects designs and points to `emmeans`/`marginaleffects`. Adding this citation turns a bare limitation into useful advice: for nested designs of the common "several trials per participant, roughly balanced" kind, averaging within participant and running a simple test on the per-participant summaries is often defensible — which brings a fair number of nominally multilevel designs back inside TOSTER's scope. + +**Caveat to include rather than citing it flat:** there is a published commentary — Bloom, Thieu & Bolger (2022), *Current Research in Neurobiology* — arguing that the summary-statistics approach breaks down under unequal cluster sizes or unequal cluster variances (it over-weights small clusters), and that it cannot recover between-cluster heterogeneity even when it gets the fixed effect right. That is directly relevant to *equivalence* testing specifically: an equivalence conclusion drawn from a summary-statistics analysis of unbalanced clusters could be driven by mis-weighting rather than by a genuinely small effect. So cite McNabb & Murayama, state the balanced-design condition, and cite the commentary in the same sentence. One or two sentences total — do not open the debate in a package paper. + +Minor scope pushback: the McNabb & Murayama argument is framed around neuroscience trial-level data, and this paper's audience is broader. Keep the phrasing conditional ("where clusters are of similar size and the question concerns the average effect") rather than implying multilevel modelling is generally avoidable. + +--- + +## Consolidated to-do + +**Manuscript (`Avocado_Update.Rmd`)** +- [ ] Intro: roadmap paragraph; flag the ANOVA generalisation as novel (also in the abstract). +- [ ] Add function-selection table. +- [ ] (Optional) Promote the estimand footnote to a short section. +- [ ] Restructure WMW subsection: no-assumption / location-shift / what-gets-bounded. +- [ ] One-line estimand statement at the head of each robust-method subsection. +- [ ] (Optional) ECDF location-shift figure alongside the symmetry plot. +- [ ] Brunner–Munzel: qualify "valid under heteroscedasticity" as a null/Type I error statement; name the nonparametric Behrens–Fisher problem. +- [ ] ANOVA: define partial η²; distinguish from η²; explain the one-sided non-inferiority framing and why it is not a TOST; state what a rejection does and does not license. +- [ ] ANOVA: bound-selection discussion; flag Δ = 0.35 as illustrative; add the f² conversion. +- [ ] ANOVA: both ncp formulas inline, plus the one-way reduction df₁ + df₂ + 1 = N. +- [ ] ANOVA: note that Campbell & Lakens ran a validating simulation; reference the new simulation for the extension. +- [ ] ANOVA: definition list for `equ_anova` output columns. +- [ ] Conclusions: McNabb & Murayama with the balance condition, plus the Bloom et al. commentary. + +**Supplement / package** +- [ ] New simulation supplement for the factorial and within-subjects extension (run the within-subjects smoke test first). +- [ ] `tests/testthat/`: exact-equality test against the one-way closed form; boundary Type I error check for a factorial design. + +**Bibliography (`interactcadsample.bib`)** +- [ ] `mcnabb2021`, `bloom2022` entries. + +**Reply to Jakub** +- [ ] Note that Campbell & Lakens did validate the one-way case, and that his comment correctly identifies the extension as the unvalidated part — this is the most useful thing in his email. +- [ ] Say what I am not doing on #1 (no restructure) and why. From e9d8bcc0c644eec6f784da8f890aeb6778f95be9 Mon Sep 17 00:00:00 2001 From: Aaron Caldwell Date: Fri, 4 Sep 2026 09:51:40 -0500 Subject: [PATCH 21/40] WMW section fixed --- papers/Avocado_Update/Avocado_Update.Rmd | 83 +++++++++++++++++-- .../revision-plan_jakub-comments.md | 6 +- 2 files changed, 81 insertions(+), 8 deletions(-) diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index 33d88bc9..1a99982b 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -589,14 +589,29 @@ Rank-based tests operate on the ranks of the data rather than the raw values, wh ### Wilcoxon-Mann-Whitney Tests -The WMW group of tests provide non-parametric tests based on ranks. Within the TOST framework, the WMW tests are comprised of two directional location-shift tests determine if the shift is within (equivalence) or outside (minimal effect) the equivalence bounds. There are many common misconceptions about the WMW tests that I will try to briefly address. +The Wilcoxon-Mann-Whitney (WMW) family --- the rank-sum test for two independent groups and the signed-rank test for paired and one-sample designs --- provides non-parametric tests based on ranks. Within the TOST framework, TOSTER performs two directional tests to determine whether the effect lies within (equivalence) or outside (minimal effect) the equivalence bounds. What that effect is depends on an assumption that researchers often do not realize they are making, and there are persistent misconceptions worth addressing directly. The three subsections below separate what the WMW tests evaluate with no additional assumptions, what they evaluate under a location-shift assumption, and what `wilcox_TOST` actually places bounds on. +#### Without a location-shift assumption: a test of stochastic equality -First, many researchers mistakenly believe these are tests of medians, but without a location-shift assumption, the WMW test is a test of stochastic equality. As @Fay_Malinovsky_2018 demonstrate, the effect size can even be presented as stochastic superiority/dominance. +Many researchers believe the WMW tests are tests of medians. They are not. With no assumption beyond independence and exchangeability, the WMW test evaluates stochastic equality which has the null hypothesis that a randomly drawn observation from one group is equally likely to exceed, or to be exceeded by, a randomly drawn observation from the other, -Second, only under a location-shift assumption does it become a test of central tendency through the Hodges-Lehmann estimator. @median_test demonstrate striking counterexamples: groups can have equal medians yet produce a significant WMW test, very different medians yet a non-significant WMW test, or even significance in the opposite direction from the median comparison. +$$ +H_0: \space P(X > Y) + \frac{1}{2} \cdot P(X = Y) = 0.5 +$$ -For equivalence testing in the TOSTER package, the function `wilcox_TOST` applies the bounds to the pseudo-median, which corresponds to the mean or median difference only under the location-shift assumption. For testing hypotheses about mean differences, `perm_t_test` or `boot_t_test` are more appropriate. For stochastic superiority, `brunner_munzel` provides a clearer framework. +As @Fay_Malinovsky_2018 demonstrate, the effect size accompanying the test is therefore most naturally expressed as stochastic superiority (or dominance) rather than as a difference in central tendency. Nothing about this null hypothesis refers to a mean or a median. + +#### Under a location-shift assumption: a test of central tendency + +The location-shift assumption holds that the two distributions are identical in every respect except location. Under this assumption, and only under it, stochastic equality and equality of medians coincide, and the Hodges-Lehmann estimator can be read as an estimate of the median difference. This is the interpretation most readers have in mind when they reach for a WMW test. + +When the assumption fails, the two interpretations can come apart dramatically. @median_test demonstrate striking counterexamples: groups can have equal medians yet produce a significant WMW test, very different medians yet a non-significant WMW test, or even significance in the direction opposite the median comparison. Critically, nothing in the test output signals that the assumption has failed --- the p-value and the Hodges-Lehmann estimate are reported the same way either way. The diagnostic described below is therefore not optional if a central-tendency interpretation is intended. + +#### What `wilcox_TOST` places bounds on + +In TOSTER, `wilcox_TOST` applies the equivalence bounds to the pseudo-median, which corresponds to the mean or median difference only under the location-shift assumption. This has a direct practical consequence for setting bounds: if you are not prepared to assert location shift, the pseudo-median is your estimand, and the smallest effect size of interest must be justified on that scale rather than imported from reasoning about mean differences. + +If the hypothesis genuinely concerns means, `perm_t_test` or `boot_t_test` are the more appropriate tools. If it concerns stochastic superiority, `brunner_munzel` provides a clearer framework and a directly interpretable effect size. ```{r} test1 = wilcox_TOST( @@ -613,7 +628,65 @@ Based on these results, we would conclude there is a significant difference and #### Checking Assumptions -The WMW tests assume independent observations and exchangeability under the null, the latter meaning that group labels carry no information about outcomes if there is no effect. For the interpretations discussed above to be clean, two additional assumptions matter. The location-shift assumption — that the two group distributions differ only in location, not shape — is what allows WMW results to be interpreted as a test of the median difference via the Hodges-Lehmann estimator. Violations are best diagnosed by overlaying the empirical cumulative density functions of the two groups; under location shift, the curves are horizontal translations of each other, whereas crossing curves or varying horizontal gaps indicate violation. For the paired and one-sample signed-rank variants used in `wilcox_TOST` or `simple_htest`, the signed-rank procedure additionally requires that the difference distribution is symmetric around the null value for the pseudo-median to coincide with the mean or median difference. Asymmetry is readily checked with a symmetry plot of the paired differences (Figure \ref{fig:symmplot}). As with the location-shift case, visible departures suggest that the test's output — the pseudo-median and associated p-value — does not map cleanly onto a hypothesis about central tendency. +The WMW tests assume independent observations and exchangeability under the null, the latter meaning that group labels carry no information about outcomes if there is no effect. Those two assumptions are all that the stochastic-equality interpretation requires. The two assumptions below determine whether the stronger central-tendency interpretation is available, and each has a straightforward visual diagnostic. + +The location-shift assumption --- that the two group distributions differ only in location, not in shape or spread --- is what allows WMW results to be interpreted as a test of the median difference via the Hodges-Lehmann estimator. It is best diagnosed by overlaying the empirical cumulative distribution functions (ECDFs) of the two groups. Under location shift, the curves are horizontal translations of one another, so the horizontal distance between them is constant across the range of the data (Figure \ref{fig:ecdfshift}A). Departures appear as a gap that widens, narrows, or reverses sign, and in the clearest cases as curves that cross (Figure \ref{fig:ecdfshift}B). In panel B, the two groups there differ in both location and spread, so there is no single number that describes "the shift," and the Hodges-Lehmann estimate that `wilcox_TOST` reports does not correspond to a difference in medians. + +For the signed-rank variants used by `wilcox_TOST` and `simple_htest`, the difference distribution must additionally be symmetric about the null value for the pseudo-median to coincide with the mean or median difference. Asymmetry is readily checked with a symmetry plot of the paired differences (Figure \ref{fig:symmplot}). As with the location-shift case, visible departures suggest that the test's output --- the pseudo-median and associated p-value --- does not map cleanly onto a hypothesis about central tendency. + +```{r } +#| label: ecdfshift +#| echo: false +#| fig.height: 3.8 +#| fig.width: 6.5 +#| fig.cap: "\\label{fig:ecdfshift}Empirical cumulative distribution functions for two simulated groups, with double-headed arrows marking the horizontal distance between the curves at the 10th, 50th, and 90th percentiles. (A) The location-shift assumption holds: the distributions differ only in location, the arrows are of equal length, and the Hodges-Lehmann estimator can be read as a median difference. (B) The assumption is violated: the groups differ in spread as well as location, so the horizontal gap varies in magnitude and reverses sign across the distribution, and the curves cross. In panel B there is no single shift to estimate, and a central-tendency interpretation of the WMW test is not available." +set.seed(19) +n_ecdf <- 200 + +ecdf_panel <- function(g1, g2, title, subtitle) { + df <- data.frame( + value = c(g1, g2), + Group = rep(c("Group 1", "Group 2"), c(length(g1), length(g2))) + ) + probs <- c(0.1, 0.5, 0.9) + seg <- data.frame( + p = probs, + x1 = quantile(g1, probs, type = 7), + x2 = quantile(g2, probs, type = 7) + ) + ggplot(df, aes(x = value, colour = Group, linetype = Group)) + + stat_ecdf(linewidth = 0.7) + + geom_segment( + data = seg, inherit.aes = FALSE, + aes(x = x1, xend = x2, y = p, yend = p), + arrow = arrow(length = unit(0.05, "in"), ends = "both"), + linewidth = 0.35, colour = "grey25" + ) + + scale_colour_grey(start = 0.1, end = 0.55) + + coord_cartesian(xlim = c(-7, 8)) + + labs(x = "Value", y = "Cumulative probability", + title = title, subtitle = subtitle) + + theme_minimal(base_size = 10) + + theme(legend.position = "bottom", + legend.title = element_blank(), + plot.title = element_text(size = 10, face = "bold"), + plot.subtitle = element_text(size = 8)) +} + +p_shift <- ecdf_panel( + rnorm(n_ecdf, 0, 1), rnorm(n_ecdf, 1.5, 1), + "A. Location shift holds", + "Horizontal gap is constant across the distribution" +) +p_viol <- ecdf_panel( + rnorm(n_ecdf, 0, 1), rnorm(n_ecdf, 0.5, 2.5), + "B. Location shift violated", + "Gap varies and reverses sign; curves cross" +) + +p_shift + p_viol + plot_layout(guides = "collect") & + theme(legend.position = "bottom") +``` ```{r } diff --git a/papers/Avocado_Update/revision-plan_jakub-comments.md b/papers/Avocado_Update/revision-plan_jakub-comments.md index 84ebc9b4..68a08d38 100644 --- a/papers/Avocado_Update/revision-plan_jakub-comments.md +++ b/papers/Avocado_Update/revision-plan_jakub-comments.md @@ -134,9 +134,9 @@ Minor scope pushback: the McNabb & Murayama argument is framed around neuroscien - [ ] Intro: roadmap paragraph; flag the ANOVA generalisation as novel (also in the abstract). - [ ] Add function-selection table. - [ ] (Optional) Promote the estimand footnote to a short section. -- [ ] Restructure WMW subsection: no-assumption / location-shift / what-gets-bounded. -- [ ] One-line estimand statement at the head of each robust-method subsection. -- [ ] (Optional) ECDF location-shift figure alongside the symmetry plot. +- [x] Restructure WMW subsection: no-assumption / location-shift / what-gets-bounded. **Done** — three `####` subsections added; H₀ for stochastic equality now stated formally; `Checking Assumptions` split into labelled location-shift and symmetry parts. +- [x] ECDF location-shift figure alongside the symmetry plot. **Done** — `fig:ecdfshift`, two panels (assumption holds / violated) with arrows marking the horizontal gap at the 10th/50th/90th percentiles. Knit-tested. +- [ ] One-line estimand statement at the head of each robust-method subsection (WMW now has one; Brunner–Munzel, bootstrap/permutation, log-TOST still need theirs). - [ ] Brunner–Munzel: qualify "valid under heteroscedasticity" as a null/Type I error statement; name the nonparametric Behrens–Fisher problem. - [ ] ANOVA: define partial η²; distinguish from η²; explain the one-sided non-inferiority framing and why it is not a TOST; state what a rejection does and does not license. - [ ] ANOVA: bound-selection discussion; flag Δ = 0.35 as illustrative; add the f² conversion. From 73cf2c59b0d899ada24cb42ee88838c81ffecf26 Mon Sep 17 00:00:00 2001 From: Aaron Caldwell Date: Fri, 4 Sep 2026 11:18:28 -0500 Subject: [PATCH 22/40] Updated ANOVA need to run quick simulation study --- papers/Avocado_Update/Avocado_Update.Rmd | 110 +- .../campbell_lakens_supplement.zip | Bin 0 -> 613335 bytes .../analysisdataset2018.csv | 12665 ++++++++++++++++ .../disregard_BF_CET_R2.R | 179 + .../disregard_hawthorne.R | 171 + .../disregard_simFigA.pdf | Bin 0 -> 219378 bytes .../disregard_simFigB.pdf | Bin 0 -> 294443 bytes .../disregard_simulation_study.R | 154 + .../disregard_simulation_study_2.R | 181 + .../campbell_lakens_supplement/lakens4.png | Bin 0 -> 69242 bytes .../revision-plan_jakub-comments.md | 16 +- 11 files changed, 13453 insertions(+), 23 deletions(-) create mode 100644 papers/Avocado_Update/campbell_lakens_supplement.zip create mode 100644 papers/Avocado_Update/campbell_lakens_supplement/analysisdataset2018.csv create mode 100644 papers/Avocado_Update/campbell_lakens_supplement/disregard_BF_CET_R2.R create mode 100644 papers/Avocado_Update/campbell_lakens_supplement/disregard_hawthorne.R create mode 100644 papers/Avocado_Update/campbell_lakens_supplement/disregard_simFigA.pdf create mode 100644 papers/Avocado_Update/campbell_lakens_supplement/disregard_simFigB.pdf create mode 100644 papers/Avocado_Update/campbell_lakens_supplement/disregard_simulation_study.R create mode 100644 papers/Avocado_Update/campbell_lakens_supplement/disregard_simulation_study_2.R create mode 100644 papers/Avocado_Update/campbell_lakens_supplement/lakens4.png diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index 1a99982b..937575a1 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -77,17 +77,47 @@ The TOSTER R package^[All updates to the package can be found on the package's w However, the original package had a number of limitations. It offered a function-per-design interface (separate functions for each t-test type), lacked robust alternatives to parametric tests, provided no support for ANOVA equivalence testing, and had no tools for comparing effect sizes across studies. The package also did not integrate with standard R workflows and usually just printed results to the console rather than returning objects that could be manipulated or plotted. -This paper provides a tutorial-style introduction to the updated TOSTER package, organized around its major capabilities: (1) a unified t-test interface (`t_TOST`) that pairs a standard significance test with an equivalence test and supports minimal effect testing, (2) a simplified hypothesis testing interface (`simple_htest`) that returns standard `htest` objects, (3) tools for computing and testing standardized effect sizes --- including standardized mean differences, rank-based effect sizes, and correlations, (4) robust and resampling-based alternatives including the Brunner-Munzel test, bootstrap and permutation t-tests, and log-transformed methods, (5) equivalence testing for ANOVAs, and (6) power analysis and sample size planning. Each section motivates when and why a researcher would use the tool before demonstrating how. +This paper provides a tutorial-style introduction to the updated TOSTER package, organized around its major capabilities: (1) a unified t-test interface (`t_TOST`) that pairs a standard significance test with an equivalence test and supports minimal effect testing, (2) a simplified hypothesis testing interface (`simple_htest`) that returns standard `htest` objects, (3) tools for computing and testing standardized effect sizes --- including standardized mean differences (SMDs), rank-based effect sizes, and correlations, (4) robust and resampling-based alternatives including the Brunner-Munzel test, bootstrap and permutation t-tests, and log-transformed methods, (5) equivalence testing for ANOVAs, and (6) power analysis and sample size planning. Each section motivates when and why a researcher would use the tool before demonstrating how. TOSTER occupies a specific niche in the R ecosystem in that it is a frequentist toolkit for equivalence testing of standard designs. Researchers interested in Bayesian approaches to evidence for the null have excellent options (e.g., `bayestestR`). For equivalence-style contrasts in more complex models --- mixed models, marginal means --- the `emmeans` and `marginaleffects` packages can accommodate non-zero null hypotheses. TOSTER fills the space of accessible, self-contained equivalence testing for common, albeit simple, research designs. For a broader introduction to the methodology, see @lakens_ori, @lakens2018equivalence, @lakens2020improving, @mazzolari2022myths, and @Tomek_Caldwell_Eisner_2026. TOSTER is also available as a [jamovi](https://www.jamovi.org/) module for researchers who prefer a graphical interface although it has limited capabilities compared to the R package. # Setting Equivalence Bounds +Two decisions precede any equivalence test: *what* quantity the hypothesis concerns, and *how much* of that quantity is too small to matter. The first is a choice of estimand, the second a choice of bounds. The two are not separable, because a bound is a magnitude on the scale of some particular quantity, and the same number can describe quite different hypotheses depending on what is being estimated. + +## Choosing an Estimand + +The estimand is the quantity a study is attempting to estimate [@Lundberg_Johnson_Stewart_2021]. In an ideal scenario, the hypothesis should determine the estimand, and estimand then narrows the set of appropriate tests. Inheriting an estimand from whichever procedure happens to be run is a reliable way to end up testing the wrong hypothesis. + +This matters more for equivalence testing than for nil-hypothesis testing. When two distributions are identical, the location-type quantities in Table \ref{tab:estimands} all take their null values at once --- the mean difference and the pseudo-median are both zero, and the probability of superiority is exactly 0.5 --- so a nil test is comparatively forgiving of an unexamined estimand. An equivalence bound makes the decision more consequential. It is a specific magnitude on a specific scale, and there is generally no way to translate a bound on one scale into a bound on another (at least without making very serious assumptions). Bounds of $\pm$ 0.5 placed on a mean difference, on a pseudo-median, and on the SMD describe three different hypotheses, only one of which is likely to be the intended one. + +Table \ref{tab:estimands} lists the estimands TOSTER can place bounds on and the functions that target each. Two entries deserve comment. The trimmed mean, available through the `tr` argument of the resampling functions (`tr = 0` by default), coincides with the mean only when the distribution is symmetric; setting `tr` therefore changes the estimand rather than merely making the same estimate more robust. The SMD is a rescaling of the mean difference by a sample-dependent quantity, which is why it warrants the separate caution given later in this paper. + +Table: \label{tab:estimands}Estimands supported by TOSTER, the scale on which equivalence bounds are specified for each, and the primary functions targeting them. + +| Estimand | Scale of the bound | Primary functions | +|:---------------------------|:--------------------|:--------------------------| +| Mean or mean difference | Original units | `t_TOST`, `tsum_TOST`, `simple_htest` | +| Mean, with empirically calibrated reference distributions | Original units | `boot_t_TOST`, `boot_t_test`, `perm_t_test` | +| Trimmed mean difference | Original units | `boot_t_TOST`, `perm_t_test` (via `tr`) | +| Ratio of geometric means | Multiplicative ratio | `log_TOST`, `boot_log_TOST` | +| Pseudo-median (Hodges-Lehmann) | Original units | `wilcox_TOST` | +| Stochastic superiority | Probability (0 to 1) | `brunner_munzel` | +| Standardized mean difference | Standard deviation units | `smd_calc`, `boot_smd_calc` | +| Correlation | Correlation ($-1$ to 1) | `z_cor_test`, `boot_cor_test` | +| Proportion of variance explained | Proportion (0 to 1) | `equ_anova`, `equ_ftest` | + +In applied settings the estimand is not fully pinned down by the choice of summary statistic either. Decisions about how intercurrent events --- post-randomization events such as dropout, treatment discontinuation, or the use of rescue medication --- and missing data are handled determine which population summary is being estimated at all. Two analyses that both report "the mean difference" may be estimating materially different things if they resolve these questions differently. + +The practical upshot is to settle the estimand first and then justify a bound on its scale, rather than selecting a test and adopting whatever quantity it happens to return. This ordering also makes the robust methods described later in this paper easier to navigate: they are not a menu of interchangeable fixes for assumption violations, but tests of different quantities, and only some of them remain answers to a hypothesis about means. + +## Justifying the Bounds + Before applying any equivalence test, the researcher must specify the equivalence bounds which defines the range of effect sizes considered too small to be meaningful. There is no single correct method for choosing these bounds; the appropriate approach depends entirely on the research question and domain of research. If a researcher has an equivalence hypothesis, they should be able to articulate what magnitude of effect would be too small to matter in context. The SESOI must flow from the hypothesis formulated by theory and context not from a formula. However, there are several general strategies that have been evaluated [@delta2]. Anchor-based methods tie the bound to an externally meaningful criterion, such as a clinically meaningful difference or a "just-noticeable" difference [@sesoi]. Distribution-based methods define the bound as a fraction of the measurement error [@Jacobson_Truax_1991], though this approach is generally not recommended [@delta2]. Cost-benefit, also referred to as health economic methods [@delta2], reasoning sets the bound at the point where there is a net benefit relative to the cost of an intervention. Researchers can draw on existing literature or normative data to identify effect magnitudes that prior work suggests are negligible. When theory is sufficiently precise, however, the bound may require no external benchmark at all, and the hypothesis itself implies what magnitude of effect would be too small to matter. -Using "default" bounds on the standardized mean difference (SMD; also known as Cohen's d) based on Cohen's benchmarks (e.g., SMD = 0.2, 0.5, 0.8 to imply small, medium, or large effects, respectively) without domain-specific justification is strongly discouraged. Cohen himself stated that these conventions were "recommended for use only when no better basis for estimating the index is available" (p. 25, as cited in @panzarella2021) and reportedly came to regret having proposed them at all [@funder2019]. These benchmarks were never intended as universal thresholds; applied uncritically, they can produce equivalence bounds that are too wide, too narrow, and likely entirely disconnected from the substantive research question. +Using "default" bounds on the SMD (also known as Cohen's d) based on Cohen's benchmarks (e.g., SMD = 0.2, 0.5, 0.8 to imply small, medium, or large effects, respectively) without domain-specific justification is strongly discouraged. Cohen himself stated that these conventions were "recommended for use only when no better basis for estimating the index is available" (p. 25, as cited in @panzarella2021) and reportedly came to regret having proposed them at all [@funder2019]. These benchmarks were never intended as universal thresholds; applied uncritically, they can produce equivalence bounds that are too wide, too narrow, and likely entirely disconnected from the substantive research question. If a researcher cannot specify meaningful bounds, this itself is informative as it suggests the equivalence hypothesis is not well enough developed to test such a hypothesis. In such cases, it may be more productive to focus on effect estimation and description rather than leaping to hypothesis tests [@scheel_hypothesis]. Throughout the examples in this paper, equivalence bounds, and the data sets themselves, are chosen for illustrative purposes; readers should not treat them as direct recommendations for their own research. @@ -97,7 +127,7 @@ The t-test is the standard tool for comparing means. Its construction is simple: What matters for equivalence testing is that $\mu_0$ is a choice rather than a fixed feature of the test. A traditional nil hypothesis test fixes $\mu_0 = 0$ and asks whether the mean, or mean difference, is exactly zero. TOST changes none of the underlying machinery; it changes only what is substituted for $\mu_0$. Two one-sided tests are performed --- one against the lower bound ($\mu_0 = \Delta_L$) and one against the upper bound ($\mu_0 = \Delta_U$) --- and equivalence is concluded only if both reject. Because each test is one-sided at $\alpha$, the interval estimate corresponding to the procedure is a $1-2\alpha$ (typically 90%) confidence interval, which is what TOSTER reports and plots. -Throughout, the estimand for a t-test --- the quantity being estimated --- is the mean, or mean difference, on the original scale of measurement.^[This describes the default analysis, not an inviolable property of the test. Several choices substitute a different estimand: trimming (the `tr` argument in the resampling functions, `tr = 0` by default) targets the trimmed mean, which coincides with the mean only under symmetry; the log-transformed methods target a ratio of geometric means; and in applied settings, decisions about how intercurrent events --- post-randomization events such as dropout or treatment discontinuation --- and missing data are handled determine which population summary is being estimated at all. The estimand is worth choosing deliberately rather than inheriting from whichever test happens to be run [@Lundberg_Johnson_Stewart_2021].] This is worth stating plainly because several of the alternatives described later in this paper --- the rank-based tests in particular --- estimate different quantities entirely, and are not interchangeable substitutes when the hypothesis concerns means. +Throughout, the estimand for a t-test is the mean, or mean difference, on the original scale of measurement. As discussed above, this is a default rather than an inviolable property of the test, and it is worth restating here because several of the alternatives described later in this paper --- the rank-based tests in particular --- estimate different quantities entirely, and are not interchangeable substitutes when the hypothesis concerns means. The `t_TOST` function in TOSTER provides a unified interface for equivalence and minimal effect testing with t-tests. It runs a standard nil significance test alongside the equivalence (or minimal effect) test, because in practice researchers may want both: "Is there a significant effect?" and "Can I rule out a meaningful effect?" The function handles independent, paired, and one-sample designs through the `paired` argument and accepts data via either a formula interface or raw vectors, making it a direct analog to base R's `t.test`. Additionally, the output includes an SMD calculation alongside the unstandarized mean difference with appropriate, albeit approximate, confidence intervals. For researchers who only want to test a single hypothesis at a time, the `simple_htest` function (described in the next section) provides a lighter alternative. @@ -256,7 +286,7 @@ res1 = t_TOST( ) ``` -The `print` method provides a verbose summary including the two-tailed test, both one-sided equivalence tests, and the standardized mean difference (Hedges' $g$ by default): +The `print` method provides a verbose summary including the two-tailed test, both one-sided equivalence tests, and the SMD (Hedges' $g$ by default): ```{r} print(res1) @@ -303,7 +333,7 @@ res4 = t_TOST( res4 ``` -The same interface handles paired designs by setting `paired = TRUE`. Standardized mean differences are reported automatically; Hedges's correction is applied by default but can be toggled with `bias_correction`, and Glass's delta can be requested via the `glass` argument. Setting equivalence bounds in SMD units (via `eqbound_type = "SMD"`) is possible but discouraged because the bound becomes dependent on sample variance (see the Standardized Effect Sizes section below). +The same interface handles paired designs by setting `paired = TRUE`. SMDs are reported automatically; Hedges's correction is applied by default but can be toggled with `bias_correction`, and Glass's delta can be requested via the `glass` argument. Setting equivalence bounds in SMD units (via `eqbound_type = "SMD"`) is possible but discouraged because the bound becomes dependent on sample variance (see the Standardized Effect Sizes section below). ## Checking Assumptions @@ -395,13 +425,13 @@ plot_htest_est(res_sh1) # Standardized Effect Sizes -This section provides a unified treatment of the standardized effect size tools in TOSTER and I cover standardized mean differences (SMDs), correlation coefficients, and tools for comparing both between studies. Before demonstrating these tools, it is worth discussing the limitations of standardized effect sizes, because these limitations are especially important for equivalence testing. +This section provides a unified treatment of the standardized effect size tools in TOSTER and I cover SMDs, correlation coefficients, and tools for comparing both between studies. Before demonstrating these tools, it is worth discussing the limitations of standardized effect sizes, because these limitations are especially important for equivalence testing. ## A Brief Warning on Standardized Effect Sizes Standardized effect sizes --- Cohen's $d$, correlation coefficients, standardized regression coefficients --- are appealing because they allow comparison across studies and scales, are required by many journals, and serve as the primary currency of meta-analysis. However, they have well-documented problems that are especially relevant to equivalence testing. -Standardized mean differences are sensitive to the sample variance. The "same" real-world effect produces different $d$ values across populations and study designs. The ubiquitously reported Cohen's $d$ conflates the magnitude of the mean difference with the precision of measurement, which means that a narrow equivalence bound on $d$ may correspond to a very different raw-scale bound depending on the variability of the sample [@Caldwell2020; @baguley2009]. This makes it unclear what, substantively, is being declared "equivalent" --- are we claiming the means are close, or that the means are close *relative to the variance*? The choice of denominator (pooled SD, control SD, within-subject SD) further complicates that same dubious interpretation. +SMDs are sensitive to the sample variance. The "same" real-world effect produces different $d$ values across populations and study designs. The ubiquitously reported Cohen's $d$ conflates the magnitude of the mean difference with the precision of measurement, which means that a narrow equivalence bound on $d$ may correspond to a very different raw-scale bound depending on the variability of the sample [@Caldwell2020; @baguley2009]. This makes it unclear what, substantively, is being declared "equivalent" --- are we claiming the means are close, or that the means are close *relative to the variance*? The choice of denominator (pooled SD, control SD, within-subject SD) further complicates that same dubious interpretation. Correlations have analogous issues. @tukey1969 noted that correlations are influenced by the range of both variables, so the "same" bivariate relationship produces different coefficients depending on sample selection. A correlation of 0.3 in a restricted-range sample may correspond to a much larger association in the full population. Standardized regression coefficients share these vulnerabilities. These depend on the variances of both the predictor and outcome, which makes them unstable across samples and difficult to interpret substantively [@greenland1991]. @@ -411,7 +441,7 @@ With these caveats in mind, the following subsections describe the tools TOSTER ## Standardized Mean Difference Functions -The `smd_calc` function computes standardized mean differences --- Cohen's $d$, Hedges' $g$ (bias-corrected), or Glass's $\Delta$ --- with confidence intervals and *optional* hypothesis testing (utilizing a z-test: $z = (d - \delta_0)/SE_d$, where $d$ is the sample SMD estimate, $\delta_0$ is the null value being tested, and $SE_d$ is its standard error.). It supports one-sample, paired, and two-sample designs, and can return either an `htest` object or a data frame. The SMD is already reported in `t_TOST` output, but `smd_calc` allows standalone computation and supports equivalence testing directly on the effect size. This may be useful as a complementary function when using `simple_htest` as the primary statistical testing function. +The `smd_calc` function computes an SMD --- Cohen's $d$, Hedges' $g$ (bias-corrected), or Glass's $\Delta$ --- with confidence intervals and *optional* hypothesis testing (utilizing a z-test: $z = (d - \delta_0)/SE_d$, where $d$ is the sample SMD estimate, $\delta_0$ is the null value being tested, and $SE_d$ is its standard error.). It supports one-sample, paired, and two-sample designs, and can return either an `htest` object or a data frame. The SMD is already reported in `t_TOST` output, but `smd_calc` allows standalone computation and supports equivalence testing directly on the effect size. This may be useful as a complementary function when using `simple_htest` as the primary statistical testing function. When estimation rather than testing is the goal, `smd_calc` can be called with `alternative = "none"` (the default), which returns only the point estimate and confidence interval without performing a hypothesis test. This is the recommended usage when the primary objective is to report the magnitude of the effect. @@ -718,7 +748,7 @@ ggplot(sp_df, aes(x = lower, y = upper)) + ### Brunner-Munzel Test -The Brunner-Munzel test [@brunner2000] estimates the relative effect $\hat{p} = P(X > Y) + 0.5 \cdot P(X = Y)$, sometimes referred to as the probability of superiority --- the probability that a randomly chosen observation from one group exceeds a randomly chosen observation from the other, with ties split evenly. This estimand is interpretable without distributional assumptions, unlike the WMW test which requires a location-shift assumption to say anything about location differences. @karch2021 argues it should be preferred as the default non-parametric procedure because it is valid under heteroscedasticity and provides an interpretable effect size. +The Brunner-Munzel test [@brunner2000] estimates the relative effect $\hat{p} = P(X > Y) + 0.5 \cdot P(X = Y)$, sometimes referred to as the probability of superiority --- the probability that a randomly chosen observation from one group exceeds a randomly chosen observation from the other, with ties split evenly. This estimand is interpretable without distributional assumptions, unlike the WMW test which requires a location-shift assumption to say anything about location differences. @karch2021 argues it should be preferred as the default non-parametric procedure. Its main advantage over WMW is that the null hypothesis of stochastic equality does not require the two distributions to be identical (i.e., the nonparametric Behrens–Fisher problem). WMW's Type I error rate can depart from nominal when the groups are stochastically equal but differ in variance or shape; the Brunner–Munzel test remains valid in that setting and provides a directly interpretable effect size. Stochastic superiority tests may be particularly well-suited to Likert-type scales which are ubiquitous in psychological research. For a single ordinal item, the spacing between response categories carries no guaranteed meaning, and a mean difference therefore has a strained interpretation. For example, what, concretely, does a 0.4-unit mean difference on a five-point scale communicate? The stochastic superiority estimand sidesteps this entirely. The probability that a randomly selected participant from one group outscores a randomly selected participant from the other is interpretable on its own terms, with no knowledge of the scale required. We are not suggesting that t-tests will *always* mislead in these contexts but it is a concern that should not be quickly dismissed either [@Liddell_Kruschke_2018]. Additionally, even if we assume traditional parametric methods are robust to the ordinal nature of the data, the preserved error rate is not the same as interpretive clarity. For researchers whose primary outcomes live on ordinal scales, the Brunner-Munzel test and its associated effect size --- as well as ordinal models more generally [@Burkner_Vuorre_2019] --- may offer a more natural and defensible default. @@ -739,7 +769,7 @@ print(bm_test) #### Checking Assumptions -The Brunner-Munzel test has unusually light assumption requirements. It requires independent observations within and between groups, and (for the t-approximation used by default) sample sizes sufficient for the asymptotic approximation to be reasonable. That's essentially it. Unlike WMW, there is no location-shift requirement for the test to be interpretable as a hypothesis about location, and unlike the proportional-odds framing of WMW as a score test from an ordinal regression model, no constant-odds-ratio assumption is needed. The relative effect is defined directly from the two marginal distributions and carries the same interpretation --— the probability that a random observation from one group exceeds one from the other --— regardless of whether the distributions are symmetric, skewed, unimodal, or shaped however they happen to be. Visual inspection of the group distributions remains useful for understanding the data and for choosing a reporting scale, but there is no distributional assumption that these checks are verifying. The studentized permutation variant (`method = "perm"`) is worth considering in small samples, where the t-approximation may be less trustworthy. +The Brunner-Munzel test has unusually light assumption requirements. It requires independent observations within and between groups, and (for the t-approximation used by default) sample sizes sufficient for the asymptotic approximation to be reasonable. That's essentially it. Unlike WMW, there is no location-shift requirement for the test to be interpretable as a hypothesis about location, and unlike the proportional-odds framing of WMW as a score test from an ordinal regression model, no constant-odds-ratio assumption is needed. The relative effect is defined directly from the two marginal distributions and carries the same interpretation --— the probability that a random observation from one group exceeds one from the other --— regardless of whether the distributions are symmetric, skewed, unimodal, or shaped however they happen to be. Visual inspection of the group distributions remains useful for understanding the data and for choosing a reporting scale, but there is no distributional assumption that these checks are verifying. The studentized permutation variant (`method = "perm"`) is worth considering in small samples, where the t-approximation method may be less trustworthy. \newpage @@ -748,9 +778,9 @@ The Brunner-Munzel test has unusually light assumption requirements. It requires When the goal is to test hypotheses about means while relaxing distributional assumptions, resampling methods provide powerful tools. TOSTER implements both bootstrap and permutation approaches through `boot_t_test` and `perm_t_test`, each with distinct inferential logic but a shared motivation to retain the mean as the estimand while obtaining reference distributions empirically --- through resampling --- rather than relying on the normal-theory derivations (i.e., Student's t-test). -Both approaches are natural extensions of the classic t-test. The test statistic itself is unchanged; what changes is how the null distribution is constructed. In the conventional t-test, the null distribution of the standardized mean difference is derived analytically under the assumption that the data are drawn from a normal distribution, with equal variances assumed in Student's t-test and unequal variances accommodated in Welch's t-test. The bootstrap approximates the sampling distribution of the same statistic by resampling from the observed data with replacement, whereas the permutation approach constructs an exact or near-exact null distribution by shuffling group labels (two-sample) or flipping signs (one-sample and paired). In both cases, the researcher continues to ask the same question about means; only the calibration of the reference distribution differs. Researchers may find the resampling methods appealing because it follows a familiar testing logic because the estimand, the null hypothesis, and the test statistic remain the same. +Both approaches are natural extensions of the classic t-test. The test statistic itself is unchanged; what changes is how the null distribution is handled. In the conventional t-test, the null distribution of the mean difference is derived analytically under the assumption that the data are drawn from a normal distribution, with equal variances assumed in Student's t-test and unequal variances accommodated in Welch's t-test. The bootstrap approximates the sampling distribution of the same statistic by resampling from the observed data with replacement, whereas the permutation approach constructs an exact or near-exact null distribution by shuffling group labels (two-sample) or flipping signs (one-sample and paired). In both cases, the researcher continues to ask the same question about means; only the calibration of the reference distribution differs. Researchers may find the resampling methods appealing because it follows a familiar testing logic because the estimand, the null hypothesis, and the test statistic remain the same. -Because the reference distribution is generated from the data rather than assumed, distributional requirements are relaxed, though not eliminated. The bootstrap relies on asymptotic arguments and assumes that the empirical distribution is a reasonable stand-in for the population distribution; it performs well with moderate-to-large samples and skewed or heavy-tailed data, but can struggle with very small samples or extreme outliers. The permutation approach is exact under the null hypothesis of exchangeability, meaning group labels carry no information about outcomes. In its basic form this exactness requires identical distributions under the null, which implicitly assumes equal variances. The studentized permutation test [@janssen1997; @chung2013] addresses this limitation by permuting a scale-invariant statistic (essentially the Welch t-statistic), yielding asymptotically valid inference even under heteroscedasticity. @Arboretti_Pesarin_Salmaso_2020 extended this logic to the equivalence testing setting, showing that studentized permutation TOST maintains nominal Type I error rates when variances differ across groups. In short, these methods are assumption-lighter rather than assumption-free, and users should still consider whether these assumptions are reasonable for their design. +Because the reference distribution is generated from the data rather than assumed, distributional requirements are relaxed, though not eliminated. The bootstrap relies on asymptotic arguments and assumes that the empirical distribution is a reasonable stand-in for the population distribution. It performs well with moderate-to-large samples and skewed or heavy-tailed data, but can struggle with very small samples or extreme outliers. The permutation approach is exact under the null hypothesis of exchangeability, meaning group labels carry no information about outcomes. In its basic form this exactness requires identical distributions under the null, which implicitly assumes equal variances. The studentized permutation test [@janssen1997; @chung2013] addresses this limitation by permuting a scale-invariant statistic (essentially the Welch t-statistic), yielding asymptotically valid inference even under heteroscedasticity. @Arboretti_Pesarin_Salmaso_2020 extended this logic to the equivalence testing setting, showing that studentized permutation TOST maintains nominal Type I error rates when variances differ across groups. In short, these methods are assumption-lighter rather than assumption-free, and users should still consider whether these assumptions are reasonable for their design. A common reflex when normality seems implausible is to switch to the WMW or other rank-based tests. However, these tests answer a different question. As discussed earlier in this manuscript, rank-based procedures test hypotheses about stochastic ordering or pseudo-medians rather than means, and under heteroscedasticity they can reject the null even when the groups have identical means. When the scientific question is genuinely about means --— as it often is in applied research where effect sizes and interventions are naturally expressed on the original scale —-- bootstrap and permutation t-tests should generally be preferred over rank-based methods for handling such violations of assumptions. The rank-based tests remain useful, but as answers to different questions, not as robust substitutes for tests of means. @@ -859,11 +889,51 @@ The log-transformed methods inherit the assumptions of the underlying test appli # Equivalence Testing with ANOVAs -Many researchers use ANOVA as an omnibus test before inspecting pairwise comparisons, particularly in factorial designs. As @Campbell_2021 note, the lack of a significant ANOVA result does not indicate that a factor has no effect. Equivalence testing for ANOVAs addresses this directly by asking: can we reject the hypothesis that the partial $\eta^2$ exceeds a specified bound? +Many researchers use ANOVA as an omnibus test before inspecting pairwise comparisons, particularly in factorial designs. As @Campbell_2021 note, the lack of a significant ANOVA result does not indicate that a factor has no effect. Equivalence testing for ANOVAs addresses this directly by asking whether the variance attributable to a factor is small enough to be declared negligible. + +## Partial Eta-Squared and the Equivalence Hypothesis + +The effect size used throughout this section is partial eta-squared, + +$$ +\eta^2_p = \frac{SS_{\text{effect}}}{SS_{\text{effect}} + SS_{\text{error}}} +$$ + +the proportion of variance attributable to an effect once the variance attributable to the other effects in the model has been removed. It is not the same quantity as $\eta^2$, which divides by the total sum of squares; the two coincide only in a one-way design. In within-subjects designs, $SS_{\text{error}}$ refers to the error term against which that particular effect is tested, so each effect in a factorial model carries its own denominator. + +Because $\eta^2_p$ cannot be negative, there is no lower bound to test against, and the procedure is therefore *not* a two one-sided tests procedure despite the framing used elsewhere in this paper. It is a single one-sided test, which @Campbell_2021 describe as an omnibus non-inferiority test: + +$$ +H_0: \eta^2_p \geq \Delta \qquad \text{versus} \qquad H_1: \eta^2_p < \Delta +$$ + +Rejecting $H_0$ supports the claim that the effect in question explains less than $\Delta$ of the variance available to it. Two limits on that conclusion are worth stating plainly. First, it is a statement about an effect as a whole rather than about any particular contrast within it: an omnibus effect can be negligible while a specific pairwise comparison nested inside it is not, so equivalence at the level of the *F*-test does not license equivalence claims about individual group differences. Second, as with any equivalence test, failing to reject $H_0$ is not evidence that the effect is large --- it means only that the data remain compatible with an effect at or beyond the bound. + +The test itself is computed from the cumulative distribution function of the non-central *F* distribution. @Campbell_2021 derived the *p*-value for a one-way ANOVA with $J$ groups and $N$ total observations as + +$$ +p = p_F \left( F; \space J-1, \space N-J, \space \frac{N \cdot \Delta}{1 - \Delta} \right) +$$ + +and evaluated the Type I error and power of the procedure by simulation. TOSTER generalizes the non-centrality parameter so that the same logic extends to multi-factor, factorial, and within-subjects designs, + +$$ +\lambda_{eq} = \frac{\Delta}{1 - \Delta} \cdot (df_1 + df_2 + 1), \qquad p_{eq} = p_F(F; \space df_1, \space df_2, \space \lambda_{eq}) +$$ + +where $df_1$ and $df_2$ are the numerator and denominator degrees of freedom for the effect being tested. For a one-way design $df_1 + df_2 + 1 = (J-1) + (N-J) + 1 = N$, so the generalization reduces exactly to the expression given by @Campbell_2021. A fuller derivation is provided in the [package documentation](https://aaroncaldwell.us/TOSTERpkg/articles/the_ftestTOSTER.html). + +The equivalence test inherits the standard ANOVA assumptions, which should be checked on the fitted model itself rather than on the equivalence output. The `performance::check_model()` function provides a comprehensive visual diagnostic suite for most linear model objects in R in a single call; the online supplement demonstrates its use. + +## Choosing a Bound on Partial Eta-Squared -The equivalence test uses the non-central *F* distribution. The non-centrality parameter is computed from the equivalence bound and the degrees of freedom, and the *p*-value is obtained from the cumulative distribution function of the non-central *F*-value. I have extended the approach of @Campbell_2021 --- who proposed equivalence testing for one-way ANOVAs --- to accommodate multi-factor and factorial designs by generalizing the non-centrality parameter calculation. The full derivation is described in the [package documentation](https://aaroncaldwell.us/TOSTERpkg/articles/the_ftestTOSTER.html); here we demonstrate usage. +Bounds on $\eta^2_p$ are harder to justify than bounds on the original measurement scale, and the difficulty is worth confronting before running the test. -The equivalence test inherits the standard ANOVA assumptions which should be checked on the fitted model itself rather than on the equivalence output. The `performance::check_model()` function provides a comprehensive visual diagnostic suite for most linear model objects in R on a single call; the online supplement demonstrates its use. +The first problem is that $\eta^2_p$ is not comparable across designs. Its denominator depends on which other effects appear in the model and, in within-subjects designs, on the error term against which the effect is tested. A partial eta-squared of 0.06 for a one-way between-subjects factor and a partial eta-squared of 0.06 for an interaction in a repeated-measures model are not estimates of the same quantity. A bound therefore cannot simply be borrowed from a published study unless that study used the same design and the same model. + +The second problem is that a proportion of variance has little direct substantive meaning. Unlike a mean difference expressed in the units of the outcome, there is rarely an anchor-based or cost-benefit argument that delivers a number on this scale directly. Researchers who find Cohen's $f$ more intuitive can move between the two with $f^2 = \eta^2_p / (1 - \eta^2_p)$, and the `power_eq_f` function can be used to determine the degrees of freedom required to achieve a given power for a specified bound, or the power attainable for a given design. The choosing-bounds section of the package's F-test vignette discusses the problem in more detail. + +The bound of $\Delta = 0.35$ used in both examples below is deliberately large. It was chosen so that the demonstrations produce interpretable output rather than because it represents a defensible threshold, and it should not be read as a default. ## Example with Summary Statistics @@ -893,7 +963,15 @@ Based on these results, there is a significant effect of "spray" and the differe ## Example with R ANOVA Objects -The `equ_anova` function accepts objects from `stats::aov`, `car::Anova`, and `afex::aov_car`, avoiding the need to extract summary statistics manually. In the example below, the traditional significance test p-value is located in the "p.null" column while to right of that value is the estimated standardized effect size (pes which stands for partial eta-squared), equivalence bound (eqb), and equivalence test p-value (p.equ). +The `equ_anova` function accepts objects from `stats::aov`, `car::Anova`, and `afex::aov_car`, avoiding the need to extract summary statistics manually. It returns the familiar ANOVA table with the equivalence test appended: + +- `effect`, `df1`, `df2`, `F.value` --- the effect being tested and the components of the traditional *F*-test. +- `p.null` --- the *p*-value for the traditional nil-hypothesis test. +- `pes` --- the estimated partial eta-squared for that effect. +- `eqbound` --- the equivalence bound that was applied. +- `p.equ` --- the *p*-value for the equivalence test. + +Because each effect in a within-subjects design is tested against its own error term, each row carries its own $df_2$ and therefore its own denominator for `pes`. Note also that an equivalence test is reported for the intercept; this is rarely a hypothesis of interest and can generally be ignored. ```{r} # Example using a purely within-subjects design diff --git a/papers/Avocado_Update/campbell_lakens_supplement.zip b/papers/Avocado_Update/campbell_lakens_supplement.zip new file mode 100644 index 0000000000000000000000000000000000000000..043b993e32f47c0c6f575a4bb04002b361a4119a GIT binary patch literal 613335 zcmZ^p(~>9(kZjw%+qP}nwr$(CZQHhO+qP}nbMD)XiI}RF`h(2MmGV-+Ajkm!{@XQw zVjY10Z?ONnjV+uUP0S1&jrE)?Y+S4joGt8Z^_-ktj6G--chqd`wpihRsed5{WdIoq zbR-=73F>)~*Ey>NVhVS_z(dXE5pT*Dw1hWW7@GTh+={KGT90b=LTv=6!#mGn$Gi@3 zS}Hv?E5bc~S;&IB^X*bK)JHr@FSzL}GU%GfXHnaTl^uZSmc&Dx=S!Y2M^^($ElU#n zk5FgUGj$iTq~?BtTwp4$9DXNnc9z9`A@16hTcyn^i$Ah2*>y-4slB8c;~Y58J!!dv zlO^erB+D8HRfseucDs`+43BTN+8j_TT$fgznFBfpy{fuAT{;MS+y zXO=tTHI}uK9fm`xG{6X+MDn0;=oBx8rf*(ADK8xa!Zb@o3iqqdx&jCHA+fb#<&+wr zy4Ww-ymtu4i{0au;$xZH&IW6X*iF1Z@Vt-~jWL|x?vA>nl{fhH@@E5CdDTZc(&G(D 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+434,B,28.00,7.00,3.00,Male,1,Baseline,21 +435,B,34.00,1.00,1.00,Male,1,Baseline,1 +436,C,18.00,0.00,0.00,Female,0,Baseline,0 +437,A,21.00,4.00,2.00,Female,1,Baseline,8 +438,B,24.00,3.00,9.00,Male,1,Baseline,27 +439,C,22.00,4.00,2.00,Male,1,Baseline,8 +440,C,19.00,5.00,3.00,Female,1,Baseline,15 +441,B,22.00,3.00,5.00,Female,1,Baseline,15 +442,A,23.00,17.0,1.00,Male,1,Baseline,17 +443,B,44.00,27.0,1.00,Female,1,Baseline,27 +444,C,20.00,4.00,10.0,Male,1,Baseline,40 +445,A,30.00,4.00,1.00,Male,1,Baseline,4 +446,A,19.00,1.00,5.00,Female,1,Baseline,5 +448,A,18.00,1.00,1.00,Female,1,Baseline,1 +449,C,19.00,2.00,4.00,Male,1,Baseline,8 +451,C,23.00,3.00,15.0,Male,1,Baseline,45 +452,B,24.00,10.0,2.00,Female,1,Baseline,20 +453,A,22.00,6.00,4.00,Female,1,Baseline,24 +454,B,23.00,2.00,10.0,Female,1,Baseline,20 +455,A,23.00,0.00,0.00,Female,0,Baseline,0 +456,A,20.00,6.00,9.00,Female,1,Baseline,54 +457,C,20.00,1.00,3.00,Female,1,Baseline,3 +458,A,25.00,22.0,4.00,Female,1,Baseline,88 +459,A,22.00,4.00,8.00,Female,1,Baseline,32 +460,C,49.00,0.00,0.00,Female,0,Baseline,0 +462,B,22.00,4.00,3.00,Female,1,Baseline,12 +463,B,29.00,12.0,2.00,Female,1,Baseline,24 +464,A,19.00,0.00,0.00,Female,0,Baseline,0 +465,B,21.00,3.00,14.0,Male,1,Baseline,42 +467,C,23.00,0.00,0.00,Female,0,Baseline,0 +468,A,21.00,4.00,8.00,Male,1,Baseline,32 +469,C,19.00,1.00,2.00,Male,1,Baseline,2 +470,A,24.00,3.00,2.00,Male,1,Baseline,6 +471,B,21.00,9.00,9.00,Male,1,Baseline,81 +472,A,19.00,5.00,2.00,Female,1,Baseline,10 +473,B,21.00,1.00,6.00,Female,1,Baseline,6 +474,B,36.00,1.00,1.00,Female,1,Baseline,1 +475,C,56.00,1.00,1.00,Male,1,Baseline,1 +476,A,23.00,0.00,0.00,Female,0,Baseline,0 +477,B,18.00,1.00,1.00,Female,1,Baseline,1 +478,C,19.00,9.00,10.0,Male,1,Baseline,90 +479,A,26.00,4.00,4.00,Male,1,Baseline,16 +480,C,41.00,1.00,1.00,Female,1,Baseline,1 +481,B,42.00,1.00,1.00,Male,1,Baseline,1 +482,C,27.00,4.00,2.00,Female,1,Baseline,8 +483,A,51.00,14.0,2.00,Male,1,Baseline,28 +484,B,23.00,6.00,2.00,Female,1,Baseline,12 +485,B,19.00,2.00,4.00,Male,1,Baseline,8 +486,C,21.00,3.00,1.00,Female,1,Baseline,3 +487,A,28.00,4.00,1.00,Female,1,Baseline,4 +489,C,28.00,1.00,1.00,Female,1,Baseline,1 +490,C,22.00,14.0,4.00,Female,1,Baseline,56 +491,C,21.00,3.00,1.00,Female,1,Baseline,3 +492,A,21.00,2.00,12.0,Female,1,Baseline,24 +493,B,20.00,2.00,2.00,Female,1,Baseline,4 +494,C,21.00,0.00,0.00,Female,0,Baseline,0 +495,C,35.00,12.0,2.00,Male,1,Baseline,24 +496,A,22.00,4.00,2.00,Female,1,Baseline,8 +497,C,20.00,4.00,2.00,Female,1,Baseline,8 +498,B,33.00,0.00,0.00,Female,0,Baseline,0 +500,B,32.00,1.00,2.00,Male,1,Baseline,2 +501,A,35.00,3.00,1.00,Male,1,Baseline,3 +502,C,45.00,0.00,0.00,Male,0,Baseline,0 +503,B,23.00,2.00,2.00,Male,1,Baseline,4 +504,A,22.00,10.0,3.00,Male,1,Baseline,30 +505,C,19.00,1.00,8.00,Female,1,Baseline,8 +506,B,20.00,6.00,8.00,Female,1,Baseline,48 +508,B,19.00,2.00,7.00,Female,1,Baseline,14 +509,C,18.00,6.00,8.00,Female,1,Baseline,48 +510,B,37.00,8.00,2.00,Female,1,Baseline,16 +511,A,34.00,14.0,1.00,Female,1,Baseline,14 +512,C,29.00,1.00,4.00,Female,1,Baseline,4 +513,A,37.00,6.00,1.00,Female,1,Baseline,6 +514,B,36.00,10.0,1.00,Male,1,Baseline,10 +515,B,19.00,5.00,8.00,Female,1,Baseline,40 +516,A,24.00,6.00,1.00,Female,1,Baseline,6 +517,A,22.00,3.00,1.00,Female,1,Baseline,3 +518,C,25.00,19.0,1.00,Female,1,Baseline,19 +519,A,17.00,4.00,3.00,Female,1,Baseline,12 +520,A,19.00,4.00,1.00,Male,1,Baseline,4 +521,B,42.00,6.00,1.00,Female,1,Baseline,6 +522,B,34.00,0.00,0.00,Female,0,Baseline,0 +523,A,22.00,2.00,7.00,Male,1,Baseline,14 +524,C,20.00,1.00,3.00,Female,1,Baseline,3 +525,A,39.00,3.00,1.00,Female,1,Baseline,3 +526,C,49.00,4.00,2.00,Female,1,Baseline,8 +528,C,20.00,2.00,1.00,Female,1,Baseline,2 +529,C,47.00,8.00,2.00,Male,1,Baseline,16 +530,A,25.00,0.00,0.00,Male,0,Baseline,0 +531,B,20.00,6.00,4.00,Male,1,Baseline,24 +532,B,20.00,0.00,0.00,Female,0,Baseline,0 +533,C,21.00,4.00,7.00,Female,1,Baseline,28 +534,A,61.00,8.00,2.00,Female,1,Baseline,16 +536,C,20.00,1.00,1.00,Male,1,Baseline,1 +537,C,18.00,3.00,7.00,Female,1,Baseline,21 +538,A,19.00,12.0,11.0,Female,1,Baseline,132 +539,B,42.00,1.00,1.00,Female,1,Baseline,1 +541,A,44.00,1.00,1.00,Female,1,Baseline,1 +542,A,18.00,4.00,8.00,Female,1,Baseline,32 +543,A,19.00,1.00,2.00,Male,1,Baseline,2 +544,B,22.00,0.00,0.00,Female,0,Baseline,0 +545,A,18.00,0.00,0.00,Female,0,Baseline,0 +546,C,18.00,3.00,5.00,Male,1,Baseline,15 +547,C,26.00,3.00,1.00,Female,1,Baseline,3 +549,B,21.00,0.00,0.00,Male,0,Baseline,0 +550,C,22.00,0.00,0.00,Male,0,Baseline,0 +552,B,21.00,5.00,9.00,Male,1,Baseline,45 +553,C,19.00,10.0,3.00,Male,1,Baseline,30 +554,C,19.00,0.00,0.00,Female,0,Baseline,0 +555,A,24.00,4.00,2.00,Female,1,Baseline,8 +556,C,21.00,0.00,0.00,Female,0,Baseline,0 +557,A,47.00,12.0,2.00,Female,1,Baseline,24 +558,B,25.00,0.00,0.00,Female,0,Baseline,0 +559,C,19.00,0.00,0.00,Female,0,Baseline,0 +560,C,42.00,0.00,0.00,Female,0,Baseline,0 +561,B,30.00,7.00,2.00,Female,1,Baseline,14 +562,B,24.00,5.00,3.00,Female,1,Baseline,15 +563,B,27.00,0.00,0.00,Male,0,Baseline,0 +564,B,19.00,10.0,12.0,Female,1,Baseline,120 +565,C,46.00,0.00,0.00,Female,0,Baseline,0 +566,A,26.00,3.00,1.00,Female,1,Baseline,3 +567,C,23.00,3.00,1.00,Female,1,Baseline,3 +568,B,26.00,3.00,1.00,Male,1,Baseline,3 +569,A,22.00,0.00,0.00,Female,0,Baseline,0 +570,B,20.00,3.00,1.00,Female,1,Baseline,3 +571,A,19.00,4.00,4.00,Female,1,Baseline,16 +572,A,20.00,1.00,9.00,Female,1,Baseline,9 +573,A,20.00,2.00,1.00,Female,1,Baseline,2 +574,B,22.00,10.0,1.00,Male,1,Baseline,10 +576,A,19.00,4.00,6.00,Male,1,Baseline,24 +577,A,19.00,6.00,8.00,Female,1,Baseline,48 +579,B,20.00,6.00,4.00,Male,1,Baseline,24 +580,C,21.00,4.00,3.00,Female,1,Baseline,12 +583,A,21.00,5.00,2.00,Female,1,Baseline,10 +584,C,25.00,3.00,2.00,Male,1,Baseline,6 +587,B,30.00,1.00,1.00,Female,1,Baseline,1 +588,A,22.00,4.00,3.00,Female,1,Baseline,12 +589,A,50.00,16.0,1.00,Male,1,Baseline,16 +591,A,35.00,1.00,4.00,Male,1,Baseline,4 +592,C,26.00,4.00,2.00,Male,1,Baseline,8 +593,B,20.00,7.00,7.00,Female,1,Baseline,49 +595,C,41.00,27.0,3.00,Female,1,Baseline,81 +596,A,20.00,4.00,6.00,Female,1,Baseline,24 +597,B,24.00,3.00,2.00,Female,1,Baseline,6 +598,C,29.00,5.00,6.00,Female,1,Baseline,30 +601,A,21.00,4.00,3.00,Female,1,Baseline,12 +603,A,19.00,0.00,0.00,Female,0,Baseline,0 +604,C,63.00,0.00,0.00,Female,0,Baseline,0 +605,C,23.00,2.00,8.00,Female,1,Baseline,16 +606,A,26.00,4.00,6.00,Female,1,Baseline,24 +608,A,23.00,2.00,2.00,Female,1,Baseline,4 +609,C,31.00,5.00,1.00,Male,1,Baseline,5 +610,A,21.00,6.00,4.00,Female,1,Baseline,24 +611,B,20.00,1.00,1.00,Female,1,Baseline,1 +612,A,19.00,0.00,0.00,Male,0,Baseline,0 +613,A,24.00,0.00,0.00,Female,0,Baseline,0 +614,B,20.00,6.00,5.00,Male,1,Baseline,30 +615,C,24.00,4.00,7.00,Male,1,Baseline,28 +616,C,22.00,2.00,3.00,Female,1,Baseline,6 +619,B,22.00,4.00,6.00,Female,1,Baseline,24 +620,A,19.00,5.00,12.0,Male,1,Baseline,60 +621,A,51.00,28.0,3.00,Female,1,Baseline,84 +622,A,18.00,5.00,4.00,Female,1,Baseline,20 +623,A,19.00,0.00,0.00,Female,0,Baseline,0 +624,A,34.00,0.00,0.00,Male,0,Baseline,0 +625,B,20.00,2.00,6.00,Female,1,Baseline,12 +628,B,24.00,8.00,1.00,Male,1,Baseline,8 +629,A,19.00,6.00,8.00,Female,1,Baseline,48 +630,B,19.00,0.00,0.00,Male,0,Baseline,0 +631,A,19.00,0.00,0.00,Female,0,Baseline,0 +632,B,22.00,1.00,10.0,Female,1,Baseline,10 +633,C,25.00,3.00,2.00,Male,1,Baseline,6 +634,A,20.00,4.00,5.00,Female,1,Baseline,20 +635,A,30.00,6.00,10.0,Female,1,Baseline,60 +636,A,22.00,5.00,2.00,Female,1,Baseline,10 +637,C,20.00,2.00,8.00,Male,1,Baseline,16 +638,C,22.00,4.00,8.00,Female,1,Baseline,32 +639,A,50.00,6.00,1.00,Female,1,Baseline,6 +640,C,21.00,3.00,3.00,Female,1,Baseline,9 +641,A,21.00,0.00,0.00,Male,0,Baseline,0 +642,B,20.00,2.00,7.00,Female,1,Baseline,14 +643,B,22.00,8.00,6.00,Male,1,Baseline,48 +644,A,20.00,0.00,0.00,Female,0,Baseline,0 +645,A,19.00,7.00,8.00,Female,1,Baseline,56 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+1459,A,24.00,13.0,3.00,Female,1,Baseline,39 +1460,A,21.00,8.00,7.00,Female,1,Baseline,56 +1461,A,22.00,11.0,7.00,Male,1,Baseline,77 +1463,A,22.00,2.00,2.00,Male,1,Baseline,4 +1464,B,19.00,7.00,7.00,Female,1,Baseline,49 +1465,B,18.00,0.00,0.00,Male,0,Baseline,0 +1466,C,21.00,2.00,3.00,Female,1,Baseline,6 +1467,B,21.00,0.00,0.00,Female,0,Baseline,0 +1468,B,21.00,7.00,4.00,Female,1,Baseline,28 +1469,C,19.00,5.00,6.00,Female,1,Baseline,30 +1470,B,30.00,0.00,0.00,Female,0,Baseline,0 +1471,A,27.00,15.0,1.00,Male,1,Baseline,15 +1472,C,18.00,4.00,1.00,Male,1,Baseline,4 +1473,A,29.00,7.00,2.00,Female,1,Baseline,14 +1474,A,21.00,5.00,1.00,Female,1,Baseline,5 +1475,A,19.00,2.00,4.00,Female,1,Baseline,8 +1476,B,22.00,5.00,8.00,Female,1,Baseline,40 +1479,B,21.00,3.00,2.00,Female,1,Baseline,6 +1480,C,31.00,13.0,6.00,Male,1,Baseline,78 +1481,C,20.00,6.00,6.00,Female,1,Baseline,36 +1482,A,24.00,5.00,3.00,Female,1,Baseline,15 +1483,C,22.00,2.00,1.00,Male,1,Baseline,2 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+1509,C,23.00,8.00,3.00,Male,1,Baseline,24 +1510,C,20.00,1.00,1.00,Female,1,Baseline,1 +1512,A,22.00,7.00,2.00,Male,1,Baseline,14 +1513,A,17.00,0.00,0.00,Male,0,Baseline,0 +1514,A,23.00,0.00,0.00,Female,0,Baseline,0 +1515,B,22.00,5.00,2.00,Male,1,Baseline,10 +1516,C,19.00,2.00,5.00,Male,1,Baseline,10 +1517,A,22.00,2.00,3.00,Female,1,Baseline,6 +1519,B,25.00,0.00,0.00,Male,0,Baseline,0 +1520,C,20.00,6.00,7.00,Female,1,Baseline,42 +1521,B,19.00,3.00,5.00,Female,1,Baseline,15 +1522,A,18.00,2.00,1.00,Male,1,Baseline,2 +1523,B,21.00,7.00,6.00,Female,1,Baseline,42 +1524,A,47.00,4.00,1.00,Female,1,Baseline,4 +1525,B,21.00,1.00,1.00,Male,1,Baseline,1 +1526,B,20.00,4.00,5.00,Male,1,Baseline,20 +1527,C,20.00,1.00,1.00,Male,1,Baseline,1 +1528,A,18.00,7.00,10.0,Female,1,Baseline,70 +1529,B,22.00,10.0,15.0,Male,1,Baseline,150 +1530,B,23.00,3.00,2.00,Male,1,Baseline,6 +1531,C,21.00,10.0,4.00,Female,1,Baseline,40 +1532,B,37.00,4.00,4.00,Male,1,Baseline,16 +1533,A,47.00,4.00,1.00,Female,1,Baseline,4 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+1559,A,19.00,6.00,1.00,Male,1,Baseline,6 +1560,A,21.00,8.00,15.0,Male,1,Baseline,120 +1561,B,29.00,4.00,6.00,Male,1,Baseline,24 +1564,B,19.00,1.00,3.00,Male,1,Baseline,3 +1566,C,21.00,7.00,5.00,Female,1,Baseline,35 +1567,B,23.00,20.0,2.00,Male,1,Baseline,40 +1568,A,21.00,2.00,4.00,Female,1,Baseline,8 +1569,B,18.00,1.00,1.00,Female,1,Baseline,1 +1570,A,29.00,3.00,2.00,Female,1,Baseline,6 +1571,A,42.00,2.00,3.00,Female,1,Baseline,6 +1572,A,20.00,3.00,8.00,Female,1,Baseline,24 +1573,C,22.00,8.00,2.00,Female,1,Baseline,16 +1574,C,21.00,3.00,12.0,Male,1,Baseline,36 +1576,A,36.00,3.00,9.00,Female,1,Baseline,27 +1578,A,24.00,8.00,1.00,Female,1,Baseline,8 +1579,C,35.00,0.00,0.00,Female,0,Baseline,0 +1580,C,21.00,4.00,2.00,Male,1,Baseline,8 +1581,C,20.00,0.00,0.00,Female,0,Baseline,0 +1582,C,19.00,5.00,8.00,Female,1,Baseline,40 +1583,A,47.00,1.00,3.00,Female,1,Baseline,3 +1584,B,19.00,4.00,1.00,Male,1,Baseline,4 +1585,B,19.00,5.00,9.00,Female,1,Baseline,45 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+2422,B,28.00,1.00,2.00,Female,1,Baseline,2 +2423,A,22.00,1.00,2.00,Female,1,Baseline,2 +2424,B,23.00,3.00,6.00,Male,1,Baseline,18 +2425,B,19.00,0.00,0.00,Female,0,Baseline,0 +2426,C,20.00,8.00,2.00,Female,1,Baseline,16 +2427,B,27.00,3.00,1.00,Male,1,Baseline,3 +2428,B,20.00,8.00,14.0,Male,1,Baseline,112 +2429,A,36.00,0.00,0.00,Female,0,Baseline,0 +2430,B,20.00,9.00,6.00,Female,1,Baseline,54 +2431,A,23.00,12.0,1.00,Female,1,Baseline,12 +2432,A,21.00,8.00,10.0,Male,1,Baseline,80 +2433,C,21.00,7.00,12.0,Male,1,Baseline,84 +2434,C,22.00,3.00,1.00,Female,1,Baseline,3 +2436,A,20.00,5.00,3.00,Female,1,Baseline,15 +2437,C,23.00,4.00,2.00,Female,1,Baseline,8 +2438,B,31.00,8.00,1.00,Female,1,Baseline,8 +2439,C,18.00,6.00,8.00,Male,1,Baseline,48 +2440,A,19.00,4.00,6.00,Male,1,Baseline,24 +2441,C,21.00,0.00,0.00,Female,0,Baseline,0 +2442,B,28.00,3.00,2.00,Female,1,Baseline,6 +2443,A,21.00,0.00,0.00,Female,0,Baseline,0 +2444,C,22.00,1.00,3.00,Female,1,Baseline,3 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+2473,B,18.00,14.0,14.0,Male,1,Baseline,196 +2475,B,18.00,9.00,9.00,Male,1,Baseline,81 +2476,B,23.00,6.00,2.00,Male,1,Baseline,12 +2477,C,20.00,1.00,6.00,Male,1,Baseline,6 +2478,C,25.00,14.0,2.00,Female,1,Baseline,28 +2479,A,19.00,4.00,8.00,Female,1,Baseline,32 +2480,C,22.00,15.0,6.00,Male,1,Baseline,90 +2482,C,19.00,5.00,18.0,Male,1,Baseline,90 +2483,C,21.00,2.00,5.00,Female,1,Baseline,10 +2484,B,22.00,1.00,8.00,Female,1,Baseline,8 +2485,B,21.00,6.00,16.0,Male,1,Baseline,96 +2486,C,26.00,2.00,1.00,Male,1,Baseline,2 +2487,B,48.00,3.00,2.00,Female,1,Baseline,6 +2488,A,23.00,3.00,4.00,Female,1,Baseline,12 +2489,C,28.00,0.00,0.00,Female,0,Baseline,0 +2490,B,25.00,4.00,2.00,Male,1,Baseline,8 +2491,C,20.00,5.00,13.0,Female,1,Baseline,65 +2492,A,18.00,4.00,5.00,Female,1,Baseline,20 +2493,B,29.00,5.00,1.00,Female,1,Baseline,5 +2494,C,22.00,0.00,0.00,Female,0,Baseline,0 +2495,C,22.00,4.00,4.00,Female,1,Baseline,16 +2496,C,19.00,2.00,4.00,Female,1,Baseline,8 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+4260,B,23.00,8.00,2.00,Male,1,Baseline,16 +4261,A,19.00,3.00,3.00,Female,1,Baseline,9 +4262,A,18.00,6.00,5.00,Female,1,Baseline,30 +4263,A,21.00,5.00,5.00,Female,1,Baseline,25 +4264,A,29.00,6.00,2.00,Female,1,Baseline,12 +4265,B,21.00,2.00,1.00,Female,1,Baseline,2 +4266,C,20.00,1.00,4.00,Female,1,Baseline,4 +4267,B,18.00,0.00,0.00,Male,0,Baseline,0 +4268,A,20.00,3.00,4.00,Female,1,Baseline,12 +4269,B,26.00,3.00,4.00,Female,1,Baseline,12 +4270,C,20.00,14.0,4.00,Male,1,Baseline,56 +4271,B,19.00,0.00,0.00,Male,0,Baseline,0 +4272,A,25.00,6.00,2.00,Female,1,Baseline,12 +4273,A,21.00,8.00,5.00,Female,1,Baseline,40 +4274,B,22.00,7.00,1.00,Male,1,Baseline,7 +4275,C,21.00,6.00,5.00,Female,1,Baseline,30 +4276,C,49.00,1.00,2.00,Female,1,Baseline,2 +4277,C,26.00,6.00,2.00,Male,1,Baseline,12 +4278,B,24.00,5.00,1.00,Female,1,Baseline,5 +4280,B,22.00,3.00,2.00,Female,1,Baseline,6 +4282,B,32.00,8.00,1.00,Male,1,Baseline,8 +4283,B,21.00,2.00,7.00,Female,1,Baseline,14 +4284,B,21.00,4.00,6.00,Female,1,Baseline,24 +4285,C,20.00,2.00,1.00,Male,1,Baseline,2 +4286,B,37.00,6.00,1.00,Female,1,Baseline,6 +4287,A,30.00,1.00,1.00,Male,1,Baseline,1 +4288,B,54.00,2.00,2.00,Female,1,Baseline,4 +4289,C,48.00,25.0,1.00,Male,1,Baseline,25 +4290,B,21.00,2.00,7.00,Female,1,Baseline,14 +4291,B,22.00,6.00,8.00,Male,1,Baseline,48 +4292,A,20.00,4.00,18.0,Male,1,Baseline,72 +4293,B,21.00,6.00,3.00,Female,1,Baseline,18 +4294,A,22.00,3.00,1.00,Female,1,Baseline,3 +4295,A,23.00,4.00,10.0,Male,1,Baseline,40 +4296,A,30.00,2.00,1.00,Female,1,Baseline,2 +4297,B,43.00,4.00,1.00,Female,1,Baseline,4 +4298,C,20.00,7.00,11.0,Male,1,Baseline,77 +4299,C,21.00,4.00,4.00,Female,1,Baseline,16 +4300,A,22.00,3.00,4.00,Male,1,Baseline,12 +4301,A,19.00,6.00,5.00,Female,1,Baseline,30 +4302,C,18.00,4.00,5.00,Female,1,Baseline,20 +4304,A,32.00,5.00,2.00,Female,1,Baseline,10 +4305,C,19.00,0.00,0.00,Female,0,Baseline,0 +4306,A,31.00,0.00,0.00,Female,0,Baseline,0 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+4354,B,58.00,8.00,6.00,Male,1,Baseline,48 +4355,C,19.00,2.00,2.00,Female,1,Baseline,4 +4356,B,20.00,0.00,0.00,Male,0,Baseline,0 +4357,C,26.00,2.00,2.00,Female,1,Baseline,4 +4358,C,25.00,5.00,2.00,Female,1,Baseline,10 +4359,C,19.00,4.00,5.00,Female,1,Baseline,20 +4360,B,20.00,6.00,5.00,Female,1,Baseline,30 +4361,A,22.00,15.0,2.00,Female,1,Baseline,30 +4362,C,21.00,0.00,0.00,Male,0,Baseline,0 +4363,B,26.00,3.00,1.00,Male,1,Baseline,3 +4364,C,21.00,0.00,0.00,Male,0,Baseline,0 +4365,B,22.00,4.00,5.00,Female,1,Baseline,20 +4366,A,18.00,0.00,0.00,Male,0,Baseline,0 +4367,B,20.00,4.00,5.00,Female,1,Baseline,20 +4368,C,24.00,12.0,1.00,Male,1,Baseline,12 +4369,B,29.00,0.00,0.00,Female,0,Baseline,0 +4370,B,21.00,3.00,1.00,Female,1,Baseline,3 +4371,B,56.00,10.0,2.00,Female,1,Baseline,20 +4372,B,23.00,4.00,2.00,Male,1,Baseline,8 +4374,B,22.00,10.0,3.00,Male,1,Baseline,30 +4375,A,18.00,0.00,0.00,Female,0,Baseline,0 +4376,C,21.00,7.00,8.00,Female,1,Baseline,56 +4378,B,20.00,10.0,10.0,Male,1,Baseline,100 +4379,A,22.00,10.0,8.00,Female,1,Baseline,80 +4381,C,19.00,0.00,0.00,Female,0,Baseline,0 +4382,A,21.00,0.00,0.00,Female,0,Baseline,0 +4383,B,21.00,0.00,0.00,Male,0,Baseline,0 +4384,B,22.00,3.00,1.00,Female,1,Baseline,3 +4385,B,21.00,8.00,4.00,Female,1,Baseline,32 +4386,C,22.00,0.00,0.00,Female,0,Baseline,0 +4387,A,19.00,4.00,6.00,Male,1,Baseline,24 +4388,A,20.00,12.0,10.0,Male,1,Baseline,120 +4389,B,23.00,2.00,1.00,Male,1,Baseline,2 +4390,A,22.00,1.00,6.00,Male,1,Baseline,6 +4391,B,20.00,4.00,10.0,Female,1,Baseline,40 +4392,B,18.00,0.00,0.00,Female,0,Baseline,0 +4393,B,22.00,1.00,1.00,Male,1,Baseline,1 +4394,B,25.00,2.00,4.00,Female,1,Baseline,8 +4395,C,20.00,2.00,3.00,Female,1,Baseline,6 +4396,A,21.00,8.00,5.00,Female,1,Baseline,40 +4397,B,21.00,4.00,8.00,Female,1,Baseline,32 +4398,C,18.00,8.00,1.00,Male,1,Baseline,8 +4399,B,19.00,0.00,0.00,Female,0,Baseline,0 +4400,A,18.00,0.00,0.00,Male,0,Baseline,0 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+4550,C,22.00,6.00,3.00,Female,1,Baseline,18 +4551,C,18.00,3.00,8.00,Female,1,Baseline,24 +4552,B,21.00,3.00,2.00,Male,1,Baseline,6 +4553,B,21.00,0.00,0.00,Female,0,Baseline,0 +4554,C,19.00,3.00,2.00,Male,1,Baseline,6 +4555,C,19.00,1.00,2.00,Female,1,Baseline,2 +4556,A,23.00,4.00,2.00,Female,1,Baseline,8 +4557,C,22.00,1.00,1.00,Female,1,Baseline,1 +4558,B,23.00,0.00,0.00,Female,0,Baseline,0 +4559,B,22.00,14.0,2.00,Male,1,Baseline,28 +4560,A,23.00,12.0,8.00,Male,1,Baseline,96 +4561,A,21.00,6.00,8.00,Male,1,Baseline,48 +4562,C,18.00,4.00,3.00,Female,1,Baseline,12 +4563,C,19.00,4.00,14.0,Male,1,Baseline,56 +4565,A,18.00,0.00,0.00,Female,0,Baseline,0 +4567,A,52.00,0.00,0.00,Male,0,Baseline,0 +4568,A,23.00,28.0,10.0,Male,1,Baseline,280 +4569,C,21.00,8.00,20.0,Male,1,Baseline,160 +4570,A,18.00,0.00,0.00,Female,0,Baseline,0 +4571,C,27.00,0.00,0.00,Female,0,Baseline,0 +4572,C,21.00,7.00,3.00,Female,1,Baseline,21 +4573,B,32.00,0.00,0.00,Female,0,Baseline,0 +4574,B,23.00,7.00,2.00,Female,1,Baseline,14 +4575,A,21.00,0.00,0.00,Female,0,Baseline,0 +4576,A,27.00,5.00,3.00,Male,1,Baseline,15 +4577,C,20.00,2.00,4.00,Male,1,Baseline,8 +4578,B,23.00,2.00,2.00,Male,1,Baseline,4 +4579,B,24.00,0.00,0.00,Female,0,Baseline,0 +4580,A,19.00,0.00,0.00,Female,0,Baseline,0 +4581,C,19.00,0.00,0.00,Male,0,Baseline,0 +4582,A,21.00,4.00,3.00,Female,1,Baseline,12 +4583,B,33.00,3.00,1.00,Female,1,Baseline,3 +4584,A,23.00,5.00,1.00,Female,1,Baseline,5 +4585,B,18.00,18.0,2.00,Female,1,Baseline,36 +4586,B,23.00,5.00,4.00,Female,1,Baseline,20 +4587,B,18.00,0.00,0.00,Female,0,Baseline,0 +4588,B,20.00,2.00,12.0,Male,1,Baseline,24 +4589,C,22.00,0.00,0.00,Male,0,Baseline,0 +4590,C,21.00,1.00,1.00,Female,1,Baseline,1 +4591,B,21.00,0.00,0.00,Male,0,Baseline,0 +4592,B,24.00,1.00,15.0,Male,1,Baseline,15 +4593,A,20.00,0.00,0.00,Female,0,Baseline,0 +4594,C,22.00,2.00,2.00,Female,1,Baseline,4 +4595,C,18.00,0.00,0.00,Female,0,Baseline,0 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+4625,B,20.00,7.00,5.00,Female,1,Baseline,35 +4626,B,20.00,1.00,2.00,Male,1,Baseline,2 +4627,A,31.00,5.00,1.00,Male,1,Baseline,5 +4629,A,20.00,4.00,5.00,Female,1,Baseline,20 +4630,B,20.00,5.00,1.00,Female,1,Baseline,5 +4631,C,21.00,8.00,2.00,Female,1,Baseline,16 +4632,B,36.00,7.00,2.00,Female,1,Baseline,14 +4633,C,20.00,0.00,0.00,Male,0,Baseline,0 +4634,A,21.00,5.00,5.00,Female,1,Baseline,25 +4635,C,44.00,10.0,1.00,Female,1,Baseline,10 +4636,C,23.00,0.00,0.00,Female,0,Baseline,0 +4637,A,22.00,0.00,0.00,Female,0,Baseline,0 +4638,A,18.00,3.00,2.00,Male,1,Baseline,6 +4639,B,22.00,4.00,4.00,Female,1,Baseline,16 +4640,C,21.00,1.00,15.0,Male,1,Baseline,15 +4641,A,18.00,0.00,0.00,Male,0,Baseline,0 +4642,C,24.00,0.00,0.00,Male,0,Baseline,0 +4643,B,21.00,6.00,5.00,Female,1,Baseline,30 +4644,C,22.00,7.00,10.0,Male,1,Baseline,70 +4645,C,20.00,2.00,3.00,Female,1,Baseline,6 +4646,B,21.00,4.00,10.0,Male,1,Baseline,40 +4647,B,20.00,3.00,2.00,Female,1,Baseline,6 +4648,C,22.00,0.00,0.00,Female,0,Baseline,0 +4649,C,51.00,4.00,1.00,Female,1,Baseline,4 +4650,B,19.00,2.00,4.00,Female,1,Baseline,8 +4651,B,24.00,3.00,3.00,Female,1,Baseline,9 +4652,A,20.00,5.00,9.00,Male,1,Baseline,45 +4653,C,20.00,1.00,3.00,Male,1,Baseline,3 +4655,C,19.00,4.00,7.00,Female,1,Baseline,28 +4656,C,49.00,20.0,1.00,Male,1,Baseline,20 +4657,C,22.00,6.00,2.00,Female,1,Baseline,12 +4658,B,20.00,4.00,3.00,Female,1,Baseline,12 +4660,B,20.00,7.00,1.00,Female,1,Baseline,7 +4661,B,19.00,4.00,2.00,Female,1,Baseline,8 +4662,C,19.00,8.00,7.00,Male,1,Baseline,56 +4663,C,20.00,4.00,3.00,Male,1,Baseline,12 +4664,A,21.00,21.0,3.00,Female,1,Baseline,63 +4665,A,21.00,0.00,0.00,Male,0,Baseline,0 +4667,B,50.00,2.00,1.00,Female,1,Baseline,2 +4669,A,21.00,4.00,6.00,Female,1,Baseline,24 +4670,A,19.00,7.00,6.00,Female,1,Baseline,42 +4672,A,18.00,5.00,1.00,Female,1,Baseline,5 +4673,B,22.00,3.00,4.00,Female,1,Baseline,12 +4674,C,33.00,1.00,4.00,Male,1,Baseline,4 +4675,B,22.00,4.00,4.00,Male,1,Baseline,16 +4676,B,19.00,6.00,2.00,Male,1,Baseline,12 +4677,C,20.00,0.00,0.00,Female,0,Baseline,0 +4678,C,20.00,1.00,1.00,Female,1,Baseline,1 +4679,C,19.00,8.00,4.00,Male,1,Baseline,32 +4680,A,21.00,2.00,6.00,Female,1,Baseline,12 +4681,B,23.00,4.00,1.00,Female,1,Baseline,4 +4682,B,21.00,2.00,4.00,Female,1,Baseline,8 +4683,B,19.00,4.00,8.00,Female,1,Baseline,32 +4684,C,20.00,8.00,15.0,Male,1,Baseline,120 +4685,B,27.00,6.00,3.00,Female,1,Baseline,18 +4687,C,19.00,3.00,2.00,Male,1,Baseline,6 +4688,B,27.00,3.00,1.00,Female,1,Baseline,3 +4689,C,19.00,9.00,3.00,Male,1,Baseline,27 +4691,C,26.00,3.00,2.00,Female,1,Baseline,6 +4692,A,37.00,20.0,2.00,Female,1,Baseline,40 +4693,C,20.00,5.00,3.00,Female,1,Baseline,15 +4694,B,37.00,3.00,4.00,Male,1,Baseline,12 +4695,C,21.00,3.00,6.00,Male,1,Baseline,18 +4696,B,59.00,8.00,2.00,Female,1,Baseline,16 +4698,C,19.00,8.00,12.0,Male,1,Baseline,96 +4699,B,21.00,5.00,3.00,Male,1,Baseline,15 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+5219,A,29.00,8.00,5.00,Male,1,Baseline,40 +5220,B,42.00,0.00,0.00,Female,0,Baseline,0 +5221,B,32.00,0.00,0.00,Male,0,Baseline,0 +5222,C,23.00,12.0,5.00,Female,1,Baseline,60 +5223,B,32.00,10.0,1.00,Male,1,Baseline,10 +5224,A,18.00,0.00,0.00,Male,0,Baseline,0 +5227,B,24.00,2.00,1.00,Female,1,Baseline,2 +5228,C,22.00,1.00,4.00,Female,1,Baseline,4 +5229,B,20.00,2.00,3.00,Male,1,Baseline,6 +5230,B,18.00,2.00,7.00,Female,1,Baseline,14 +5231,C,21.00,5.00,8.00,Female,1,Baseline,40 +5232,A,29.00,8.00,2.00,Male,1,Baseline,16 +5233,B,23.00,3.00,3.00,Female,1,Baseline,9 +5235,C,25.00,10.0,7.00,Male,1,Baseline,70 +5236,C,20.00,5.00,12.0,Male,1,Baseline,60 +5237,C,18.00,1.00,1.00,Female,1,Baseline,1 +5238,B,25.00,0.00,0.00,Male,0,Baseline,0 +5239,B,22.00,0.00,0.00,Female,0,Baseline,0 +5240,B,48.00,8.00,1.00,Female,1,Baseline,8 +5241,C,18.00,10.0,3.00,Female,1,Baseline,30 +5243,C,18.00,3.00,7.00,Female,1,Baseline,21 +5244,C,25.00,10.0,2.00,Male,1,Baseline,20 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+5433,C,24.00,1.00,1.00,Female,1,Baseline,1 +5434,C,21.00,6.00,14.0,Male,1,Baseline,84 +5435,C,23.00,7.00,2.00,Female,1,Baseline,14 +5436,B,20.00,5.00,8.00,Female,1,Baseline,40 +5437,C,25.00,9.00,2.00,Female,1,Baseline,18 +5438,C,26.00,1.00,1.00,Female,1,Baseline,1 +5439,B,24.00,9.00,3.00,Female,1,Baseline,27 +5442,C,21.00,4.00,9.00,Female,1,Baseline,36 +5443,B,20.00,6.00,3.00,Female,1,Baseline,18 +5444,C,21.00,4.00,7.00,Female,1,Baseline,28 +5445,B,21.00,2.00,10.0,Male,1,Baseline,20 +5446,A,18.00,7.00,2.00,Female,1,Baseline,14 +5447,A,55.00,20.0,2.00,Female,1,Baseline,40 +5448,B,19.00,0.00,0.00,Female,0,Baseline,0 +5449,C,44.00,5.00,2.00,Female,1,Baseline,10 +5450,C,22.00,3.00,1.00,Male,1,Baseline,3 +5453,C,46.00,16.0,2.00,Female,1,Baseline,32 +5454,A,19.00,0.00,0.00,Male,0,Baseline,0 +5455,A,22.00,3.00,7.00,Female,1,Baseline,21 +5456,C,26.00,1.00,6.00,Female,1,Baseline,6 +5457,B,26.00,2.00,1.00,Female,1,Baseline,2 +5458,B,31.00,9.00,4.00,Female,1,Baseline,36 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+5486,A,24.00,2.00,1.00,Female,1,Baseline,2 +5487,C,21.00,4.00,6.00,Female,1,Baseline,24 +5488,A,20.00,4.00,1.00,Female,1,Baseline,4 +5489,A,30.00,0.00,0.00,Male,0,Baseline,0 +5490,C,22.00,2.00,4.00,Female,1,Baseline,8 +5491,B,24.00,0.00,0.00,Female,0,Baseline,0 +5492,A,52.00,1.00,1.00,Female,1,Baseline,1 +5493,A,21.00,14.0,3.00,Male,1,Baseline,42 +5494,A,21.00,2.00,2.00,Female,1,Baseline,4 +5495,A,20.00,3.00,7.00,Male,1,Baseline,21 +5496,C,21.00,6.00,2.00,Male,1,Baseline,12 +5497,B,23.00,6.00,8.00,Female,1,Baseline,48 +5498,B,19.00,0.00,0.00,Female,0,Baseline,0 +5499,A,24.00,0.00,0.00,Male,0,Baseline,0 +5500,B,18.00,8.00,6.00,Female,1,Baseline,48 +5501,A,20.00,3.00,4.00,Male,1,Baseline,12 +5502,A,20.00,3.00,6.00,Female,1,Baseline,18 +5503,A,22.00,10.0,8.00,Male,1,Baseline,80 +5504,A,35.00,4.00,2.00,Female,1,Baseline,8 +5505,C,55.00,8.00,2.00,Female,1,Baseline,16 +5506,C,23.00,4.00,1.00,Male,1,Baseline,4 +5507,B,28.00,4.00,3.00,Female,1,Baseline,12 +5510,A,19.00,11.0,10.0,Female,1,Baseline,110 +5511,C,31.00,4.00,3.00,Male,1,Baseline,12 +5512,C,19.00,3.00,4.00,Female,1,Baseline,12 +5513,A,35.00,1.00,4.00,Female,1,Baseline,4 +5514,A,52.00,0.00,0.00,Female,0,Baseline,0 +5515,C,19.00,0.00,0.00,Male,0,Baseline,0 +5516,A,20.00,0.00,0.00,Female,0,Baseline,0 +5517,A,29.00,1.00,2.00,Female,1,Baseline,2 +5518,B,53.00,19.0,3.00,Male,1,Baseline,57 +5520,C,21.00,4.00,3.00,Female,1,Baseline,12 +5521,B,20.00,0.00,0.00,Female,0,Baseline,0 +5522,B,20.00,3.00,10.0,Female,1,Baseline,30 +5523,A,20.00,0.00,0.00,Female,0,Baseline,0 +5524,A,41.00,4.00,1.00,Male,1,Baseline,4 +5525,A,24.00,4.00,6.00,Female,1,Baseline,24 +5527,B,19.00,2.00,5.00,Female,1,Baseline,10 +5528,A,32.00,4.00,2.00,Female,1,Baseline,8 +5529,C,18.00,1.00,2.00,Female,1,Baseline,2 +5530,C,57.00,0.00,0.00,Female,0,Baseline,0 +5531,B,19.00,0.00,0.00,Male,0,Baseline,0 +5532,C,44.00,12.0,2.00,Female,1,Baseline,24 +5533,C,44.00,2.00,1.00,Female,1,Baseline,2 +5537,C,23.00,1.00,1.00,Male,1,Baseline,1 +5538,B,19.00,2.00,7.00,Female,1,Baseline,14 +5540,B,21.00,1.00,1.00,Female,1,Baseline,1 +5541,A,56.00,21.0,1.00,Male,1,Baseline,21 +5542,B,19.00,0.00,0.00,Male,0,Baseline,0 +5543,C,19.00,3.00,1.00,Female,1,Baseline,3 +5544,A,25.00,22.0,2.00,Female,1,Baseline,44 +5546,B,22.00,5.00,18.0,Female,1,Baseline,90 +5547,C,21.00,4.00,9.00,Female,1,Baseline,36 +5548,A,22.00,2.00,5.00,Male,1,Baseline,10 +5549,C,18.00,0.00,0.00,Female,0,Baseline,0 +5550,B,18.00,9.00,6.00,Female,1,Baseline,54 +5553,A,19.00,2.00,1.00,Male,1,Baseline,2 +5555,B,22.00,0.00,0.00,Female,0,Baseline,0 +5556,C,24.00,15.0,1.00,Female,1,Baseline,15 +5557,B,25.00,0.00,0.00,Female,0,Baseline,0 +5558,B,20.00,2.00,4.00,Female,1,Baseline,8 +5559,A,42.00,2.00,4.00,Female,1,Baseline,8 +5560,B,20.00,0.00,0.00,Female,0,Baseline,0 +5562,C,30.00,1.00,1.00,Female,1,Baseline,1 +5563,A,20.00,4.00,2.00,Female,1,Baseline,8 +5565,C,39.00,7.00,1.00,Male,1,Baseline,7 +5566,A,26.00,4.00,2.00,Male,1,Baseline,8 +5568,A,21.00,0.00,0.00,Female,0,Baseline,0 +5569,A,24.00,4.00,2.00,Female,1,Baseline,8 +5571,A,26.00,18.0,10.0,Female,1,Baseline,180 +5572,A,44.00,4.00,2.00,Female,1,Baseline,8 +5574,B,18.00,4.00,14.0,Male,1,Baseline,56 +5575,A,18.00,0.00,0.00,Female,0,Baseline,0 +5576,B,53.00,14.0,1.00,Female,1,Baseline,14 +5578,A,23.00,2.00,10.0,Male,1,Baseline,20 +5579,C,43.00,0.00,0.00,Female,0,Baseline,0 +5580,B,48.00,2.00,1.00,Male,1,Baseline,2 +5581,B,22.00,3.00,1.00,Female,1,Baseline,3 +5582,B,21.00,2.00,2.00,Male,1,Baseline,4 +5583,B,34.00,10.0,8.00,Female,1,Baseline,80 +5584,A,25.00,1.00,1.00,Female,1,Baseline,1 +5585,C,23.00,4.00,4.00,Female,1,Baseline,16 +5586,B,19.00,0.00,0.00,Female,0,Baseline,0 +5587,A,42.00,14.0,1.00,Female,1,Baseline,14 +5588,B,44.00,28.0,1.00,Female,1,Baseline,28 +5589,C,21.00,2.00,2.00,Female,1,Baseline,4 +5590,A,25.00,0.00,0.00,Female,0,Baseline,0 +5591,A,26.00,3.00,2.00,Male,1,Baseline,6 +5592,C,21.00,5.00,2.00,Male,1,Baseline,10 +5593,C,22.00,6.00,8.00,Male,1,Baseline,48 +5594,B,22.00,8.00,8.00,Male,1,Baseline,64 +5595,B,21.00,4.00,6.00,Female,1,Baseline,24 +5596,B,25.00,19.0,4.00,Female,1,Baseline,76 +5597,A,35.00,3.00,1.00,Female,1,Baseline,3 +5599,B,38.00,0.00,0.00,Male,0,Baseline,0 +5601,B,18.00,0.00,0.00,Male,0,Baseline,0 +5602,B,35.00,10.0,7.00,Male,1,Baseline,70 +5603,B,23.00,1.00,1.00,Female,1,Baseline,1 +5604,A,23.00,4.00,2.00,Female,1,Baseline,8 +5605,C,18.00,1.00,5.00,Female,1,Baseline,5 +5606,C,21.00,2.00,8.00,Female,1,Baseline,16 +5607,A,22.00,0.00,0.00,Male,0,Baseline,0 +5608,A,36.00,28.0,1.00,Female,1,Baseline,28 +5610,A,35.00,3.00,1.00,Male,1,Baseline,3 +5611,A,40.00,7.00,2.00,Male,1,Baseline,14 +5612,B,40.00,0.00,0.00,Female,0,Baseline,0 +5613,B,22.00,3.00,3.00,Female,1,Baseline,9 +5615,C,33.00,0.00,0.00,Male,0,Baseline,0 +5616,A,26.00,2.00,6.00,Male,1,Baseline,12 +5617,B,19.00,4.00,8.00,Female,1,Baseline,32 +5618,C,18.00,2.00,1.00,Male,1,Baseline,2 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+5878,C,28.00,4.00,2.00,Female,1,Baseline,8 +5879,A,20.00,6.00,1.00,Male,1,Baseline,6 +5880,A,18.00,4.00,5.00,Female,1,Baseline,20 +5881,B,21.00,10.0,3.00,Female,1,Baseline,30 +5882,B,20.00,2.00,8.00,Male,1,Baseline,16 +5883,C,45.00,10.0,3.00,Female,1,Baseline,30 +5884,C,18.00,3.00,2.00,Female,1,Baseline,6 +5885,A,22.00,8.00,3.00,Male,1,Baseline,24 +5886,B,21.00,4.00,7.00,Female,1,Baseline,28 +5887,A,20.00,2.00,2.00,Female,1,Baseline,4 +5888,A,24.00,3.00,1.00,Female,1,Baseline,3 +5889,A,21.00,14.0,4.00,Female,1,Baseline,56 +5890,B,33.00,0.00,0.00,Female,0,Baseline,0 +5891,B,53.00,7.00,3.00,Female,1,Baseline,21 +5892,C,18.00,9.00,3.00,Male,1,Baseline,27 +5893,B,19.00,0.00,0.00,Female,0,Baseline,0 +5894,C,37.00,8.00,1.00,Male,1,Baseline,8 +5895,A,19.00,6.00,13.0,Male,1,Baseline,78 +5896,C,19.00,1.00,2.00,Female,1,Baseline,2 +5898,B,43.00,8.00,1.00,Female,1,Baseline,8 +5899,C,21.00,0.00,0.00,Female,0,Baseline,0 +5901,A,29.00,6.00,2.00,Female,1,Baseline,12 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+6493,C,19.00,2.00,2.00,Female,1,Baseline,4 +6494,C,20.00,2.00,6.00,Male,1,Baseline,12 +6495,C,28.00,1.00,2.00,Male,1,Baseline,2 +6497,A,31.00,13.0,2.00,Female,1,Baseline,26 +6498,B,21.00,8.00,5.00,Female,1,Baseline,40 +6499,A,22.00,0.00,0.00,Male,0,Baseline,0 +6500,B,25.00,20.0,1.00,Female,1,Baseline,20 +6502,A,20.00,10.0,5.00,Female,1,Baseline,50 +6503,C,27.00,5.00,8.00,Male,1,Baseline,40 +6504,A,18.00,2.00,4.00,Female,1,Baseline,8 +6505,C,21.00,4.00,1.00,Female,1,Baseline,4 +6506,C,18.00,2.00,4.00,Female,1,Baseline,8 +6507,B,21.00,3.00,7.00,Male,1,Baseline,21 +6508,C,21.00,7.00,6.00,Female,1,Baseline,42 +6509,A,19.00,2.00,1.00,Male,1,Baseline,2 +6510,A,22.00,7.00,10.0,Female,1,Baseline,70 +6511,B,21.00,4.00,2.00,Female,1,Baseline,8 +6513,B,18.00,8.00,6.00,Female,1,Baseline,48 +6514,B,44.00,12.0,4.00,Female,1,Baseline,48 +6515,C,20.00,0.00,0.00,Male,0,Baseline,0 +6516,A,22.00,4.00,4.00,Male,1,Baseline,16 +6517,C,19.00,4.00,7.00,Male,1,Baseline,28 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+6547,C,28.00,3.00,3.00,Male,1,Baseline,9 +6549,C,22.00,6.00,12.0,Male,1,Baseline,72 +6550,B,24.00,1.00,4.00,Female,1,Baseline,4 +6551,B,21.00,10.0,2.00,Female,1,Baseline,20 +6552,C,30.00,8.00,6.00,Male,1,Baseline,48 +6553,B,20.00,3.00,4.00,Female,1,Baseline,12 +6554,A,24.00,6.00,2.00,Male,1,Baseline,12 +6556,C,21.00,3.00,10.0,Male,1,Baseline,30 +6557,A,20.00,20.0,3.00,Male,1,Baseline,60 +6558,C,19.00,1.00,1.00,Female,1,Baseline,1 +6559,B,21.00,0.00,0.00,Female,0,Baseline,0 +6562,A,24.00,5.00,1.00,Male,1,Baseline,5 +6563,C,22.00,2.00,2.00,Female,1,Baseline,4 +6564,B,21.00,0.00,0.00,Male,0,Baseline,0 +6565,A,19.00,4.00,4.00,Male,1,Baseline,16 +6566,C,22.00,4.00,8.00,Female,1,Baseline,32 +6567,C,21.00,1.00,7.00,Male,1,Baseline,7 +6569,C,18.00,6.00,4.00,Female,1,Baseline,24 +6570,C,42.00,10.0,2.00,Male,1,Baseline,20 +6571,C,23.00,0.00,0.00,Male,0,Baseline,0 +6572,A,20.00,4.00,4.00,Female,1,Baseline,16 +6573,C,31.00,0.00,0.00,Female,0,Baseline,0 +6574,C,23.00,6.00,4.00,Male,1,Baseline,24 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+6738,B,18.00,0.00,0.00,Male,0,Baseline,0 +6739,C,38.00,0.00,0.00,Female,0,Baseline,0 +6740,A,18.00,3.00,4.00,Female,1,Baseline,12 +6742,B,25.00,2.00,1.00,Female,1,Baseline,2 +6743,C,20.00,0.00,0.00,Female,0,Baseline,0 +6745,B,20.00,0.00,0.00,Female,0,Baseline,0 +6746,C,21.00,3.00,8.00,Female,1,Baseline,24 +6747,C,36.00,20.0,1.00,Female,1,Baseline,20 +6749,A,18.00,3.00,8.00,Female,1,Baseline,24 +6750,B,20.00,2.00,10.0,Male,1,Baseline,20 +6751,A,32.00,0.00,0.00,Male,0,Baseline,0 +6752,B,23.00,1.00,1.00,Female,1,Baseline,1 +6753,C,19.00,6.00,3.00,Female,1,Baseline,18 +6754,B,34.00,0.00,0.00,Male,0,Baseline,0 +6755,A,19.00,8.00,20.0,Male,1,Baseline,160 +6757,A,21.00,0.00,0.00,Female,0,Baseline,0 +6758,A,25.00,5.00,2.00,Male,1,Baseline,10 +6759,C,21.00,2.00,2.00,Male,1,Baseline,4 +6760,A,18.00,4.00,5.00,Female,1,Baseline,20 +6761,B,21.00,10.0,2.00,Male,1,Baseline,20 +6762,A,22.00,0.00,0.00,Female,0,Baseline,0 +6764,B,22.00,14.0,1.00,Female,1,Baseline,14 +6765,A,20.00,3.00,1.00,Female,1,Baseline,3 +6766,C,21.00,6.00,6.00,Female,1,Baseline,36 +6767,B,22.00,7.00,2.00,Male,1,Baseline,14 +6768,B,18.00,0.00,0.00,Male,0,Baseline,0 +6769,B,18.00,4.00,4.00,Male,1,Baseline,16 +6770,A,29.00,12.0,5.00,Male,1,Baseline,60 +6771,B,19.00,8.00,3.00,Male,1,Baseline,24 +6772,A,24.00,2.00,2.00,Female,1,Baseline,4 +6773,A,20.00,1.00,1.00,Female,1,Baseline,1 +6775,A,35.00,1.00,1.00,Male,1,Baseline,1 +6776,C,20.00,0.00,0.00,Female,0,Baseline,0 +6777,C,22.00,8.00,4.00,Male,1,Baseline,32 +6778,C,21.00,0.00,0.00,Male,0,Baseline,0 +6779,C,24.00,8.00,2.00,Male,1,Baseline,16 +6780,A,32.00,22.0,1.00,Female,1,Baseline,22 +6781,B,21.00,6.00,3.00,Female,1,Baseline,18 +6783,B,19.00,3.00,3.00,Male,1,Baseline,9 +6785,C,23.00,4.00,2.00,Female,1,Baseline,8 +6786,C,20.00,3.00,2.00,Male,1,Baseline,6 +6788,A,20.00,5.00,2.00,Female,1,Baseline,10 +6789,B,33.00,0.00,0.00,Female,0,Baseline,0 +6790,C,19.00,3.00,5.00,Male,1,Baseline,15 +6791,B,34.00,4.00,1.00,Female,1,Baseline,4 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+7783,A,46.00,3.00,1.00,Male,1,Baseline,3 +7784,C,39.00,0.00,0.00,Female,0,Baseline,0 +7787,A,23.00,5.00,11.0,Male,1,Baseline,55 +7788,C,25.00,0.00,0.00,Male,0,Baseline,0 +7789,A,34.00,0.00,0.00,Female,0,Baseline,0 +7790,C,39.00,12.0,2.00,Female,1,Baseline,24 +7792,C,22.00,4.00,2.00,Female,1,Baseline,8 +7793,A,35.00,1.00,3.00,Female,1,Baseline,3 +7794,B,55.00,12.0,1.00,Female,1,Baseline,12 +7795,A,55.00,5.00,1.00,Female,1,Baseline,5 +7796,A,21.00,1.00,1.00,Female,1,Baseline,1 +7797,C,30.00,20.0,4.00,Male,1,Baseline,80 +7798,B,20.00,3.00,1.00,Female,1,Baseline,3 +7799,C,20.00,14.0,2.00,Female,1,Baseline,28 +7800,A,23.00,7.00,12.0,Female,1,Baseline,84 +7801,A,30.00,0.00,0.00,Female,0,Baseline,0 +7802,C,23.00,13.0,1.00,Female,1,Baseline,13 +7803,B,18.00,10.0,4.00,Male,1,Baseline,40 +7804,A,20.00,6.00,2.00,Female,1,Baseline,12 +7805,B,49.00,8.00,2.00,Female,1,Baseline,16 +7807,A,22.00,0.00,0.00,Male,0,Baseline,0 +7808,B,21.00,15.0,2.00,Female,1,Baseline,30 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+7994,B,19.00,16.0,5.00,Female,1,Baseline,80 +7995,B,18.00,7.00,2.00,Female,1,Baseline,14 +7996,A,19.00,12.0,12.0,Male,1,Baseline,144 +7997,B,43.00,18.0,2.00,Female,1,Baseline,36 +7998,B,18.00,20.0,1.00,Female,1,Baseline,20 +7999,C,18.00,10.0,9.00,Female,1,Baseline,90 +8000,B,26.00,0.00,0.00,Male,0,Baseline,0 +8001,A,40.00,0.00,0.00,Male,0,Baseline,0 +8002,B,19.00,5.00,2.00,Female,1,Baseline,10 +8003,C,23.00,19.0,4.00,Male,1,Baseline,76 +8004,A,21.00,20.0,6.00,Male,1,Baseline,120 +8005,C,20.00,8.00,8.00,Female,1,Baseline,64 +8006,A,22.00,18.0,2.00,Male,1,Baseline,36 +8007,B,23.00,14.0,1.00,Male,1,Baseline,14 +8008,C,21.00,2.00,4.00,Female,1,Baseline,8 +8009,A,38.00,2.00,2.00,Female,1,Baseline,4 +8010,C,18.00,4.00,5.00,Male,1,Baseline,20 +8011,B,21.00,12.0,3.00,Female,1,Baseline,36 +8012,A,49.00,15.0,1.00,Female,1,Baseline,15 +8013,A,22.00,2.00,2.00,Male,1,Baseline,4 +8014,C,22.00,3.00,4.00,Female,1,Baseline,12 +8015,C,19.00,2.00,13.0,Female,1,Baseline,26 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must be provided with non-empirical curves") + } + + if (!empirical & (length(domain) != 2 | !is.numeric(domain))) { + stop("'domain' must be a two-value numeric vector, like c(0, 10)") + } + + if (empirical) { + # Approximate the functional form of both curves + curve1_f <- approxfun(curve1$x, curve1$y, rule = 2) + curve2_f <- approxfun(curve2$x, curve2$y, rule = 2) + + # Calculate the intersection of curve 1 and curve 2 along the x-axis + point_x <- uniroot(function(x) curve1_f(x) - curve2_f(x), + c(min(curve1$x), max(curve1$x)))$root + + # Find where point_x is in curve 2 + point_y <- curve2_f(point_x) + } else { + # Calculate the intersection of curve 1 and curve 2 along the x-axis + # within the given domain + point_x <- uniroot(function(x) curve1(x) - curve2(x), domain)$root + + # Find where point_x is in curve 2 + point_y <- curve2(point_x) + } + + return(list(x = point_x, y = point_y)) +} + +#### + +library(BayesFactor) + +# Set colours # +green<-rgb(0.1, 0.8, 0.1, alpha=0.25) +blue<-rgb(0.1, 0.8, 0.2, alpha=0.25) +yellow<-rgb(0.3, 0.8, 0.2, alpha=0.25) +clear<-rgb(1, 0, 0, alpha=0.0001) +red<-rgb(1, 0, 0.1, alpha=0.25) + + +################################# +lseq<-function(from, to, length.out){10^(seq(log10(from), log10(to), length.out = length.out))} + + + + + + +resultmat<-data.frame() +Nvec<-c(lseq(30,200,50),200,240:260, 300, 500,750,1000) +BFvec<-c(round(1/BFthres,3),BFthres) + +k<-1 +for(i in 1:length(BFvec)){ + for(j in 1:length(Nvec)){ +N <- Nvec[j] +Delta <- 0.1 +BF <- BFvec[i] + +matchBF<-function(x){(BF-linearReg.R2stat(N, p, x, rscale = "medium", simple = TRUE))} + +R2 <- tryCatch(uniroot(matchBF, interval=c(0.0001,0.999))$root, error=function(x) 0) + +resultmat[k,"BF"]<-BF +resultmat[k,"N"]<-N +resultmat[k,"R2"]<-R2 +resultmat[k,"p"]<-p +resultmat[k,"Delta"]<-Delta + +R2_CET<-function(x){ +Fstat <- (x/K)/( (1-x)/(N-K-1) ) +pval1 <- pf(Fstat,df1 = K, df2 = N-K-1, lower.tail=FALSE) +pval2 <- pf(Fstat, df1 = K, df2 = N-K-1, ncp = (N*Delta)/(1-Delta), lower.tail=TRUE ) +if(pval1=alpha1){return(list(pval=pval2,result=-1)) } + } + +R2_NHST<-function(x){ + Fstat <- (x/K)/( (1-x)/(N-K-1) ) + pf(Fstat,df1 = K, df2 = N-K-1,lower.tail=FALSE) +} + +R2_EQUIV<-function(x){ + Fstat <- (x/K)/( (1-x)/(N-K-1) ) + pf(Fstat, df1 = K, df2 = N-K-1, ncp = (N*Delta)/(1-Delta) ,lower.tail=TRUE) +} + + +if(BF>=1){ +R2_freq <- uniroot(function(x){R2_NHST(x)-alpha1}, interval=c(0,1))$root} + + +if(BF<1){ +R2_freq <- uniroot(function(x){R2_EQUIV(x)-alpha2}, interval=c(0,1))$root} + + + +resultmat[k, "R2_freq"] <- R2_freq + +k <- k+1 +print(k) +}} + +library(ggplot2) + + + +###################### +x <-data.frame(N=resultmat[resultmat$BF==unique(resultmat $BF)[1],]$N) +x$R2<-resultmat[resultmat $BF==unique(resultmat $BF)[1],]$R2_freq +x$x2<-resultmat[resultmat $BF==unique(resultmat $BF)[2],]$R2_freq + +interxect_x<-round(curve_intersect(data.frame(x=x$N, y=x$R2), data.frame(x=x$N, y=x$x2) )$x) + + +equivplot<- ggplot(x, aes(x=N, y=R2)) + + geom_ribbon(data=subset(x, 0 <= N & N <= interxect_x), + aes(ymin=R2,ymax=x2), fill="yellow", alpha="0.5") + + geom_ribbon(data=subset(x, 0 <= (N+1) & (N-1) <= interxect_x+1), + aes(ymin=0,ymax= R2), fill="red", alpha="0.5") + + geom_ribbon(data=subset(x, interxect_x <= (N) & (N-1) <=10000), + aes(ymin=0,ymax= x2), fill="red", alpha="0.5") + + geom_ribbon(data=subset(x, interxect_x <= N & N <=10000), + aes(ymin= x2,ymax= R2), fill="palegreen1", alpha="0.5")+ + geom_ribbon(data=subset(x, interxect_x <= (N+1) & (N-3) <=10000), + aes(ymin= R2,ymax= 1), fill="green", alpha="0.5")+ + geom_ribbon(data=subset(x, 0<= (N+2) & (N-2) <=interxect_x), + aes(ymin= x2,ymax= 1), fill="green", alpha="0.5")+ + geom_line(aes(y = R2)) + + geom_line(aes(y = x2))+ geom_line(aes(y = 0.10), lty=3)+ + scale_y_log10(breaks=c(.001,.1,0.5,1), limits=c(0.0001,1))+ scale_x_log10(breaks=c(10,30,50,100, interxect_x ,500,1000,5000,10000)) + + + + + + + + +##### + + + +x <-data.frame(N=resultmat[resultmat$BF==unique(resultmat $BF)[1],]$N) +x$R2<-resultmat[resultmat$BF==unique(resultmat $BF)[1],]$R2 +x$x2<-resultmat[resultmat$BF==unique(resultmat $BF)[2],]$R2 + +BFplot<- ggplot(x, aes(x=N, y=R2)) + geom_line(aes(y = R2)) + + geom_line(aes(y = x2)) + + scale_y_log10(breaks=c(.001,.1,0.5,1), limits=c(0.0001,1))+ scale_x_log10(breaks=c(10,30,50,100,500,1000,5000,10000)) + + geom_ribbon(data=subset(x, 0 <= (N+1) & (N-1) <= 10000), + aes(ymin=x2,ymax=1), fill="green", alpha="0.5") + + geom_ribbon(data=subset(x, 0 <= N & N <= 10000), + aes(ymin=R2,ymax= x2), fill="yellow", alpha="0.5") + + geom_ribbon(data=subset(x, 0 <= (N+1) & (N-1) <=10000), + aes(ymin=0,ymax= R2), fill="red", alpha="0.5") + + + + require(gridExtra) + + grid.arrange(equivplot, BFplot, ncol=2) + + +## This behaviour may seem odd but: +#plot(apply(cbind(lseq(50,500,90)), 1, function(x){ci.R2(R2=0.12,N=x, K=12, alpha.upper=0.05, alpha.lower=0.0, Random.Regressors=FALSE)$Upper.Conf.Limit.R2})~lseq(50,500,90), ylim=c(0,0.20)); abline(0.12,0) \ No newline at end of file diff --git a/papers/Avocado_Update/campbell_lakens_supplement/disregard_hawthorne.R b/papers/Avocado_Update/campbell_lakens_supplement/disregard_hawthorne.R new file mode 100644 index 00000000..7cd86241 --- /dev/null +++ b/papers/Avocado_Update/campbell_lakens_supplement/disregard_hawthorne.R @@ -0,0 +1,171 @@ +library(gee) +library(geepack) +library(BayesFactor) +library(RCurl) +library(httr) + +############################################################# +# This loads custom code for ANOVA non-inferiority testing from github repo: +script <- getURL("https://raw.githubusercontent.com/harlanhappydog/noninfANOVAlm/master/noninfANOVA.R", ssl.verifypeer = FALSE) +eval(parse(text = script)) + + +############################################################# +## This reads in the data, and formats it appropriately: +Hdata <-read.csv(text= getURL("https://raw.githubusercontent.com/harlanhappydog/noninfANOVAlm/master/analysisdataset2018.csv")) + +Hdata$group<-as.factor(Hdata$group) +Hdata$t<-as.factor(Hdata$t) +Hdata$participant_ID <-as.factor(Hdata$participant_ID) + +Hdata<- Hdata[!(Hdata$group)=="",] +Hdata$group<-as.factor(as.character(Hdata$group)) + +Hdata<-Hdata[order(Hdata$participant_ID),] +Hdata<-Hdata[Hdata$itt==1,] + +############################################################# +## This reads reduces the data to subset of complete cases (i.e., participants that have both baseline and follow-up recorded for at least one outcome) : + +Hdata_complete<-Hdata[Hdata$participant_ID%in%(c(names(table(Hdata$participant_ID)[table(Hdata$participant_ID)==2]))),] + +Hdata_complete$participant_ID<-as.factor(as.character((Hdata_complete$participant_ID))) +Hdata_complete<-Hdata_complete[order(Hdata_complete$participant_ID),] + +Hdata_complete <- Hdata_complete[order(Hdata_complete$participant_ID),] + + + +base_data<-na.omit(Hdata[Hdata$t=="Baseline",][,c("participant_ID","totaldrinking", "group")]) + +followup_data<-na.omit(Hdata[Hdata$t=="Followup",][,c("participant_ID","totaldrinking")]) + +side_data<-merge(base_data , followup_data, by="participant_ID", all=TRUE) +side_data$totaldrinking.diff<-side_data$totaldrinking.y-side_data$totaldrinking.x + + +############################################################# +## This creates a "wide" version of the complete cases data: + + + +#### sample sizes don't quite line up with what is published for Followup outcomes: +library(xtable) + + +### Group A +n_A <- length(na.omit(side_data[side_data$group=="A",]$totaldrinking.x)) +m_A <- mean(na.omit(side_data[ side_data$group=="A",]$totaldrinking.x)) +sd_A <- sd(na.omit(side_data[ side_data$group=="A",]$totaldrinking.x)) + + +### Group B +n_B <- length(na.omit(side_data[ side_data$group=="B",]$totaldrinking.x)) +m_B <- mean(na.omit(side_data[ side_data$group=="B",]$totaldrinking.x)) +sd_B <- sd(na.omit(side_data[ side_data$group=="B",]$totaldrinking.x)) + +### Group C +n_C <- length(na.omit(side_data[ side_data$group=="C",]$totaldrinking.x)) +m_C <- mean(na.omit(side_data[ side_data$group=="C",]$totaldrinking.x)) +sd_C <- sd(na.omit(side_data[ side_data$group=="C",]$totaldrinking.x)) + +### Total +n_Total<- length(na.omit(side_data[,]$totaldrinking.x)) +m_Total <- mean(na.omit(side_data[,]$totaldrinking.x)) +sd_Total <- sd(na.omit(side_data[,]$totaldrinking.x)) + + +baseline <-c(c(n_A,m_A,sd_A),c(n_B,m_B,sd_B),c(n_C,m_C,sd_C),c(n_Total,m_Total,sd_Total)) +### Group A +n_A <- length(na.omit(side_data[side_data$group=="A",]$totaldrinking.y)) +m_A <- mean(na.omit(side_data[ side_data$group=="A",]$totaldrinking.y)) +sd_A <- sd(na.omit(side_data[ side_data$group=="A",]$totaldrinking.y)) + + +### Group B +n_B <- length(na.omit(side_data[ side_data$group=="B",]$totaldrinking.y)) +m_B <- mean(na.omit(side_data[ side_data$group=="B",]$totaldrinking.y)) +sd_B <- sd(na.omit(side_data[ side_data$group=="B",]$totaldrinking.y)) + +### Group C +n_C <- length(na.omit(side_data[ side_data$group=="C",]$totaldrinking.y)) +m_C <- mean(na.omit(side_data[ side_data$group=="C",]$totaldrinking.y)) +sd_C <- sd(na.omit(side_data[ side_data$group=="C",]$totaldrinking.y)) + +### Total +n_Total<- length(na.omit(side_data[,]$totaldrinking.y)) +m_Total <- mean(na.omit(side_data[,]$totaldrinking.y)) +sd_Total <- sd(na.omit(side_data[,]$totaldrinking.y)) + + +followup <-c(c(n_A,m_A,sd_A),c(n_B,m_B,sd_B),c(n_C,m_C,sd_C),c(n_Total,m_Total,sd_Total)) + + +### Group A +n_A <- length(na.omit(side_data[side_data$group=="A",]$totaldrinking.diff)) +m_A <- mean(na.omit(side_data[ side_data$group=="A",]$totaldrinking.diff)) +sd_A <- sd(na.omit(side_data[ side_data$group=="A",]$totaldrinking.diff)) + + +### Group B +n_B <- length(na.omit(side_data[ side_data$group=="B",]$totaldrinking.diff)) +m_B <- mean(na.omit(side_data[ side_data$group=="B",]$totaldrinking.diff)) +sd_B <- sd(na.omit(side_data[ side_data$group=="B",]$totaldrinking.diff)) + +### Group C +n_C <- length(na.omit(side_data[ side_data$group=="C",]$totaldrinking.diff)) +m_C <- mean(na.omit(side_data[ side_data$group=="C",]$totaldrinking.diff)) +sd_C <- sd(na.omit(side_data[ side_data$group=="C",]$totaldrinking.diff)) + +### Total +n_Total<- length(na.omit(side_data[,]$totaldrinking.diff)) +m_Total <- mean(na.omit(side_data[,]$totaldrinking.diff)) +sd_Total <- sd(na.omit(side_data[,]$totaldrinking.diff)) + + +diff<-c(c(n_A,m_A,sd_A),c(n_B,m_B,sd_B),c(n_C,m_C,sd_C),c(n_Total,m_Total,sd_Total)) + +table_for_paper<-t(rbind(baseline, followup, diff)) + +rownames(table_for_paper)<-c(c("n_A","m_A","sd_A"),c("n_B","m_B","sd_B"),c("n_C","m_C","sd_C"),c("n_Total","m_Total","sd_Total")) + +xtable(table_for_paper) + + + + +#### Analysis: + + +Xmatrix <- model.matrix(totaldrinking.diff ~ group, data= side_data) +lmmodel <- lm(totaldrinking.diff ~ group , data= side_data) + +R2 <- summary(lmmodel)$r.squared +Fstat <- summary(lmmodel)$fstatistic[1] +K <- dim(Xmatrix)[2] - 1 +N <- dim(Xmatrix)[1] +Delta <- 0.01 + +pf(Fstat,df1=K,df2=N-K-1,ncp=(N*Delta)/(1-Delta),lower.tail=TRUE) + +linearReg.R2stat(N=N, p=K, R2= R2, simple=TRUE) + + + + +### The code bellow replicates the results published in McCambridge et al. (2019) +### Note: there appears to be a typo in McCambridge et al. (2019) Table 2: +### p-values 0.89 and 0.86 are switched. + +Hdata$group<-relevel(Hdata$group,"A") + +mod0 <- geeglm(totaldrinking ~ + group+t, id= participant_ID, corstr="independence", data= Hdata, x=TRUE) +mod1 <- geeglm(totaldrinking ~ group*t + group+t, id= participant_ID, corstr="independence", data= Hdata) +(anova(mod1,mod0)) +summary(mod1)$coefficients + +Hdata$group<-relevel(Hdata$group,"C") +mod1a <- geeglm(totaldrinking ~ group*t + group+t, id= participant_ID, corstr="independence", data= Hdata) +summary(mod1a) + + diff --git a/papers/Avocado_Update/campbell_lakens_supplement/disregard_simFigA.pdf b/papers/Avocado_Update/campbell_lakens_supplement/disregard_simFigA.pdf new file mode 100644 index 0000000000000000000000000000000000000000..8eca39095fe376282597e4fa60e6a78a39e00a23 GIT binary patch literal 219378 zcmXV$WmFtZ*RJsl5Znf5aCZ+926xv0!QEYhySo!y27)`m-Q5Wg+}(m6p7%SydUdZ= z-Bs1K_P+MMZz?4*2_`mX4kW7nz3aV`(wn^L{y`*8GFCEsBP%2U0WuajQ#*4P3o@>c zM=E425|%bDrcNJM8$%aUF;ioE6H_uFAtYxPCsRXPBoFAboUut8NvE8HJM2B}hp&E1 z8?K*DBM3tMxeb;rU(|K4My7IC{dcSRc?%H_o`|*!gm-*qu_I+yp z{ma{{XoJPWG&$E*bJrC0+go#kR{&+ok?_K(qA=XOsHSTkiuy%WWO>5te=n5A;zBLm z8LHD~PJ4Cg`k{LsX4*CT+IQ9RojELS$4U51GRd7#evyns=={W~bDjD7YjICbdtD(z z^gU!|<2W(P^^I5WqV#6eh6pV!ZqKLdJjS<+qe*v?bm

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zeE)XO`cN1=R7Vh>HK2S7FfbsbFB$=u5{dzLgQOh}O~Nl|91bd52;eJ|oEtDbP+tJ7 zWcMk0&pv>E57n^<3V^(p6mDjvRqfWBw%O@JQx~njd>nmpxy0pk@NZ2$lO literal 0 HcmV?d00001 diff --git a/papers/Avocado_Update/campbell_lakens_supplement/disregard_simulation_study.R b/papers/Avocado_Update/campbell_lakens_supplement/disregard_simulation_study.R new file mode 100644 index 00000000..21a7d549 --- /dev/null +++ b/papers/Avocado_Update/campbell_lakens_supplement/disregard_simulation_study.R @@ -0,0 +1,154 @@ +set.seed(123) + + +# Functions for rounding: +ceiling_dec <- function(x, level=1) round((x + 5*10^(-level-1))/0.5, level)*0.5 + +# number of simulations: +nSim<-10000 + +resultsmat <- data.frame( +JJJindex=NA, +trueP2_num= NA, +power=NA, +as.matrix(expand.grid(Nvar=c(2,4),sigma2=c(0.4,0.5,1), Nsample=c(120, 1200,12000)))) + +resultsmat$trueP2 <- NA + +resultsmat$alpha_sig <- 0.05 + +dim(resultsmat)[1] + + +resultsmat<-resultsmat[order(resultsmat$Nvar, resultsmat$sigma2),] +pval_list<-list() +problist<-list() +for(jjj in 1:dim(resultsmat)[1]){ + +print(jjj) + +resultsmat[jjj, "JJJindex"] <- paste("jjj",jjj,sep="") +sigma2 <- resultsmat[jjj, "sigma2"] +N <- resultsmat[jjj, "Nsample"] +nVar <- resultsmat[jjj, "Nvar"] + +basematrix <- data.frame(expand.grid( + X1=c(0,1), + X2=c(0,1), + X3=c(0,1), + X4=c(0,1))) + +X1 <- rep(basematrix$X1,2000000) +X2 <- rep(basematrix$X2,2000000) +X3 <- rep(basematrix$X3,2000000) +X4 <- rep(basematrix$X4,2000000) + +epsilon <- rnorm(length(X1), 0, sqrt(sigma2)) + + +if(nVar==2){ +X <- as.matrix(cbind(1, X1, X2)) +betavec <- c(0.0, 0.2, 0.3) + +Y <- c(X%*%betavec) + epsilon +sigmaXY <-(c(cov(X[,-1], Y))) +SIGMAX <- cov(X[,-1]) +(trueP2_num <-(t(sigmaXY)%*%solve(SIGMAX)%*%sigmaXY)) + +resultsmat[jjj, "trueP2_num"] <- trueP2_num +resultsmat[jjj, "trueP2"] <-(resultsmat[jjj, "trueP2_num"]/var(Y)) + +confirmmod<-lm(Y~ X[,-1]) +print(c(summary(confirmmod)$r.squared,resultsmat[jjj, "trueP2"])) + +resultsmat[jjj, "trueP2_v2"] <- summary(confirmmod)$r.squared + +Xmatrix_ <- X[1:N,] +} + +if(nVar==4){ +X <- as.matrix(cbind(1, X1, X2, X3, X4)) +betavec <- c(0.0, 0.2, 0.2, -0.1, -0.2) + +Y <- c(X%*%betavec) + epsilon +sigmaXY <- c(cov(X[,-1], Y)) +SIGMAX <- cov(X[,-1]) +(trueP2_num <- (t(sigmaXY)%*%solve(SIGMAX)%*%sigmaXY)) + +resultsmat[jjj, "trueP2_num"] <- trueP2_num +resultsmat[jjj, "trueP2"] <-(resultsmat[jjj, "trueP2_num"]/var(Y)) + + +confirmmod<-lm(Y~ X[,-1]) +print(c(summary(confirmmod)$r.squared,resultsmat[jjj, "trueP2"])) + +resultsmat[jjj, "trueP2_v2"] <- summary(confirmmod)$r.squared + +Xmatrix_ <- X[1:N,] +} + + +pval_list[[jjj]]<-list() +Deltavec<-seq(0.01,0.10,0.005) + +for(iii in 1:nSim){ + + + + epsilon <- rnorm(N, 0, sqrt(sigma2)) + Y <- Xmatrix_%*%betavec + epsilon + + lmmodel<-lm(Y~ Xmatrix_[,-1]) + Xmatrix <- model.matrix(lmmodel) + + R2 <- summary(lmmodel)$r.squared + Fstat <- summary(lmmodel)$fstatistic[1] + K <- dim(Xmatrix)[2] - 1 + N <- dim(Xmatrix)[1] + + pval_list[[jjj]][[iii]]<-vector() + + for(kk in 1:length(Deltavec)){ + Delta<-Deltavec[kk] + pval <- pf(Fstat,df1=K,df2=N-K-1,ncp=(N*Delta)/(1-Delta),lower.tail=TRUE) + pval_list[[jjj]][[iii]][kk]<-pval + } + + } + +pvalmat<-as.data.frame(pval_list[[jjj]], col.names=NA, row.names=Deltavec) +problist[[jjj]]<-rowMeans(pvalmat0.03 & resultsmatall$trueP2_v2<0.05]<-round( mean(resultsmatall$trueP2_v2[resultsmatall$trueP2_v2>0.03 & resultsmatall$trueP2_v2<0.05]),3) + +resultsmatall$trueP2_v2[resultsmatall$trueP2_v2>0.06 & resultsmatall$trueP2_v2<0.07]<-round( mean(resultsmatall$trueP2_v2[resultsmatall$trueP2_v2>0.06 & resultsmatall$trueP2_v2<0.07]),3) + +resultsmatall$trueP2_v2[resultsmatall$trueP2_v2>0.07]<-round( mean(resultsmatall$trueP2_v2[resultsmatall$trueP2_v2>0.07]),3) + +resultsmatall$trueP2_v2[resultsmatall$trueP2_v2<0.02]<-round( mean(resultsmatall$trueP2_v2[resultsmatall$trueP2_v2<0.02]),3) + +resultsmatall$P2 <-as.factor(round((resultsmatall$trueP2_v2),3)) +resultsmatall$N <-as.factor(resultsmatall$Nsample) + +resultsmatall$g <- as.factor(paste(as.character(resultsmatall$P2),as.character(resultsmatall$N), sep="_")) + +resultsmatall <- transform(resultsmatall, + Nvar = factor(Nvar, levels = sort(unique(resultsmatall$Nvar)), c( (("K = 2")),(("K = 4"))) )) + +#saveRDS(resultsmatall,'~/Desktop/UBC/ThesisProposal/Rcode/lakens_project/resultsmatall.rds') + +resultsmatall<- readRDS('~/Desktop/UBC/ThesisProposal/Rcode/lakens_project/resultsmatall.rds') + +resultsmatall$KK <- as.numeric(resultsmatall$Nvar)*2 +resultsmatall$NN <- as.numeric(as.character(resultsmatall$N)) + + +resultsmatall$power <-NA +power_estimate<-function(Delta, K, N){ +Fstat_star <- qf(0.05,df1=K,df2=N-K-1,ncp=(N*Delta)/(1-Delta),lower.tail=TRUE) +power<-pf(Fstat_star,df1=K,df2=N-K-1,lower.tail=TRUE) +return(round(power,3))} + + +resultsmatall[resultsmatall$P2==0,]$power <- apply(resultsmatall[resultsmatall$P2==0, c("Delta","KK","NN")],1, function(y){ power_estimate(y[1],y[2],y[3])}) + + +#### plot with truncated axis: +library(ggplot2) +qplot(x=Delta, y= pr_less_alpha, group= g, pch=N, lty=N, col= P2, data= resultsmatall)+geom_line()+ geom_hline(yintercept = 0.05)+ facet_grid(Nvar ~ . ) + scale_x_continuous(breaks = seq(0, 0.1, by = 0.01)) + scale_y_continuous(breaks = seq(-0.01, 1, by = 0.05))+ coord_cartesian(ylim = c(0, 0.20))+ labs(x = expression(Delta), y = expression("probability of p" < alpha))+geom_line(aes(y = power, x= Delta), color = "black", linetype = "dotted") + +#### plot with full axis: +library(ggplot2) +qplot(x=Delta, y= pr_less_alpha, group= g, pch=N, lty=N, col= P2, data= resultsmatall)+geom_line()+ geom_hline(yintercept = 0.05)+ facet_grid(Nvar ~. ) + scale_x_continuous(breaks = seq(0, 0.1, by = 0.01)) + scale_y_continuous(breaks = seq(0, 1, by = 0.1))+ labs(x = expression(Delta),y = expression("probability of p" < alpha))+ coord_cartesian(ylim = c(0, 1)) +geom_line(aes(y = power, x= Delta), color = "black", linetype = "dotted") + + + + diff --git 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68a08d38..f3eee9e6 100644 --- a/papers/Avocado_Update/revision-plan_jakub-comments.md +++ b/papers/Avocado_Update/revision-plan_jakub-comments.md @@ -95,6 +95,8 @@ The section (~lines 784–840) is about five paragraphs for the paper's genuinel **Verdict: accept, with a correction to the premise. This is the big one.** +> **Status (2026-09-04): deferred.** The manuscript now states that Campbell & Lakens validated their procedure by simulation, but says nothing about the validation status of the generalisation. The simulation work described below has *not* been done and is not currently planned for this revision. Revisit if a reviewer raises it. + Correcting the premise: Campbell & Lakens (2021) *did* validate their method — the paper includes a simulation study of Type I error and power for the one-way ANOVA and multivariable regression R² cases, plus a comparison against a Bayesian alternative. So the published method is not unvalidated, and the manuscript should say so (one clause, currently absent, worth adding regardless). What is **not** validated is my generalisation. The λ_eq = [Δ/(1−Δ)]·(df₁ + df₂ + 1) substitution is what extends the method to factorial and within-subjects designs, and it reduces to the published formula only in the one-way case. Honest framing: the base method is validated, the extension is not. A reviewer is likely to find this too, so better to close it now. @@ -132,17 +134,17 @@ Minor scope pushback: the McNabb & Murayama argument is framed around neuroscien **Manuscript (`Avocado_Update.Rmd`)** - [ ] Intro: roadmap paragraph; flag the ANOVA generalisation as novel (also in the abstract). -- [ ] Add function-selection table. -- [ ] (Optional) Promote the estimand footnote to a short section. +- [x] Add function-selection table. **Done** — realized as Table \ref{tab:estimands} inside the new estimand subsection, keyed on estimand rather than on design. First table in the paper; verified it builds under `rticles::tf_article` (the `tf` template loads `longtable`/`booktabs` conditionally on the document containing a table). +- [x] Promote the estimand footnote to a short section. **Done** — `# Setting Equivalence Bounds` now opens with a two-decision lead and splits into `## Choosing an Estimand` and `## Justifying the Bounds`. The long footnote in "TOST with t-tests" is gone; its content (trimming, log ratios, intercurrent events, Lundberg et al.) is absorbed into the new subsection, and the sentence that carried it now back-references. - [x] Restructure WMW subsection: no-assumption / location-shift / what-gets-bounded. **Done** — three `####` subsections added; H₀ for stochastic equality now stated formally; `Checking Assumptions` split into labelled location-shift and symmetry parts. - [x] ECDF location-shift figure alongside the symmetry plot. **Done** — `fig:ecdfshift`, two panels (assumption holds / violated) with arrows marking the horizontal gap at the 10th/50th/90th percentiles. Knit-tested. - [ ] One-line estimand statement at the head of each robust-method subsection (WMW now has one; Brunner–Munzel, bootstrap/permutation, log-TOST still need theirs). - [ ] Brunner–Munzel: qualify "valid under heteroscedasticity" as a null/Type I error statement; name the nonparametric Behrens–Fisher problem. -- [ ] ANOVA: define partial η²; distinguish from η²; explain the one-sided non-inferiority framing and why it is not a TOST; state what a rejection does and does not license. -- [ ] ANOVA: bound-selection discussion; flag Δ = 0.35 as illustrative; add the f² conversion. -- [ ] ANOVA: both ncp formulas inline, plus the one-way reduction df₁ + df₂ + 1 = N. -- [ ] ANOVA: note that Campbell & Lakens ran a validating simulation; reference the new simulation for the extension. -- [ ] ANOVA: definition list for `equ_anova` output columns. +- [x] ANOVA: define partial η²; distinguish from η²; explain the one-sided non-inferiority framing and why it is not a TOST; state what a rejection does and does not license. +- [x] ANOVA: bound-selection discussion; flag Δ = 0.35 as illustrative; add the f² conversion. +- [x] ANOVA: both ncp formulas inline, plus the one-way reduction df₁ + df₂ + 1 = N. +- [x] ANOVA: note that Campbell & Lakens ran a validating simulation. **Done.** Per decision on 2026-09-04, the manuscript does *not* flag the factorial/within-subjects extension as separately unvalidated — see the note under comment #5 below. +- [x] ANOVA: definition list for `equ_anova` output columns. - [ ] Conclusions: McNabb & Murayama with the balance condition, plus the Bloom et al. commentary. **Supplement / package** From b444a2d4eceab77d8983c5d819629d81b040d0c5 Mon Sep 17 00:00:00 2001 From: Aaron Caldwell Date: Fri, 4 Sep 2026 11:46:10 -0500 Subject: [PATCH 23/40] ANOVA updates Need to work through simulations more --- papers/Avocado_Update/Avocado_Update.Rmd | 6 +- .../campbell_lakens_verification.html | 4345 +++++++++++++++++ .../campbell_lakens_verification.qmd | 564 +++ 3 files changed, 4912 insertions(+), 3 deletions(-) create mode 100644 papers/Avocado_Update/campbell_lakens_verification.html create mode 100644 papers/Avocado_Update/campbell_lakens_verification.qmd diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index 937575a1..7a603ce1 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -893,7 +893,7 @@ Many researchers use ANOVA as an omnibus test before inspecting pairwise compari ## Partial Eta-Squared and the Equivalence Hypothesis -The effect size used throughout this section is partial eta-squared, +The effect size used throughout this section is "partial eta-squared" ($\eta^2_p$), $$ \eta^2_p = \frac{SS_{\text{effect}}}{SS_{\text{effect}} + SS_{\text{error}}} @@ -907,7 +907,7 @@ $$ H_0: \eta^2_p \geq \Delta \qquad \text{versus} \qquad H_1: \eta^2_p < \Delta $$ -Rejecting $H_0$ supports the claim that the effect in question explains less than $\Delta$ of the variance available to it. Two limits on that conclusion are worth stating plainly. First, it is a statement about an effect as a whole rather than about any particular contrast within it: an omnibus effect can be negligible while a specific pairwise comparison nested inside it is not, so equivalence at the level of the *F*-test does not license equivalence claims about individual group differences. Second, as with any equivalence test, failing to reject $H_0$ is not evidence that the effect is large --- it means only that the data remain compatible with an effect at or beyond the bound. +Rejecting $H_0$ supports the claim that the effect in question explains less than $\Delta$ of the variance available to it. There are two limits on that conclusion worth consideration. First, it is a statement about an effect as a whole rather than about any particular contrast within it: an omnibus effect can be negligible while a specific pairwise comparison nested inside it is not, so equivalence at the level of the *F*-test does not license equivalence claims about individual group differences. Second, as with any equivalence test, failing to reject $H_0$ is not evidence that the effect is large and it means only that the data remain compatible with an effect at or beyond the bound. The test itself is computed from the cumulative distribution function of the non-central *F* distribution. @Campbell_2021 derived the *p*-value for a one-way ANOVA with $J$ groups and $N$ total observations as @@ -927,7 +927,7 @@ The equivalence test inherits the standard ANOVA assumptions, which should be ch ## Choosing a Bound on Partial Eta-Squared -Bounds on $\eta^2_p$ are harder to justify than bounds on the original measurement scale, and the difficulty is worth confronting before running the test. +Bounds on $\eta^2_p$ are harder to justify than bounds on the original measurement scale (much like the SMD), and the difficulty is worth confronting before running the test. The first problem is that $\eta^2_p$ is not comparable across designs. Its denominator depends on which other effects appear in the model and, in within-subjects designs, on the error term against which the effect is tested. A partial eta-squared of 0.06 for a one-way between-subjects factor and a partial eta-squared of 0.06 for an interaction in a repeated-measures model are not estimates of the same quantity. A bound therefore cannot simply be borrowed from a published study unless that study used the same design and the same model. diff --git a/papers/Avocado_Update/campbell_lakens_verification.html b/papers/Avocado_Update/campbell_lakens_verification.html new file mode 100644 index 00000000..6a134013 --- /dev/null +++ b/papers/Avocado_Update/campbell_lakens_verification.html @@ -0,0 +1,4345 @@ + + + + + + + + + + + +Verifying TOSTER’s F-test Equivalence Procedure + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/papers/Avocado_Update/campbell_lakens_verification.qmd b/papers/Avocado_Update/campbell_lakens_verification.qmd new file mode 100644 index 00000000..703b7ed2 --- /dev/null +++ b/papers/Avocado_Update/campbell_lakens_verification.qmd @@ -0,0 +1,564 @@ +--- +title: "Verifying TOSTER's *F*-test Equivalence Procedure" +subtitle: "A reproduction of the simulation study in Campbell & Lakens (2021)" +author: "Aaron R. Caldwell" +date: today +format: + html: + toc: true + toc-depth: 3 + toc-location: left + code-fold: show + code-tools: true + theme: cosmo + embed-resources: true + fig-width: 8 + fig-height: 6 +execute: + warning: false + message: false + cache: false +--- + +```{r} +#| label: setup +#| code-fold: true +library(TOSTER) +library(ggplot2) +library(dplyr) +library(tidyr) +library(knitr) + +theme_set(theme_bw(base_size = 12)) +set.seed(123) +``` + +## Purpose + +@Campbell_2021 proposed an omnibus non-inferiority ("equivalence") test for $R^2$ in +multivariable linear regression and for the omnibus effect in a one-way ANOVA, and +evaluated its operating characteristics by simulation. `TOSTER::equ_ftest()` implements +a generalization of that procedure. + +This document does three things: + +1. Establishes that TOSTER's non-centrality parameter is **algebraically identical** to + the published one in the setting Campbell & Lakens studied, and confirms this + numerically. +2. **Reproduces their simulation study** using TOSTER's implementation, and checks that + the recovered operating characteristics match what the paper reports. +3. Notes a few things about the published supplementary code that are worth knowing if + anyone tries to run it directly. + +This is a verification exercise. Nothing here is currently cited in the manuscript. + +::: {.callout-note} +## Summary of findings + +- TOSTER's non-centrality parameter reduces **exactly** to the published expression + whenever $df_1 + df_2 + 1 = N$, which holds for both the one-way ANOVA and the + multivariable regression settings Campbell & Lakens simulated. Numerical agreement is + at machine precision (max absolute difference $\approx 2 \times 10^{-16}$). +- The reproduced simulation recovers the published behaviour: the rejection rate is + at or below $\alpha$ everywhere the null hypothesis is true, and rises to exactly + $\alpha$ at the boundary $\Delta = \rho^2$. +- `power_eq_f()` reproduces the analytic power function used in their supplement + **exactly** (difference of 0 across every combination checked). +::: + +## Background: the procedure + +The hypotheses are one-sided, because a proportion of variance cannot be negative: + +$$ +H_0: \rho^2 \geq \Delta \qquad \text{versus} \qquad H_1: \rho^2 < \Delta +$$ + +@Campbell_2021 give the *p*-value for a model with $K$ predictors and $N$ observations as + +$$ +p = p_F\left(F; \space K, \space N - K - 1, \space \frac{N \cdot \Delta}{1 - \Delta}\right) +$$ + +where $p_F(\cdot)$ is the CDF of the non-central *F* distribution. TOSTER instead +computes the non-centrality parameter from the degrees of freedom directly, so that the +same logic can be applied to designs where the relationship between $N$ and the degrees +of freedom is less direct: + +$$ +\lambda_{eq} = \frac{\Delta}{1 - \Delta} \cdot (df_1 + df_2 + 1) +$$ + +### The two are identical in this setting + +For a regression with $K$ predictors, $df_1 = K$ and $df_2 = N - K - 1$, so + +$$ +df_1 + df_2 + 1 = K + (N - K - 1) + 1 = N +$$ + +and the two expressions coincide. The same holds for a one-way ANOVA with $J$ groups, +where $df_1 = J - 1$ and $df_2 = N - J$. Any divergence between the two implementations +is therefore floating-point noise, not a difference in method. + +```{r} +#| label: ncp-identity +#| code-fold: false +grid <- expand.grid( + Fstat = c(0.5, 1.2, 3.4, 8.9), + K = c(2, 4, 5), + N = c(60, 180, 450, 1000), + Delta = c(0.01, 0.035, 0.06, 0.10) +) +grid$df1 <- grid$K +grid$df2 <- grid$N - grid$K - 1 + +# Campbell & Lakens, verbatim from their supplementary script +grid$p_CL <- pf(grid$Fstat, + df1 = grid$df1, df2 = grid$df2, + ncp = (grid$N * grid$Delta) / (1 - grid$Delta), + lower.tail = TRUE) + +# TOSTER +grid$p_TOSTER <- mapply( + function(f, d1, d2, e) equ_ftest(Fstat = f, df1 = d1, df2 = d2, eqb = e)$p.value, + grid$Fstat, grid$df1, grid$df2, grid$Delta +) + +c(comparisons = nrow(grid), + max_abs_diff = max(abs(grid$p_CL - grid$p_TOSTER)), + machine_epsilon = .Machine$double.eps) +``` + +Agreement is at machine precision across all `r nrow(grid)` combinations. + +## The simulation design + +The design below follows `disregard_simulation_study_2.R` from the published +supplement, which is the more complete of the two scripts (it adds a true-null condition +and overlays the analytic power curve). + +| Factor | Levels | +|:-------|:-------| +| Predictors $K$ | 2, 4 | +| Sample size $N$ | 60, 180, 450, 1000 | +| Residual variance $\sigma^2$ | 0.4, 0.5, 1 (effect present); 1 with $\beta = 0$ (true null) | +| Equivalence bound $\Delta$ | 0.010 to 0.100 in steps of 0.005 | +| Replications | 10,000 per cell | + +Predictors are balanced binary variables from a $2^4$ factorial grid. Coefficients are +$\beta = (0.2,\ 0.3)$ for $K = 2$ and $\beta = (0.2,\ 0.2,\ -0.1,\ -0.2)$ for $K = 4$. +Both give $\sum \beta_j^2 = 0.13$, so the two values of $K$ produce the same true +$\rho^2$ at a given $\sigma^2$ — an elegant feature of their design that makes the two +facets directly comparable. + +### The true $\rho^2$ + +Campbell & Lakens estimate the true $\rho^2$ by Monte Carlo over a 32-million-row design +matrix. It is available in closed form. Because the predictors are independent +Bernoulli(0.5), $\mathrm{Var}(X\beta) = 0.25 \sum \beta_j^2 = 0.0325$, giving +$\rho^2 = 0.0325 / (0.0325 + \sigma^2)$. + +One wrinkle is worth handling carefully. Their code builds the design with +`X[1:N, ]`, taking the first $N$ rows of a repeating 16-row block. Since none of +$N \in \{60, 180, 450, 1000\}$ is a multiple of 16, the final block is truncated and the +four predictors are not perfectly balanced. The quantity that actually determines +whether $H_0$ holds is therefore the **design-specific** $\rho^2$, computed from the +non-centrality the fixed design implies: + +$$ +\lambda_{\text{true}} = \frac{\lVert (I - P_1) X\beta \rVert^2}{\sigma^2}, +\qquad +\rho^2_{\text{true}} = \frac{\lambda_{\text{true}}}{\lambda_{\text{true}} + N} +$$ + +```{r} +#| label: design-fns +build_X <- function(K, N) { + base <- as.matrix(expand.grid(X1 = c(0, 1), X2 = c(0, 1), + X3 = c(0, 1), X4 = c(0, 1))) + reps <- ceiling(N / nrow(base)) + Xf <- cbind(1, base[rep(seq_len(nrow(base)), reps), , drop = FALSE]) + Xf[seq_len(N), c(1, seq_len(K) + 1), drop = FALSE] +} + +beta_for <- function(K, null = FALSE) { + if (null) return(rep(0, K + 1)) + if (K == 2) c(0, 0.2, 0.3) else c(0, 0.2, 0.2, -0.1, -0.2) +} + +rho2_design <- function(K, N, sigma2, betavec) { + mu <- as.vector(build_X(K, N) %*% betavec) + lambda <- sum((mu - mean(mu))^2) / sigma2 + lambda / (lambda + N) +} +``` + +```{r} +#| label: rho2-table +#| code-fold: true +expand.grid(K = c(2, 4), N = c(60, 180, 450, 1000), sigma2 = c(0.4, 0.5, 1)) |> + rowwise() |> + mutate( + rho2_population = 0.0325 / (0.0325 + sigma2), + rho2_design = rho2_design(K, N, sigma2, beta_for(K)) + ) |> + ungroup() |> + mutate(across(starts_with("rho2"), ~ round(.x, 5))) |> + pivot_wider(names_from = N, values_from = rho2_design, + names_prefix = "N = ") |> + kable(caption = "Population vs. design-specific true rho-squared. The K = 2 design stays balanced under truncation; K = 4 drifts slightly.") +``` + +The $K = 2$ design happens to remain balanced under truncation (the first two columns of +the factorial grid cycle with periods 2 and 4). The $K = 4$ design drifts by up to about +0.002 in $\rho^2$. This is minor, but it is the reason the boundary check below uses the +design-specific value rather than the population value. + +## Reproducing the simulation + +The implementation below is a vectorized rewrite rather than a transcription. For a +fixed design matrix the QR decomposition can be reused across all replications, so the +whole study runs in well under a minute instead of fitting 320,000 separate models with +`lm()`. The statistic computed is identical. + +```{r} +#| label: sim-fn +sim_condition <- function(K, N, sigma2, betavec, Deltavec, + nSim = 10000, alpha = 0.05, block = 2500) { + X <- build_X(K, N) + mu <- as.vector(X %*% betavec) + qrX <- qr(X) + df2 <- N - K - 1 + + Fstats <- numeric(nSim) + done <- 0L + while (done < nSim) { + b <- min(block, nSim - done) + Y <- mu + matrix(rnorm(N * b, 0, sqrt(sigma2)), nrow = N) + RSS <- colSums(qr.resid(qrX, Y)^2) + TSS <- colSums(sweep(Y, 2, colMeans(Y))^2) + R2 <- 1 - RSS / TSS + Fstats[(done + 1):(done + b)] <- (R2 / K) / ((1 - R2) / df2) + done <- done + b + } + + data.frame( + Delta = Deltavec, + pr_reject = vapply(Deltavec, function(D) { + mean(pf(Fstats, K, df2, ncp = N * D / (1 - D), lower.tail = TRUE) < alpha) + }, numeric(1)) + ) +} +``` + +```{r} +#| label: run-sim +Deltavec <- seq(0.01, 0.10, 0.005) +nSim <- 10000 + +conditions <- expand.grid( + K = c(2, 4), + N = c(60, 180, 450, 1000), + sigma2 = c(0.4, 0.5, 1), + is_null = c(FALSE, TRUE), + stringsAsFactors = FALSE +) |> + # the true-null condition is defined only once, at sigma2 = 1 + filter(!(is_null & sigma2 != 1)) + +set.seed(123) +results <- conditions |> + rowwise() |> + group_split() |> + lapply(function(cond) { + b <- beta_for(cond$K, null = cond$is_null) + out <- sim_condition(cond$K, cond$N, cond$sigma2, b, Deltavec, nSim = nSim) + out$K <- cond$K + out$N <- cond$N + out$sigma2 <- cond$sigma2 + out$is_null <- cond$is_null + out$rho2 <- if (cond$is_null) 0 else rho2_design(cond$K, cond$N, cond$sigma2, b) + out + }) |> + bind_rows() + +results <- results |> + mutate( + rho2_lab = factor(round(rho2, 3)), + N_lab = factor(N, levels = c(60, 180, 450, 1000)), + K_lab = factor(K, levels = c(2, 4), labels = c("K = 2", "K = 4")), + grp = interaction(rho2_lab, N_lab, drop = TRUE) + ) + +dim(results) +``` + +### Figure 1: rejection rate, truncated axis + +This is their main figure. The horizontal line is $\alpha = 0.05$; the vertical dashed +lines mark each true $\rho^2$. **To the left of a vertical line, $H_0$ is true**, and the +curve of the matching colour must sit at or below 0.05. + +```{r} +#| label: fig-truncated +#| fig-cap: "Probability of rejecting the equivalence null as a function of the bound. Colour distinguishes the true rho-squared; point shape and line type distinguish sample size. Vertical dashed lines mark each true rho-squared, i.e. the boundary of the null hypothesis." +bounds <- results |> distinct(K_lab, rho2, rho2_lab) |> filter(rho2 > 0) + +ggplot(results, aes(x = Delta, y = pr_reject, + group = grp, colour = rho2_lab)) + + geom_vline(data = bounds, aes(xintercept = rho2, colour = rho2_lab), + linetype = "dashed", linewidth = 0.4, alpha = 0.7) + + geom_hline(yintercept = 0.05, linewidth = 0.4) + + geom_line(aes(linetype = N_lab)) + + geom_point(aes(shape = N_lab), size = 1.6) + + facet_grid(K_lab ~ .) + + scale_x_continuous(breaks = seq(0, 0.10, by = 0.01)) + + scale_y_continuous(breaks = seq(0, 0.20, by = 0.05)) + + coord_cartesian(ylim = c(0, 0.20)) + + scale_colour_viridis_d(end = 0.85) + + labs(x = expression(Delta), + y = expression("Pr(" * p < alpha * ")"), + colour = expression("true " * rho^2), + shape = "N", linetype = "N") +``` + +### Figure 2: rejection rate, full axis + +```{r} +#| label: fig-full +#| fig-cap: "The same simulation on the full vertical scale, showing how power accumulates once the bound exceeds the true rho-squared." +ggplot(results, aes(x = Delta, y = pr_reject, + group = grp, colour = rho2_lab)) + + geom_vline(data = bounds, aes(xintercept = rho2, colour = rho2_lab), + linetype = "dashed", linewidth = 0.4, alpha = 0.7) + + geom_hline(yintercept = 0.05, linewidth = 0.4) + + geom_line(aes(linetype = N_lab)) + + geom_point(aes(shape = N_lab), size = 1.6) + + facet_grid(K_lab ~ .) + + scale_x_continuous(breaks = seq(0, 0.10, by = 0.01)) + + scale_y_continuous(breaks = seq(0, 1, by = 0.1)) + + coord_cartesian(ylim = c(0, 1)) + + scale_colour_viridis_d(end = 0.85) + + labs(x = expression(Delta), + y = expression("Pr(" * p < alpha * ")"), + colour = expression("true " * rho^2), + shape = "N", linetype = "N") +``` + +### Type I error control + +The strongest single check is behaviour at the boundary. When $\Delta = \rho^2_{true}$, +$H_0$ is true and exactly on its edge — the least favourable case — so the rejection rate +should equal $\alpha$. Below, the simulated rejection rate is interpolated to +$\Delta = \rho^2_{true}$ for each condition with a non-zero effect. + +```{r} +#| label: tbl-boundary +#| code-fold: true +boundary <- results |> + filter(!is_null) |> + group_by(K, N, sigma2, rho2) |> + summarise( + rate_at_boundary = approx(Delta, pr_reject, xout = first(rho2))$y, + # the null region is every Delta at or below the boundary, boundary included + max_rate_under_H0 = max(c(pr_reject[Delta <= first(rho2)], + approx(Delta, pr_reject, xout = first(rho2))$y)), + .groups = "drop" + ) |> + mutate(z = (rate_at_boundary - 0.05) / sqrt(0.05 * 0.95 / nSim)) + +boundary |> + mutate(across(c(rho2, rate_at_boundary, max_rate_under_H0), ~ round(.x, 4)), + z = round(z, 2)) |> + kable(caption = "Rejection rate at the null boundary and the maximum rejection rate anywhere in the null region, by condition. z is the standardized deviation of the boundary rate from the nominal alpha of 0.05.") +``` + +```{r} +#| label: boundary-summary +#| code-fold: true +mcse <- sqrt(0.05 * 0.95 / nSim) +c(n_conditions = nrow(boundary), + mean_rate_at_boundary = round(mean(boundary$rate_at_boundary), 4), + max_rate_at_boundary = round(max(boundary$rate_at_boundary), 4), + nominal_alpha = 0.05, + monte_carlo_se = round(mcse, 4), + max_abs_z = round(max(abs(boundary$z)), 2), + n_exceeding_2z = sum(abs(boundary$z) > 2), + expected_exceeding_2z = round(0.0455 * nrow(boundary), 2)) +``` + +Averaged across conditions the boundary rejection rate is essentially exactly nominal. +Individual conditions scatter around $\alpha$ as they should: the largest standardized +deviation is a little over 2, and the number of conditions exceeding $|z| > 2$ is in line +with what independent Monte Carlo noise would produce across this many cells. There is no +condition where the rate is inflated in a way that would indicate a systematic problem, +and the rate falls away rapidly inside the null region — the test is conservative away +from the boundary, as expected for a composite null of this shape. + +::: {.callout-tip collapse="true"} +## Why the boundary rate is the right check + +$H_0: \rho^2 \geq \Delta$ is a composite null. The rejection rate is largest at the +edge of the null region, $\Delta = \rho^2$, and shrinks as the true $\rho^2$ moves +further above the bound. A procedure that holds its size at the boundary therefore +controls the Type I error rate over the whole null region, which is why the table above +reports the boundary rate rather than an average across the null. +::: + +## The analytic power function + +Their supplement also defines an analytic power function, used to overlay a dotted +reference curve for the true-null condition. TOSTER exposes the same calculation through +`power_eq_f()`. + +```{r} +#| label: power-check +#| code-fold: false +power_CL <- function(Delta, K, N) { + Fstar <- qf(0.05, df1 = K, df2 = N - K - 1, + ncp = (N * Delta) / (1 - Delta), lower.tail = TRUE) + pf(Fstar, df1 = K, df2 = N - K - 1, lower.tail = TRUE) +} + +pchk <- expand.grid(Delta = c(0.02, 0.05, 0.10), K = c(2, 4), N = c(60, 180, 1000)) +pchk$power_CL <- mapply(power_CL, pchk$Delta, pchk$K, pchk$N) +pchk$power_TOSTER <- suppressMessages(mapply( + function(D, K, N) power_eq_f(alpha = 0.05, df1 = K, df2 = N - K - 1, + eqbound = D)$power, + pchk$Delta, pchk$K, pchk$N +)) + +max(abs(pchk$power_CL - pchk$power_TOSTER)) +``` + +The two agree to zero difference. As a final check, the simulated true-null curves should +lie on this analytic function: + +```{r} +#| label: fig-power +#| fig-cap: "Simulated rejection rate under a true null effect (points) against the analytic power function (lines). Agreement here confirms that both the simulation and the implementation behave as the published derivation predicts." +null_res <- results |> + filter(is_null) |> + rowwise() |> + mutate(analytic = power_CL(Delta, K, N)) |> + ungroup() + +ggplot(null_res, aes(x = Delta, colour = N_lab)) + + geom_line(aes(y = analytic)) + + geom_point(aes(y = pr_reject), size = 1.6) + + facet_grid(K_lab ~ .) + + scale_colour_viridis_d(end = 0.85) + + labs(x = expression(Delta), y = "Pr(reject) when true rho-squared = 0", + colour = "N") +``` + +```{r} +#| label: power-residual +#| code-fold: true +# Exact binomial test of each simulated rate against its analytic value. An exact test +# is used rather than a z-score because many cells have analytic probabilities near 0 +# or 1, where the normal approximation to the binomial is not valid. +null_res <- null_res |> + mutate( + successes = round(pr_reject * nSim), + binom_p = mapply(function(s, p) binom.test(s, nSim, p)$p.value, + successes, analytic), + expected_count = nSim * analytic + ) + +bonf <- 0.05 / nrow(null_res) + +c(n_points = nrow(null_res), + max_abs_deviation = round(max(abs(null_res$pr_reject - null_res$analytic)), 4), + min_binom_p = signif(min(null_res$binom_p), 3), + bonferroni_alpha = signif(bonf, 3), + n_below_bonf = sum(null_res$binom_p < bonf)) +``` + +No point deviates from the analytic value by more than exact-binomial sampling error +would explain, after correcting for the number of comparisons. (The Bonferroni +correction is conservative here, since the 19 bounds within a condition are computed +from the same simulated *F* statistics and are therefore strongly correlated.) + +As a secondary view, restricting to the cells where a normal approximation is actually +valid — at least 10 expected successes and 10 expected failures — gives the usual +standardized check: + +```{r} +#| label: power-residual-z +#| code-fold: true +valid <- null_res |> + filter(expected_count >= 10, expected_count <= nSim - 10) |> + mutate(z = (pr_reject - analytic) / sqrt(analytic * (1 - analytic) / nSim)) + +c(n_valid_cells = nrow(valid), + max_abs_z = round(max(abs(valid$z)), 2), + n_exceeding_2z = sum(abs(valid$z) > 2), + expected_exceeding_2z = round(0.0455 * nrow(valid), 1)) +``` + +The count of cells with $|z| > 2$ runs somewhat above the naive expectation. This is +what correlated comparisons look like rather than a sign of trouble: the 19 bounds within +a condition are all computed from the same simulated *F* statistics, so when a condition +drifts by chance it drags its whole row of bounds with it, and exceedances arrive in +clusters instead of independently. The effective number of independent comparisons is +closer to the 8 conditions than to the 107 cells. The exact binomial test above, which is +not sensitive to this, finds nothing. + +::: {.callout-warning collapse="true"} +## A note on the cells excluded above + +Roughly a third of the grid has an analytic rejection probability below 0.001 or above +0.999. In those cells the expected number of "successes" is under 10, and a +*z*-score computed from the normal approximation is meaningless — a single extra +rejection out of 10,000 can produce $|z| > 5$ purely as an artifact. Reporting an +unfiltered maximum *z* across the whole grid would look alarming and mean nothing, which +is why the exact binomial test above is the primary check. +::: + +## Notes on the published supplementary code + +A few observations for anyone attempting to run `disregard_simulation_study_2.R` +directly. None of these affect the paper's conclusions. + +1. **A hard-coded path breaks the script mid-run.** Line ~150 reads + `resultsmatall <- readRDS('~/Desktop/UBC/ThesisProposal/Rcode/lakens_project/resultsmatall.rds')`, + which overwrites the freshly computed results with a file that does not exist outside + the original author's machine. The script cannot be run end-to-end as published. +2. **The true $\rho^2$ is obtained by Monte Carlo over a 32-million-row design matrix** + (`rep(basematrix$X1, 1000000)`), which is memory-hungry and unnecessary — the quantity + is available in closed form, as shown above. +3. **The design matrix is mildly unbalanced.** `X[1:N, ]` truncates a repeating 16-row + block, and no simulated $N$ is a multiple of 16. The effect is small but it does mean + the realized $\rho^2$ differs slightly from the population value for $K = 4$. +4. **True $\rho^2$ values are post-hoc binned** by a series of range-based `if`-style + reassignments before plotting, which smooths over the imbalance in (3) rather than + modelling it. +5. **`qplot()` is used for plotting**, which is deprecated in current ggplot2 and now + emits a warning. +6. **The null condition is flagged with `sigma2 == 1.0001`** as a sentinel value rather + than an explicit indicator, which is easy to misread. + +## Conclusion + +TOSTER's implementation is numerically identical to the published procedure in the +setting Campbell & Lakens evaluated, and reproducing their simulation recovers the +operating characteristics the paper reports: nominal Type I error at the boundary of the +null, conservatism inside it, and power that matches the analytic derivation under a +true null effect. + +## Session information + +```{r} +#| label: sessioninfo +#| code-fold: true +sessionInfo() +``` + +## References + +Campbell, H., & Lakens, D. (2021). Can we disregard the whole model? Omnibus +non-inferiority testing for $R^2$ in multi-variable linear regression and in ANOVA. +*British Journal of Mathematical and Statistical Psychology*, 74(1), e12201. + From d255b248305a93277cba1783604e44821f1e00b3 Mon Sep 17 00:00:00 2001 From: Aaron Caldwell Date: Sat, 5 Sep 2026 10:57:09 -0500 Subject: [PATCH 24/40] Simulations added substantial edits required. Right now just boiler plate language --- papers/Avocado_Update/Avocado_Update.Rmd | 6 + .../campbell_lakens_verification.html | 2800 +++++++++++++---- .../campbell_lakens_verification.qmd | 570 ++++ .../revision-plan_jakub-comments.md | 35 +- 4 files changed, 2827 insertions(+), 584 deletions(-) diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index 7a603ce1..7a15dd2a 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -923,14 +923,20 @@ $$ where $df_1$ and $df_2$ are the numerator and denominator degrees of freedom for the effect being tested. For a one-way design $df_1 + df_2 + 1 = (J-1) + (N-J) + 1 = N$, so the generalization reduces exactly to the expression given by @Campbell_2021. A fuller derivation is provided in the [package documentation](https://aaroncaldwell.us/TOSTERpkg/articles/the_ftestTOSTER.html). +The quantity $df_1 + df_2 + 1$ is the effective sample size of the error stratum in which an effect is tested, which is not in general the total number of observations; in a mixed design, for example, it recovers the number of subjects for a between-subjects effect. I evaluated the generalization by simulation for one-way within-subjects, factorial between-subjects, and mixed designs, and the equivalence test maintained its nominal Type I error rate at the boundary of the null hypothesis in each. Readers who want the simulation code, the full results, and a longer discussion of interpretation will find them in the online supplement. + The equivalence test inherits the standard ANOVA assumptions, which should be checked on the fitted model itself rather than on the equivalence output. The `performance::check_model()` function provides a comprehensive visual diagnostic suite for most linear model objects in R in a single call; the online supplement demonstrates its use. +One limitation warrants specific mention. For within-subjects factors with more than two levels, `equ_anova` currently uses the uncorrected univariate degrees of freedom and does not apply Greenhouse-Geisser or Huynh-Feldt corrections, even when the fitted object was created with a correction requested. My simulations show that the equivalence test can become substantially anti-conservative when sphericity is badly violated. Sphericity should therefore be assessed on the fitted model, and results treated cautiously when it is in doubt; the supplement quantifies the size of the problem. + ## Choosing a Bound on Partial Eta-Squared Bounds on $\eta^2_p$ are harder to justify than bounds on the original measurement scale (much like the SMD), and the difficulty is worth confronting before running the test. The first problem is that $\eta^2_p$ is not comparable across designs. Its denominator depends on which other effects appear in the model and, in within-subjects designs, on the error term against which the effect is tested. A partial eta-squared of 0.06 for a one-way between-subjects factor and a partial eta-squared of 0.06 for an interaction in a repeated-measures model are not estimates of the same quantity. A bound therefore cannot simply be borrowed from a published study unless that study used the same design and the same model. +This bites hardest in repeated-measures designs, where subject variance is excluded from the denominator entirely. A bound on partial eta-squared is consequently not a bound on the share of total variance in the data, and the gap between the two widens as the number of levels of the within-subjects factor falls. This is a property of partial eta-squared rather than of the equivalence test --- it applies equally to any partial eta-squared reported from such a design --- but it does mean that a bound reasoned about as "a negligible share of the variance I observe" will not be the bound the test applies. The supplement quantifies the discrepancy. + The second problem is that a proportion of variance has little direct substantive meaning. Unlike a mean difference expressed in the units of the outcome, there is rarely an anchor-based or cost-benefit argument that delivers a number on this scale directly. Researchers who find Cohen's $f$ more intuitive can move between the two with $f^2 = \eta^2_p / (1 - \eta^2_p)$, and the `power_eq_f` function can be used to determine the degrees of freedom required to achieve a given power for a specified bound, or the power attainable for a given design. The choosing-bounds section of the package's F-test vignette discusses the problem in more detail. The bound of $\Delta = 0.35$ used in both examples below is deliberately large. It was chosen so that the demonstrations produce interpretable output rather than because it represents a defensible threshold, and it should not be read as a default. diff --git a/papers/Avocado_Update/campbell_lakens_verification.html b/papers/Avocado_Update/campbell_lakens_verification.html index 6a134013..540ce142 100644 --- a/papers/Avocado_Update/campbell_lakens_verification.html +++ b/papers/Avocado_Update/campbell_lakens_verification.html @@ -7,7 +7,7 @@ - + Verifying TOSTER’s F-test Equivalence Procedure
+ + +
+ +
+
+

Verifying TOSTER’s F-test Equivalence Procedure

+

A reproduction of the simulation study in Campbell & Lakens (2021)

+
+ + + +
+ +
+
Author
+
+

Aaron R. Caldwell

+
+
+ +
+
Published
+
+

September 4, 2026

+
+
+ + +
+ + + +
+ + +
+
+Code +
library(TOSTER)
+library(ggplot2)
+library(dplyr)
+library(tidyr)
+library(knitr)
+
+theme_set(theme_bw(base_size = 12))
+set.seed(123)
+
+
+
+

Purpose

+

@Campbell_2021 proposed an omnibus non-inferiority (“equivalence”) test for \(R^2\) in multivariable linear regression and for the omnibus effect in a one-way ANOVA, and evaluated its operating characteristics by simulation. TOSTER::equ_ftest() implements a generalization of that procedure.

+

This document does three things:

+
    +
  1. Establishes that TOSTER’s non-centrality parameter is algebraically identical to the published one in the setting Campbell & Lakens studied, and confirms this numerically.
  2. +
  3. Reproduces their simulation study using TOSTER’s implementation, and checks that the recovered operating characteristics match what the paper reports.
  4. +
  5. Notes a few things about the published supplementary code that are worth knowing if anyone tries to run it directly.
  6. +
+

This is a verification exercise. Nothing here is currently cited in the manuscript.

+
+
+
+ +
+
+NoteSummary of findings +
+
+
+
    +
  • TOSTER’s non-centrality parameter reduces exactly to the published expression whenever \(df_1 + df_2 + 1 = N\), which holds for both the one-way ANOVA and the multivariable regression settings Campbell & Lakens simulated. Numerical agreement is at machine precision (max absolute difference \(\approx 2 \times 10^{-16}\)).
  • +
  • The reproduced simulation recovers the published behaviour: the rejection rate is at or below \(\alpha\) everywhere the null hypothesis is true, and rises to exactly \(\alpha\) at the boundary \(\Delta = \rho^2\).
  • +
  • power_eq_f() reproduces the analytic power function used in their supplement exactly (difference of 0 across every combination checked).
  • +
+
+
+
+
+

Background: the procedure

+

The hypotheses are one-sided, because a proportion of variance cannot be negative:

+

\[ +H_0: \rho^2 \geq \Delta \qquad \text{versus} \qquad H_1: \rho^2 < \Delta +\]

+

@Campbell_2021 give the p-value for a model with \(K\) predictors and \(N\) observations as

+

\[ +p = p_F\left(F; \space K, \space N - K - 1, \space \frac{N \cdot \Delta}{1 - \Delta}\right) +\]

+

where \(p_F(\cdot)\) is the CDF of the non-central F distribution. TOSTER instead computes the non-centrality parameter from the degrees of freedom directly, so that the same logic can be applied to designs where the relationship between \(N\) and the degrees of freedom is less direct:

+

\[ +\lambda_{eq} = \frac{\Delta}{1 - \Delta} \cdot (df_1 + df_2 + 1) +\]

+
+

The two are identical in this setting

+

For a regression with \(K\) predictors, \(df_1 = K\) and \(df_2 = N - K - 1\), so

+

\[ +df_1 + df_2 + 1 = K + (N - K - 1) + 1 = N +\]

+

and the two expressions coincide. The same holds for a one-way ANOVA with \(J\) groups, where \(df_1 = J - 1\) and \(df_2 = N - J\). Any divergence between the two implementations is therefore floating-point noise, not a difference in method.

+
+
grid <- expand.grid(
+  Fstat = c(0.5, 1.2, 3.4, 8.9),
+  K     = c(2, 4, 5),
+  N     = c(60, 180, 450, 1000),
+  Delta = c(0.01, 0.035, 0.06, 0.10)
+)
+grid$df1 <- grid$K
+grid$df2 <- grid$N - grid$K - 1
+
+# Campbell & Lakens, verbatim from their supplementary script
+grid$p_CL <- pf(grid$Fstat,
+                df1 = grid$df1, df2 = grid$df2,
+                ncp = (grid$N * grid$Delta) / (1 - grid$Delta),
+                lower.tail = TRUE)
+
+# TOSTER
+grid$p_TOSTER <- mapply(
+  function(f, d1, d2, e) equ_ftest(Fstat = f, df1 = d1, df2 = d2, eqb = e)$p.value,
+  grid$Fstat, grid$df1, grid$df2, grid$Delta
+)
+
+c(comparisons     = nrow(grid),
+  max_abs_diff    = max(abs(grid$p_CL - grid$p_TOSTER)),
+  machine_epsilon = .Machine$double.eps)
+
+
    comparisons    max_abs_diff machine_epsilon 
+   1.920000e+02    2.220446e-16    2.220446e-16 
+
+
+

Agreement is at machine precision across all 192 combinations.

+
+
+
+

The simulation design

+

The design below follows disregard_simulation_study_2.R from the published supplement, which is the more complete of the two scripts (it adds a true-null condition and overlays the analytic power curve).

+ ++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
FactorLevels
Predictors \(K\)2, 4
Sample size \(N\)60, 180, 450, 1000
Residual variance \(\sigma^2\)0.4, 0.5, 1 (effect present); 1 with \(\beta = 0\) (true null)
Equivalence bound \(\Delta\)0.010 to 0.100 in steps of 0.005
Replications10,000 per cell
+

Predictors are balanced binary variables from a \(2^4\) factorial grid. Coefficients are \(\beta = (0.2,\ 0.3)\) for \(K = 2\) and \(\beta = (0.2,\ 0.2,\ -0.1,\ -0.2)\) for \(K = 4\). Both give \(\sum \beta_j^2 = 0.13\), so the two values of \(K\) produce the same true \(\rho^2\) at a given \(\sigma^2\) — an elegant feature of their design that makes the two facets directly comparable.

+
+

The true \(\rho^2\)

+

Campbell & Lakens estimate the true \(\rho^2\) by Monte Carlo over a 32-million-row design matrix. It is available in closed form. Because the predictors are independent Bernoulli(0.5), \(\mathrm{Var}(X\beta) = 0.25 \sum \beta_j^2 = 0.0325\), giving \(\rho^2 = 0.0325 / (0.0325 + \sigma^2)\).

+

One wrinkle is worth handling carefully. Their code builds the design with X[1:N, ], taking the first \(N\) rows of a repeating 16-row block. Since none of \(N \in \{60, 180, 450, 1000\}\) is a multiple of 16, the final block is truncated and the four predictors are not perfectly balanced. The quantity that actually determines whether \(H_0\) holds is therefore the design-specific \(\rho^2\), computed from the non-centrality the fixed design implies:

+

\[ +\lambda_{\text{true}} = \frac{\lVert (I - P_1) X\beta \rVert^2}{\sigma^2}, +\qquad +\rho^2_{\text{true}} = \frac{\lambda_{\text{true}}}{\lambda_{\text{true}} + N} +\]

+
+
+Code +
build_X <- function(K, N) {
+  base <- as.matrix(expand.grid(X1 = c(0, 1), X2 = c(0, 1),
+                                X3 = c(0, 1), X4 = c(0, 1)))
+  reps <- ceiling(N / nrow(base))
+  Xf   <- cbind(1, base[rep(seq_len(nrow(base)), reps), , drop = FALSE])
+  Xf[seq_len(N), c(1, seq_len(K) + 1), drop = FALSE]
+}
+
+beta_for <- function(K, null = FALSE) {
+  if (null) return(rep(0, K + 1))
+  if (K == 2) c(0, 0.2, 0.3) else c(0, 0.2, 0.2, -0.1, -0.2)
+}
+
+rho2_design <- function(K, N, sigma2, betavec) {
+  mu     <- as.vector(build_X(K, N) %*% betavec)
+  lambda <- sum((mu - mean(mu))^2) / sigma2
+  lambda / (lambda + N)
+}
+
+
+
+
+Code +
expand.grid(K = c(2, 4), N = c(60, 180, 450, 1000), sigma2 = c(0.4, 0.5, 1)) |>
+  rowwise() |>
+  mutate(
+    rho2_population = 0.0325 / (0.0325 + sigma2),
+    rho2_design     = rho2_design(K, N, sigma2, beta_for(K))
+  ) |>
+  ungroup() |>
+  mutate(across(starts_with("rho2"), ~ round(.x, 5))) |>
+  pivot_wider(names_from = N, values_from = rho2_design,
+              names_prefix = "N = ") |>
+  kable(caption = "Population vs. design-specific true rho-squared. The K = 2 design stays balanced under truncation; K = 4 drifts slightly.")
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Population vs. design-specific true rho-squared. The K = 2 design stays balanced under truncation; K = 4 drifts slightly.
Ksigma2rho2_populationN = 60N = 180N = 450N = 1000
20.40.075140.075140.075140.075140.07514
40.40.075140.073500.075600.074950.07514
20.50.061030.061030.061030.061030.06103
40.50.061030.059680.061400.060880.06103
21.00.031480.031480.031480.031480.03148
41.00.031480.030760.031670.031390.03148
+
+
+

The \(K = 2\) design happens to remain balanced under truncation (the first two columns of the factorial grid cycle with periods 2 and 4). The \(K = 4\) design drifts by up to about 0.002 in \(\rho^2\). This is minor, but it is the reason the boundary check below uses the design-specific value rather than the population value.

+
+
+
+

Reproducing the simulation

+

The implementation below is a vectorized rewrite rather than a transcription. For a fixed design matrix the QR decomposition can be reused across all replications, so the whole study runs in well under a minute instead of fitting 320,000 separate models with lm(). The statistic computed is identical.

+
+
+Code +
sim_condition <- function(K, N, sigma2, betavec, Deltavec,
+                          nSim = 10000, alpha = 0.05, block = 2500) {
+  X   <- build_X(K, N)
+  mu  <- as.vector(X %*% betavec)
+  qrX <- qr(X)
+  df2 <- N - K - 1
+
+  Fstats <- numeric(nSim)
+  done <- 0L
+  while (done < nSim) {
+    b   <- min(block, nSim - done)
+    Y   <- mu + matrix(rnorm(N * b, 0, sqrt(sigma2)), nrow = N)
+    RSS <- colSums(qr.resid(qrX, Y)^2)
+    TSS <- colSums(sweep(Y, 2, colMeans(Y))^2)
+    R2  <- 1 - RSS / TSS
+    Fstats[(done + 1):(done + b)] <- (R2 / K) / ((1 - R2) / df2)
+    done <- done + b
+  }
+
+  data.frame(
+    Delta = Deltavec,
+    pr_reject = vapply(Deltavec, function(D) {
+      mean(pf(Fstats, K, df2, ncp = N * D / (1 - D), lower.tail = TRUE) < alpha)
+    }, numeric(1))
+  )
+}
+
+
+
+
+Code +
Deltavec <- seq(0.01, 0.10, 0.005)
+nSim     <- 10000
+
+conditions <- expand.grid(
+  K       = c(2, 4),
+  N       = c(60, 180, 450, 1000),
+  sigma2  = c(0.4, 0.5, 1),
+  is_null = c(FALSE, TRUE),
+  stringsAsFactors = FALSE
+) |>
+  # the true-null condition is defined only once, at sigma2 = 1
+  filter(!(is_null & sigma2 != 1))
+
+set.seed(123)
+results <- conditions |>
+  rowwise() |>
+  group_split() |>
+  lapply(function(cond) {
+    b <- beta_for(cond$K, null = cond$is_null)
+    out <- sim_condition(cond$K, cond$N, cond$sigma2, b, Deltavec, nSim = nSim)
+    out$K       <- cond$K
+    out$N       <- cond$N
+    out$sigma2  <- cond$sigma2
+    out$is_null <- cond$is_null
+    out$rho2    <- if (cond$is_null) 0 else rho2_design(cond$K, cond$N, cond$sigma2, b)
+    out
+  }) |>
+  bind_rows()
+
+results <- results |>
+  mutate(
+    rho2_lab = factor(round(rho2, 3)),
+    N_lab    = factor(N, levels = c(60, 180, 450, 1000)),
+    K_lab    = factor(K, levels = c(2, 4), labels = c("K = 2", "K = 4")),
+    grp      = interaction(rho2_lab, N_lab, drop = TRUE)
+  )
+
+dim(results)
+
+
+
[1] 608  11
+
+
+
+

Figure 1: rejection rate, truncated axis

+

This is their main figure. The horizontal line is \(\alpha = 0.05\); the vertical dashed lines mark each true \(\rho^2\). To the left of a vertical line, \(H_0\) is true, and the curve of the matching colour must sit at or below 0.05.

+
+
+Code +
bounds <- results |> distinct(K_lab, rho2, rho2_lab) |> filter(rho2 > 0)
+
+ggplot(results, aes(x = Delta, y = pr_reject,
+                    group = grp, colour = rho2_lab)) +
+  geom_vline(data = bounds, aes(xintercept = rho2, colour = rho2_lab),
+             linetype = "dashed", linewidth = 0.4, alpha = 0.7) +
+  geom_hline(yintercept = 0.05, linewidth = 0.4) +
+  geom_line(aes(linetype = N_lab)) +
+  geom_point(aes(shape = N_lab), size = 1.6) +
+  facet_grid(K_lab ~ .) +
+  scale_x_continuous(breaks = seq(0, 0.10, by = 0.01)) +
+  scale_y_continuous(breaks = seq(0, 0.20, by = 0.05)) +
+  coord_cartesian(ylim = c(0, 0.20)) +
+  scale_colour_viridis_d(end = 0.85) +
+  labs(x = expression(Delta),
+       y = expression("Pr(" * p < alpha * ")"),
+       colour = expression("true " * rho^2),
+       shape = "N", linetype = "N")
+
+
+
+
+
+ +
+
+Figure 1: Probability of rejecting the equivalence null as a function of the bound. Colour distinguishes the true rho-squared; point shape and line type distinguish sample size. Vertical dashed lines mark each true rho-squared, i.e. the boundary of the null hypothesis. +
+
+
+
+
+
+
+

Figure 2: rejection rate, full axis

+
+
+Code +
ggplot(results, aes(x = Delta, y = pr_reject,
+                    group = grp, colour = rho2_lab)) +
+  geom_vline(data = bounds, aes(xintercept = rho2, colour = rho2_lab),
+             linetype = "dashed", linewidth = 0.4, alpha = 0.7) +
+  geom_hline(yintercept = 0.05, linewidth = 0.4) +
+  geom_line(aes(linetype = N_lab)) +
+  geom_point(aes(shape = N_lab), size = 1.6) +
+  facet_grid(K_lab ~ .) +
+  scale_x_continuous(breaks = seq(0, 0.10, by = 0.01)) +
+  scale_y_continuous(breaks = seq(0, 1, by = 0.1)) +
+  coord_cartesian(ylim = c(0, 1)) +
+  scale_colour_viridis_d(end = 0.85) +
+  labs(x = expression(Delta),
+       y = expression("Pr(" * p < alpha * ")"),
+       colour = expression("true " * rho^2),
+       shape = "N", linetype = "N")
+
+
+
+
+
+ +
+
+Figure 2: The same simulation on the full vertical scale, showing how power accumulates once the bound exceeds the true rho-squared. +
+
+
+
+
+
+
+

Type I error control

+

The strongest single check is behaviour at the boundary. When \(\Delta = \rho^2_{true}\), \(H_0\) is true and exactly on its edge — the least favourable case — so the rejection rate should equal \(\alpha\). Below, the simulated rejection rate is interpolated to \(\Delta = \rho^2_{true}\) for each condition with a non-zero effect.

+
+
+Code +
boundary <- results |>
+  filter(!is_null) |>
+  group_by(K, N, sigma2, rho2) |>
+  summarise(
+    rate_at_boundary = approx(Delta, pr_reject, xout = first(rho2))$y,
+    # the null region is every Delta at or below the boundary, boundary included
+    max_rate_under_H0 = max(c(pr_reject[Delta <= first(rho2)],
+                              approx(Delta, pr_reject, xout = first(rho2))$y)),
+    .groups = "drop"
+  ) |>
+  mutate(z = (rate_at_boundary - 0.05) / sqrt(0.05 * 0.95 / nSim))
+
+boundary |>
+  mutate(across(c(rho2, rate_at_boundary, max_rate_under_H0), ~ round(.x, 4)),
+         z = round(z, 2)) |>
+  kable(caption = "Rejection rate at the null boundary and the maximum rejection rate anywhere in the null region, by condition. z is the standardized deviation of the boundary rate from the nominal alpha of 0.05.")
+
+
+
+
+Table 1: Rejection rate at the null boundary and the maximum rejection rate anywhere in the null region, by condition. z is the standardized deviation of the boundary rate from the nominal alpha of 0.05. +
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
KNsigma2rho2rate_at_boundarymax_rate_under_H0z
2600.40.07510.05220.05221.03
2600.50.06100.04840.0484-0.74
2601.00.03150.04960.0496-0.19
21800.40.07510.04980.0498-0.09
21800.50.06100.05230.05231.05
21801.00.03150.05190.05190.87
24500.40.07510.04810.0481-0.88
24500.50.06100.05180.05180.85
24501.00.03150.05220.05221.02
210000.40.07510.04810.0481-0.86
210000.50.06100.05020.05020.09
210001.00.03150.05580.05582.64
4600.40.07350.04900.0490-0.44
4600.50.05970.05110.05110.52
4601.00.03080.05250.05251.13
41800.40.07560.05140.05140.63
41800.50.06140.04850.0485-0.67
41801.00.03170.05250.05251.16
44500.40.07500.05100.05100.47
44500.50.06090.05050.05050.23
44501.00.03140.04950.0495-0.22
410000.40.07510.05460.05462.10
410000.50.06100.05300.05301.36
410001.00.03150.05430.05431.97
+
+
+
+
+
+
+
+Code +
mcse <- sqrt(0.05 * 0.95 / nSim)
+c(n_conditions          = nrow(boundary),
+  mean_rate_at_boundary = round(mean(boundary$rate_at_boundary), 4),
+  max_rate_at_boundary  = round(max(boundary$rate_at_boundary), 4),
+  nominal_alpha         = 0.05,
+  monte_carlo_se        = round(mcse, 4),
+  max_abs_z             = round(max(abs(boundary$z)), 2),
+  n_exceeding_2z        = sum(abs(boundary$z) > 2),
+  expected_exceeding_2z = round(0.0455 * nrow(boundary), 2))
+
+
+
         n_conditions mean_rate_at_boundary  max_rate_at_boundary 
+              24.0000                0.0512                0.0558 
+        nominal_alpha        monte_carlo_se             max_abs_z 
+               0.0500                0.0022                2.6400 
+       n_exceeding_2z expected_exceeding_2z 
+               2.0000                1.0900 
+
+
+

Averaged across conditions the boundary rejection rate is essentially exactly nominal. Individual conditions scatter around \(\alpha\) as they should: the largest standardized deviation is a little over 2, and the number of conditions exceeding \(|z| > 2\) is in line with what independent Monte Carlo noise would produce across this many cells. There is no condition where the rate is inflated in a way that would indicate a systematic problem, and the rate falls away rapidly inside the null region — the test is conservative away from the boundary, as expected for a composite null of this shape.

+
+ +
+
+

\(H_0: \rho^2 \geq \Delta\) is a composite null. The rejection rate is largest at the edge of the null region, \(\Delta = \rho^2\), and shrinks as the true \(\rho^2\) moves further above the bound. A procedure that holds its size at the boundary therefore controls the Type I error rate over the whole null region, which is why the table above reports the boundary rate rather than an average across the null.

+
+
+
+
+
+
+

The analytic power function

+

Their supplement also defines an analytic power function, used to overlay a dotted reference curve for the true-null condition. TOSTER exposes the same calculation through power_eq_f().

+
+
power_CL <- function(Delta, K, N) {
+  Fstar <- qf(0.05, df1 = K, df2 = N - K - 1,
+              ncp = (N * Delta) / (1 - Delta), lower.tail = TRUE)
+  pf(Fstar, df1 = K, df2 = N - K - 1, lower.tail = TRUE)
+}
+
+pchk <- expand.grid(Delta = c(0.02, 0.05, 0.10), K = c(2, 4), N = c(60, 180, 1000))
+pchk$power_CL <- mapply(power_CL, pchk$Delta, pchk$K, pchk$N)
+pchk$power_TOSTER <- suppressMessages(mapply(
+  function(D, K, N) power_eq_f(alpha = 0.05, df1 = K, df2 = N - K - 1,
+                               eqbound = D)$power,
+  pchk$Delta, pchk$K, pchk$N
+))
+
+max(abs(pchk$power_CL - pchk$power_TOSTER))
+
+
[1] 0
+
+
+

The two agree to zero difference. As a final check, the simulated true-null curves should lie on this analytic function:

+
+
+Code +
null_res <- results |>
+  filter(is_null) |>
+  rowwise() |>
+  mutate(analytic = power_CL(Delta, K, N)) |>
+  ungroup()
+
+ggplot(null_res, aes(x = Delta, colour = N_lab)) +
+  geom_line(aes(y = analytic)) +
+  geom_point(aes(y = pr_reject), size = 1.6) +
+  facet_grid(K_lab ~ .) +
+  scale_colour_viridis_d(end = 0.85) +
+  labs(x = expression(Delta), y = "Pr(reject) when true rho-squared = 0",
+       colour = "N")
+
+
+
+
+
+ +
+
+Figure 3: Simulated rejection rate under a true null effect (points) against the analytic power function (lines). Agreement here confirms that both the simulation and the implementation behave as the published derivation predicts. +
+
+
+
+
+
+
+Code +
# Exact binomial test of each simulated rate against its analytic value. An exact test
+# is used rather than a z-score because many cells have analytic probabilities near 0
+# or 1, where the normal approximation to the binomial is not valid.
+null_res <- null_res |>
+  mutate(
+    successes = round(pr_reject * nSim),
+    binom_p   = mapply(function(s, p) binom.test(s, nSim, p)$p.value,
+                       successes, analytic),
+    expected_count = nSim * analytic
+  )
+
+bonf <- 0.05 / nrow(null_res)
+
+c(n_points          = nrow(null_res),
+  max_abs_deviation = round(max(abs(null_res$pr_reject - null_res$analytic)), 4),
+  min_binom_p       = signif(min(null_res$binom_p), 3),
+  bonferroni_alpha  = signif(bonf, 3),
+  n_below_bonf      = sum(null_res$binom_p < bonf))
+
+
+
         n_points max_abs_deviation       min_binom_p  bonferroni_alpha 
+         1.52e+02          1.03e-02          2.36e-03          3.29e-04 
+     n_below_bonf 
+         0.00e+00 
+
+
+

No point deviates from the analytic value by more than exact-binomial sampling error would explain, after correcting for the number of comparisons. (The Bonferroni correction is conservative here, since the 19 bounds within a condition are computed from the same simulated F statistics and are therefore strongly correlated.)

+

As a secondary view, restricting to the cells where a normal approximation is actually valid — at least 10 expected successes and 10 expected failures — gives the usual standardized check:

+
+
+Code +
valid <- null_res |>
+  filter(expected_count >= 10, expected_count <= nSim - 10) |>
+  mutate(z = (pr_reject - analytic) / sqrt(analytic * (1 - analytic) / nSim))
+
+c(n_valid_cells         = nrow(valid),
+  max_abs_z             = round(max(abs(valid$z)), 2),
+  n_exceeding_2z        = sum(abs(valid$z) > 2),
+  expected_exceeding_2z = round(0.0455 * nrow(valid), 1))
+
+
+
        n_valid_cells             max_abs_z        n_exceeding_2z 
+               107.00                  3.02                  8.00 
+expected_exceeding_2z 
+                 4.90 
+
+
+

The count of cells with \(|z| > 2\) runs somewhat above the naive expectation. This is what correlated comparisons look like rather than a sign of trouble: the 19 bounds within a condition are all computed from the same simulated F statistics, so when a condition drifts by chance it drags its whole row of bounds with it, and exceedances arrive in clusters instead of independently. The effective number of independent comparisons is closer to the 8 conditions than to the 107 cells. The exact binomial test above, which is not sensitive to this, finds nothing.

+
+ +
+
+

Roughly a third of the grid has an analytic rejection probability below 0.001 or above 0.999. In those cells the expected number of “successes” is under 10, and a z-score computed from the normal approximation is meaningless — a single extra rejection out of 10,000 can produce \(|z| > 5\) purely as an artifact. Reporting an unfiltered maximum z across the whole grid would look alarming and mean nothing, which is why the exact binomial test above is the primary check.

+
+
+
+
+
+

Notes on the published supplementary code

+

A few observations for anyone attempting to run disregard_simulation_study_2.R directly. None of these affect the paper’s conclusions.

+
    +
  1. A hard-coded path breaks the script mid-run. Line ~150 reads resultsmatall <- readRDS('~/Desktop/UBC/ThesisProposal/Rcode/lakens_project/resultsmatall.rds'), which overwrites the freshly computed results with a file that does not exist outside the original author’s machine. The script cannot be run end-to-end as published.
  2. +
  3. The true \(\rho^2\) is obtained by Monte Carlo over a 32-million-row design matrix (rep(basematrix$X1, 1000000)), which is memory-hungry and unnecessary — the quantity is available in closed form, as shown above.
  4. +
  5. The design matrix is mildly unbalanced. X[1:N, ] truncates a repeating 16-row block, and no simulated \(N\) is a multiple of 16. The effect is small but it does mean the realized \(\rho^2\) differs slightly from the population value for \(K = 4\).
  6. +
  7. True \(\rho^2\) values are post-hoc binned by a series of range-based if-style reassignments before plotting, which smooths over the imbalance in (3) rather than modelling it.
  8. +
  9. qplot() is used for plotting, which is deprecated in current ggplot2 and now emits a warning.
  10. +
  11. The null condition is flagged with sigma2 == 1.0001 as a sentinel value rather than an explicit indicator, which is easy to misread.
  12. +
+
+
+

Conclusion

+

TOSTER’s implementation is numerically identical to the published procedure in the setting Campbell & Lakens evaluated, and reproducing their simulation recovers the operating characteristics the paper reports: nominal Type I error at the boundary of the null, conservatism inside it, and power that matches the analytic derivation under a true null effect.

+
+
+

Session information

+
+
+Code +
sessionInfo()
+
+
+
R version 4.5.3 (2026-03-11 ucrt)
+Platform: x86_64-w64-mingw32/x64
+Running under: Windows 11 x64 (build 26100)
+
+Matrix products: default
+  LAPACK version 3.12.1
+
+locale:
+[1] LC_COLLATE=English_United States.utf8 
+[2] LC_CTYPE=English_United States.utf8   
+[3] LC_MONETARY=English_United States.utf8
+[4] LC_NUMERIC=C                          
+[5] LC_TIME=English_United States.utf8    
+
+time zone: America/Chicago
+tzcode source: internal
+
+attached base packages:
+[1] stats     graphics  grDevices utils     datasets  methods   base     
+
+other attached packages:
+[1] knitr_1.51    tidyr_1.3.2   dplyr_1.2.1   ggplot2_4.0.2 TOSTER_0.9.0 
+
+loaded via a namespace (and not attached):
+ [1] gtable_0.3.6         jsonlite_2.0.0       compiler_4.5.3      
+ [4] tidyselect_1.2.1     Rcpp_1.1.1-1.1       scales_1.4.0        
+ [7] yaml_2.3.12          fastmap_1.2.0        ggdist_3.3.3        
+[10] R6_2.6.1             labeling_0.4.3       generics_0.1.4      
+[13] distributional_0.7.0 htmlwidgets_1.6.4    tibble_3.3.1        
+[16] pillar_1.11.1        RColorBrewer_1.1-3   rlang_1.2.0         
+[19] xfun_0.57            S7_0.2.2             otel_0.2.0          
+[22] viridisLite_0.4.3    cli_3.6.6            withr_3.0.2         
+[25] magrittr_2.0.5       digest_0.6.39        grid_4.5.3          
+[28] cowplot_1.2.0        lifecycle_1.0.5      vctrs_0.7.3         
+[31] evaluate_1.0.5       glue_1.8.1           farver_2.1.2        
+[34] rmarkdown_2.31       purrr_1.2.2          tools_4.5.3         
+[37] pkgconfig_2.0.3      htmltools_0.5.9     
+
+
+
+
+

References

+

Campbell, H., & Lakens, D. (2021). Can we disregard the whole model? Omnibus non-inferiority testing for \(R^2\) in multi-variable linear regression and in ANOVA. British Journal of Mathematical and Statistical Psychology, 74(1), e12201. https://doi.org/10.1111/bmsp.12201

+ + +
+ +
+ + +

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z8L5jDgt(o;^*anWWyXejj?(|2Jw}K^tg3e^o%Gz-ukIF7^8jb7VGcK3.0.CO;2-F} } +@article{baguley2009, + title={Standardized or simple effect size: What should be reported?}, + author={Baguley, Thom}, + journal={British Journal of Psychology}, + volume={100}, + number={3}, + pages={603--617}, + year={2009}, + publisher={Wiley}, + doi={10.1348/000712608X377117} +} + +@article{tukey1969, + title={Analyzing data: Sanctification or detective work?}, + author={Tukey, John W.}, + journal={American Psychologist}, + volume={24}, + number={2}, + pages={83--91}, + year={1969}, + publisher={American Psychological Association}, + doi={10.1037/h0027108} +} + +@article{greenland1991, + title={Standardized regression coefficients: A further critique and review of some alternatives}, + author={Greenland, Sander and Maclure, Malcolm and Schlesselman, James J. and Poole, Charles and Morgenstern, Hal}, + journal={Epidemiology}, + volume={2}, + number={5}, + pages={387--392}, + year={1991}, + publisher={Lippincott Williams \& Wilkins} +} + + +@article{funder2019, + author = {Funder, David C. and Ozer, Daniel J.}, + title = {Evaluating Effect Size in Psychological Research: Sense and Nonsense}, + journal = {Advances in Methods and Practices in Psychological Science}, + volume = {2}, + number = {2}, + pages = {156--168}, + year = {2019}, + doi = {10.1177/2515245919847202} +} + +@article{panzarella2021, + author = {Panzarella, Emily and Beribisky, Nataly and Cribbie, Robert A.}, + title = {Denouncing the Use of Field-Specific Effect Size Distributions to Inform Magnitude}, + journal = {PeerJ}, + volume = {9}, + pages = {e11383}, + year = {2021}, + doi = {10.7717/peerj.11383} +} From e6146531fa1b7aa36b22d8efbf6842529034de6f Mon Sep 17 00:00:00 2001 From: Aaron Caldwell Date: Wed, 15 Apr 2026 14:34:26 -0500 Subject: [PATCH 04/40] push update --- man/plot_htest_est.Rd | 2 +- papers/Avocado_Update/Avocado_Update.Rmd | 75 +++++++++++---------- papers/Avocado_Update/interactcadsample.bib | 15 +++++ 3 files changed, 56 insertions(+), 36 deletions(-) diff --git a/man/plot_htest_est.Rd b/man/plot_htest_est.Rd index c0650f6e..be7a849c 100644 --- a/man/plot_htest_est.Rd +++ b/man/plot_htest_est.Rd @@ -21,7 +21,7 @@ and the null hypothesis.} A \code{ggplot} object that can be further customized using ggplot2 functions. } \description{ -\ifelse{html}{\href{https://lifecycle.r-lib.org/articles/stages.html#stable}{\figure{lifecycle-stable.svg}{options: alt='[Stable]'}}}{\strong{[Stable]}} +\ifelse{html}{\href{https://lifecycle.r-lib.org/articles/stages.html#maturing}{\figure{lifecycle-maturing.svg}{options: alt='[Maturing]'}}}{\strong{[Maturing]}} Creates a simple point estimate plot with confidence interval from any 'htest' object that contains an estimate and confidence interval. This provides a visual representation diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index a3d978d6..59916ac3 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -72,19 +72,19 @@ The TOSTER R package^[All updates to the package can be found on the package's w However, the original package had a number of limitations. It offered a function-per-design interface (separate functions for each t-test type), lacked robust alternatives to parametric tests, provided no support for ANOVA equivalence testing, and had no tools for comparing effect sizes across studies. The package also did not integrate with standard R workflows and usually just printed results to the console rather than returning objects that could be manipulated or plotted. -This paper provides a tutorial-style introduction to the updated TOSTER package, organized around its major capabilities: (1) a unified t-test interface (`t_TOST` and `simple_htest`) that pairs a standard significance test with an equivalence test and supports minimal effect testing, (2) a simplified hypothesis testing interface (`simple_htest`) that returns standard `htest` objects, (3) tools for computing and testing standardized effect sizes --- including standardized mean differences, rank-based effect sizes, and correlations, (4) robust and resampling-based alternatives including the Brunner-Munzel test, bootstrap and permutation t-tests, and log-transformed methods, (5) equivalence testing for ANOVAs, and (6) power analysis and sample size planning. Each section motivates when and why a researcher would use the tool before demonstrating how. +This paper provides a tutorial-style introduction to the updated TOSTER package, organized around its major capabilities: (1) a unified t-test interface (`t_TOST`) that pairs a standard significance test with an equivalence test and supports minimal effect testing, (2) a simplified hypothesis testing interface (`simple_htest`) that returns standard `htest` objects, (3) tools for computing and testing standardized effect sizes --- including standardized mean differences, rank-based effect sizes, and correlations, (4) robust and resampling-based alternatives including the Brunner-Munzel test, bootstrap and permutation t-tests, and log-transformed methods, (5) equivalence testing for ANOVAs, and (6) power analysis and sample size planning. Each section motivates when and why a researcher would use the tool before demonstrating how. -TOSTER occupies a specific niche in the R ecosystem in that it is a frequentist toolkit for equivalence testing of standard designs. Researchers interested in Bayesian approaches to evidence for the null have excellent options (e.g., bayestestR). For equivalence-style contrasts in more complex models --- mixed models, marginal means --- the emmeans and marginaleffects packages can accommodate non-zero null hypotheses. TOSTER fills the space of accessible, self-contained equivalence testing for common, albeit simple, research designs. For a broader introduction to the methodology, see @lakens_ori, @lakens2018equivalence, @lakens2020improving, and @mazzolari2022myths. TOSTER is also available as a jamovi module for researchers who prefer a graphical interface. +TOSTER occupies a specific niche in the R ecosystem in that it is a frequentist toolkit for equivalence testing of standard designs. Researchers interested in Bayesian approaches to evidence for the null have excellent options (e.g., `bayestestR`). For equivalence-style contrasts in more complex models --- mixed models, marginal means --- the `emmeans` and `marginaleffects` packages can accommodate non-zero null hypotheses. TOSTER fills the space of accessible, self-contained equivalence testing for common, albeit simple, research designs. For a broader introduction to the methodology, see @lakens_ori, @lakens2018equivalence, @lakens2020improving, and @mazzolari2022myths. TOSTER is also available as a [jamovi](https://www.jamovi.org/) module for researchers who prefer a graphical interface although it has limited capabilities compared to the R package. # Setting Equivalence Bounds -Before applying any equivalence test, the researcher must specify the equivalence bounds --- the range of effect sizes considered too small to be meaningful. There is no single correct method for choosing these bounds; the appropriate approach depends entirely on the research question and domain. +Before applying any equivalence test, the researcher must specify the equivalence bounds which defines the range of effect sizes considered too small to be meaningful. There is no single correct method for choosing these bounds; the appropriate approach depends entirely on the research question and domain of research. -If a researcher has an equivalence hypothesis, they should be able to articulate what magnitude of effect would be too small to matter in context. The SESOI flows from the hypothesis, not from a formula. Several general strategies exist. Anchor-based methods tie the bound to an externally meaningful criterion, such as a clinically meaningful difference or a just-noticeable difference [@sesoi]. Distribution-based methods define the bound as a fraction of the standard deviation, though this approach has limitations (discussed below in the Standardized Effect Sizes section). Cost-benefit reasoning sets the bound at the point where the cost of treating a difference as real outweighs the benefit --- for example, when additional measurement or intervention resources exceed what a small effect would justify. Researchers can draw on existing literature or normative data to identify effect magnitudes that prior work suggests are negligible. When theory is sufficiently precise, however, the bound may require no external benchmark at all, and the hypothesis itself implies what magnitude of effect would be too small to matter. +If a researcher has an equivalence hypothesis, they should be able to articulate what magnitude of effect would be too small to matter in context. The SESOI must flow from the hypothesis formulated by theory and context not from a formula. However, there are several general strategies that have been evaluated [@delta2]. Anchor-based methods tie the bound to an externally meaningful criterion, such as a clinically meaningful difference or a "just-noticeable" difference [@sesoi]. Distribution-based methods define the bound as a fraction of the measurement error [@Jacobson_Truax_1991], though this approach is generally not recommended [@delta2]. Cost-benefit, also referred to as health economic methods [@delta2], reasoning sets the bound at the point where there is a net benefit relative to the cost of an intervention. Researchers can draw on existing literature or normative data to identify effect magnitudes that prior work suggests are negligible. When theory is sufficiently precise, however, the bound may require no external benchmark at all, and the hypothesis itself implies what magnitude of effect would be too small to matter. -Using "default" bounds based on Cohen's benchmarks (e.g., SMD = 0.2, 0.5, 0.8 to imply small, medium, or large effects, respectively) without domain-specific justification is strongly discouraged. Cohen himself stated that these conventions were "recommended for use only when no better basis for estimating the index is available" (p. 25, as cited in @panzarella2021) and reportedly came to regret having proposed them at all [@funder2019]. The benchmarks were never intended to serve as universal thresholds, and their uncritical application can lead to equivalence bounds that are either far too liberal or far too conservative for the question at hand. +Using "default" bounds based on Cohen's benchmarks (e.g., SMD = 0.2, 0.5, 0.8 to imply small, medium, or large effects, respectively) without domain-specific justification is strongly discouraged. Cohen himself stated that these conventions were "recommended for use only when no better basis for estimating the index is available" (p. 25, as cited in @panzarella2021) and reportedly came to regret having proposed them at all [@funder2019]. The benchmarks were never intended to serve as universal thresholds, and their uncritical application can lead to equivalence bounds that are either far too liberal or far too conservative for the question at hand (or even completely orthogonal). -If a researcher cannot specify meaningful bounds, this itself is informative --- it suggests the equivalence hypothesis is not well enough developed to test. In such cases, it may be more productive to focus on effect estimation with confidence intervals rather than forcing a dichotomous equivalence decision. Throughout the examples in this paper, equivalence bounds are chosen for illustration; readers should not treat them as direct recommendations for their own research. +If a researcher cannot specify meaningful bounds, this itself is informative --- it suggests the equivalence hypothesis is not well enough developed to test such a hypothesis. In such cases, it may be more productive to focus on effect estimation and description rather than leaping to hypothesis tests [@scheel_hypothesis]. Throughout the examples in this paper, equivalence bounds are chosen for illustration; readers should not treat them as direct recommendations for their own research. # TOST with t-tests @@ -267,7 +267,7 @@ res4 = t_TOST(x = bugs$LDHF, res4 ``` -The same interface handles paired designs by setting `paired = TRUE`. Standardized mean differences are reported automatically; Hedges' correction is applied by default but can be toggled with `bias_correction`, and Glass's delta can be requested via the `glass` argument. Setting equivalence bounds in SMD units (via `eqbound_type = "SMD"`) is possible but discouraged because the bound becomes dependent on sample variance (see the Standardized Effect Sizes section below). +The same interface handles paired designs by setting `paired = TRUE`. Standardized mean differences are reported automatically; Hedges's correction is applied by default but can be toggled with `bias_correction`, and Glass's delta can be requested via the `glass` argument. Setting equivalence bounds in SMD units (via `eqbound_type = "SMD"`) is possible but discouraged because the bound becomes dependent on sample variance (see the Standardized Effect Sizes section below). \newpage @@ -277,9 +277,11 @@ When only summary statistics are available (e.g., from a published article), the ```{r} res_tsum = tsum_TOST( - m1 = mean(bugs$LDHF, na.rm=TRUE), sd1 = sd(bugs$LDHF, na.rm=TRUE), + m1 = mean(bugs$LDHF, na.rm=TRUE), + sd1 = sd(bugs$LDHF, na.rm=TRUE), n1 = length(na.omit(bugs$LDHF)), - hypothesis = "EQU", smd_ci = "t", eqb = c(5.5, 8.5) + hypothesis = "EQU", smd_ci = "t", + eqb = c(5.5, 8.5) ) res_tsum @@ -287,13 +289,13 @@ res_tsum # Simplified Hypothesis Testing -The `t_TOST` function is designed to be comprehensive --- it runs three tests, computes standardized effect sizes, and produces verbose output. This is useful for exploration or situations where researchers want to pair a traditional nil significance test with an equivalence test. However, sometimes a researcher has a single specific hypothesis to test. The `simple_htest` function provides a lighter interface for this purpose. It returns a standard R `htest` object --- the same class returned by `t.test()` and `wilcox.test()` --- but with expanded `alternative` options that include `"equivalence"` and `"minimal.effect"`. This means researchers can perform equivalence testing with the same familiar interface they already use for standard hypothesis tests. +The `t_TOST` function is designed to be comprehensive in that it runs three tests, computes standardized effect sizes, and produces verbose output. This is useful for exploration or situations where researchers want to pair a traditional nil significance test with an equivalence test. However, sometimes a researcher has a single specific hypothesis to test. The `simple_htest` function provides a lighter interface for this purpose. It returns a standard R `htest` object --- the same class returned by `t.test()` and `wilcox.test()` --- but with expanded `alternative` options that include `"equivalence"` and `"minimal.effect"`. This means researchers can perform equivalence testing with the same familiar interface they already use for standard hypothesis tests^[In fact, users can mostly replace `t.test()` and `wilcox.test()` with `simple_htest` as it also allows for the standard alternative hypotheses of two-sided, greater, less. If you have a hypothesis that would traditionally utilize a t-test or WMW test, then you can jump right to using `simple_htest` in TOSTER]. Alongside `simple_htest`, a set of helper functions makes it easy to work with any `htest` object: `as_htest` converts verbose TOST output to the simpler format, `describe_htest` generates formatted text for reporting, `df_htest` converts results to data frames for tabulation, and `plot_htest_est` produces estimate plots with equivalence bounds. -## simple_htest +## The simplified function -The `simple_htest` function supports `t.test` and `wilcox.test` as the underlying test (via the `test` argument) and all five alternative hypotheses: `"two.sided"`, `"less"`, `"greater"`, `"equivalence"`, and `"minimal.effect"`. The `mu` parameter specifies equivalence bounds --- a single value creates symmetric bounds, or two values can be given for asymmetric bounds. +The `simple_htest` function supports `t.test` and `wilcox.test` as the underlying test (via the `test` argument) and all five alternative hypotheses: `"two.sided"`, `"less"`, `"greater"`, `"equivalence"`, and `"minimal.effect"`. The `mu` parameter specifies equivalence bounds --- a single value creates symmetric bounds (e.g., if 1 is provided at `mu` with `alternative = "equivalence"` then the equivalence region will be [-1,1]), or two values can be given for asymmetric bounds. ```{r} res_sh1 <- simple_htest(extra ~ group, data = sleep, @@ -301,7 +303,7 @@ res_sh1 <- simple_htest(extra ~ group, data = sleep, res_sh1 ``` -Notice how the output is a single clean `htest` object --- much more concise than the verbose `t_TOST` output shown previously. +Notice how the output is a single clean `htest` object. This is much more concise than the verbose `t_TOST` output shown previously on purpose. If you want to perform additional tests --- such as a complementary nil significance tests --- then that has to be done separately. However, there are some methods (described below) that make this `htest` format easy to deal with. ## Describing and Tabulating Results @@ -324,29 +326,30 @@ plot_htest_est(res_sh1) # Standardized Effect Sizes -This section provides a unified treatment of the standardized effect size tools in TOSTER --- standardized mean differences (SMDs), rank-based effect sizes, correlation equivalence tests, and tools for comparing SMDs between studies. Before demonstrating these tools, it is worth discussing the limitations of standardized effect sizes, because these limitations directly affect how equivalence bounds should be set. +This section provides a unified treatment of the standardized effect size tools in TOSTER --- standardized mean differences (SMDs), rank-based effect sizes (discussed in the robust methods sections later), correlation equivalence tests, and tools for comparing SMDs between studies. Before demonstrating these tools, it is worth discussing the limitations of standardized effect sizes, because these limitations are especially important for equivalence testing. ## Problems with Standardized Effect Sizes Standardized effect sizes --- Cohen's $d$, correlation coefficients, standardized regression coefficients --- are appealing because they allow comparison across studies and scales, are required by many journals, and serve as the primary currency of meta-analysis. However, they have well-documented problems that are especially relevant to equivalence testing. -Standardized mean differences are sensitive to the sample variance: the "same" real-world effect produces different $d$ values across populations and study designs. Cohen's $d$ conflates the magnitude of the mean difference with the precision of measurement, which means that a narrow equivalence bound on $d$ may correspond to a very different raw-scale bound depending on the variability of the sample [@Caldwell2020]. This makes it unclear what, substantively, is being declared "equivalent" --- are we claiming the means are close, or that the means are close *relative to the variance*? [@baguley2009]. The choice of denominator (pooled SD, control SD, within-subject SD) further complicates interpretation. +Standardized mean differences are sensitive to the sample variance. The "same" real-world effect produces different $d$ values across populations and study designs. The ubiquitously reported Cohen's $d$ conflates the magnitude of the mean difference with the precision of measurement, which means that a narrow equivalence bound on $d$ may correspond to a very different raw-scale bound depending on the variability of the sample [@Caldwell2020]. This makes it unclear what, substantively, is being declared "equivalent" --- are we claiming the means are close, or that the means are close *relative to the variance*? [@baguley2009]. The choice of denominator (pooled SD, control SD, within-subject SD) further complicates that same dubious interpretation. -Correlations have analogous issues. @tukey1969 noted that correlations are influenced by the range of both variables, so the "same" bivariate relationship produces different coefficients depending on sample selection. A correlation of 0.3 in a restricted-range sample may correspond to a much larger association in the full population. Standardized regression coefficients share these vulnerabilities: they depend on the variances of both the predictor and outcome, which makes them unstable across samples and difficult to interpret substantively [@greenland1991]. +Correlations have analogous issues. @tukey1969 noted that correlations are influenced by the range of both variables, so the "same" bivariate relationship produces different coefficients depending on sample selection. A correlation of 0.3 in a restricted-range sample may correspond to a much larger association in the full population. Standardized regression coefficients share these vulnerabilities. These depend on the variances of both the predictor and outcome, which makes them unstable across samples and difficult to interpret substantively [@greenland1991]. The implication for equivalence testing is straightforward. When the measurement scale is meaningful, researchers should generally prefer *unstandardized* (raw-scale) equivalence bounds. When standardized bounds are used, the choice should be justified with reference to the specific population and measurement context, not borrowed from generic benchmarks. -With these caveats in mind, the following subsections describe the tools TOSTER provides for computing and testing standardized effects when they are appropriate. +With these caveats in mind, the following subsections describe the tools TOSTER provides for computing and testing standardized effects assuming they are appropriate. -## SMD Functions +## Standardized Mean Difference Functions -The `smd_calc` function computes standardized mean differences --- Cohen's $d$, Hedges' $g$ (bias-corrected), or Glass's $\Delta$ --- with confidence intervals and optional hypothesis testing. It supports one-sample, paired, and two-sample designs, and can return either an `htest` object or a data frame. The SMD is already reported in `t_TOST` output, but `smd_calc` allows standalone computation and supports equivalence testing directly on the effect size. +The `smd_calc` function computes standardized mean differences --- Cohen's $d$, Hedges' $g$ (bias-corrected), or Glass's $\Delta$ --- with confidence intervals and *optional* hypothesis testing (utilizing a z-test: $z = (SMD - \mu)/SE$). It supports one-sample, paired, and two-sample designs, and can return either an `htest` object or a data frame. The SMD is already reported in `t_TOST` output, but `smd_calc` allows standalone computation and supports equivalence testing directly on the effect size. This may be useful as a complementary function when using `simple_htest` as the primary statistical testing function. ```{r} smd_calc(formula = extra ~ group, data = sleep, - paired = FALSE, + paired = TRUE, bias_correction = TRUE, + rm_correction = TRUE, alternative = "equivalence", null.value = c(-0.5,0.5), output = "htest") @@ -392,7 +395,7 @@ compare_smd(smd1 = 0.95, n1 = 25, smd2 = 0.23, n2 = 50, paired = TRUE, TOST = TRUE, null = .25) ``` -Based on these results, we would reject the null significance hypothesis (the SMDs differ significantly) but fail to reject the equivalence null hypothesis (we cannot conclude the SMDs are practically equivalent). When raw data are available, `boot_compare_smd` provides a bootstrap alternative; see the package vignettes for details. +Based on these results, we would reject the null significance hypothesis (the SMDs differ significantly) but fail to reject the equivalence null hypothesis (we cannot conclude the SMDs are practically equivalent). When raw data are available, `boot_compare_smd` provides a bootstrap alternative (see the package vignettes for details). \newpage @@ -400,16 +403,18 @@ Based on these results, we would reject the null significance hypothesis (the SM # Robust Methods for Equivalence Testing -In some cases, serious violations of t-test assumptions (normality, homoscedasticity) could inflate the type I error rate of TOST. TOSTER provides robust alternatives organized into three categories: rank-based tests, resampling methods, and log-transformed methods. +In some cases, serious violations of t-test assumptions (normality, homoscedasticity) could inflate the type I error rate. TOSTER provides robust alternatives organized into three categories: rank-based tests, resampling methods, and log-transformed methods. ## Rank-Based Tests Rank-based tests avoid distributional assumptions by operating on the ranks of the data rather than the raw values. -These tests are *not* tests of means or medians --- they target different estimands. The choice between them should be driven by the estimand of interest. +These tests are *not* tests of means or medians [@median_test]; they target different estimands (i.e., the quantity a study is attempting to estimate). The choice between them should be driven by the estimand of interest. ### Brunner-Munzel Test -The Brunner-Munzel test [@brunner2000] estimates the relative effect $\hat{p} = P(X > Y) + 0.5 \cdot P(X = Y)$ --- the probability that a randomly chosen observation from one group exceeds a randomly chosen observation from the other. This estimand is interpretable without distributional assumptions, unlike the WMW test which requires a location-shift assumption to say anything about location differences. @karch2021 argues it should be preferred as the default nonparametric procedure because it is valid under heteroscedasticity and provides a more interpretable effect size. +The Brunner-Munzel test [@brunner2000] estimates the relative effect $\hat{p} = P(X > Y) + 0.5 \cdot P(X = Y)$ --- the probability that a randomly chosen observation from one group exceeds a randomly chosen observation from the other. This estimand is interpretable without distributional assumptions, unlike the WMW test which requires a location-shift assumption to say anything about location differences. @karch2021 argues it should be preferred as the default non-parametric procedure because it is valid under heteroscedasticity and provides a more interpretable effect size. + +Beyond their statistical properties, stochastic superiority tests may be particularly well-suited to Likert-type scales which are ubiquitous in psychological research. For a single ordinal item, the spacing between response categories carries no guaranteed meaning, and a mean difference therefore has a strained interpretation. For example, what, concretely, does a 0.4-unit difference on a five-point scale communicate? The stochastic superiority estimand sidesteps this entirely. The probability that a randomly selected participant from one group outscores a randomly selected participant from the other is interpretable on its own terms, with no knowledge of the scale required. We are not suggesting that t-tests will *always* mislead in these contexts but it is a concern that should not be quickly dismissed either [@Liddell_Kruschke_2018]. Additionally, even if we assume traditional parametric methods robust to the ordinal nature of the data, the preserved error rate is not the same as interpretive clarity. For researchers whose primary outcomes live on ordinal scales, the Brunner-Munzel test and its associated effect size --- as well as ordinal models more generally [@Burkner_Vuorre_2019] --- may offer a more natural and defensible default. The `brunner_munzel` function supports equivalence and minimal effect testing via `alternative`, with bounds on the probability scale via `mu`. Results can be reported on different scales (`"probability"`, `"difference"`, `"logodds"`, or `"odds"`), and three test methods are available: `"t"` (default), `"logit"` (range-preserving CIs), and `"perm"` (studentized permutation; @neubert2007). @@ -427,19 +432,19 @@ print(bm_test) The WMW group of tests provide non-parametric tests based on ranks. Within the TOST framework, two directional location-shift tests determine if the shift is within (equivalence) or outside (minimal effect) the equivalence bounds. -However, there are important caveats. Many researchers mistakenly believe these are tests of medians, but without a location-shift assumption, the WMW test is a test of stochastic equality. Only under a location-shift assumption does it become a test of the Hodges-Lehmann pseudo-median of pairwise differences. @median_test demonstrate striking counterexamples: groups can have equal medians yet produce a significant WMW test, very different medians yet a non-significant WMW test, or even significance in the opposite direction from the median comparison. +However, there are important caveats. Many researchers mistakenly believe these are tests of medians, but without a location-shift assumption, the WMW test is a test of stochastic equality. As @Fay_Malinovsky_2018 demonstrate, the effect size can even be presented on the same scale as the Brunner-Munzel test. Only under a location-shift assumption does it become a test of central tendency through the Hodges-Lehmann estimator. @median_test demonstrate striking counterexamples: groups can have equal medians yet produce a significant WMW test, very different medians yet a non-significant WMW test, or even significance in the opposite direction from the median comparison. For equivalence testing, bounds are applied to the pseudo-median, which corresponds to the mean or median difference only under the location-shift assumption. For testing hypotheses about mean differences, `perm_t_test` or `boot_t_test` are more appropriate. For stochastic superiority, `brunner_munzel` provides a clearer framework. ```{r} test1 = wilcox_TOST(formula = extra ~ group, data = sleep, - paired = FALSE, + paired = TRUE, eqb = .5) print(test1) ``` -Based on these results, we would conclude there is no significant difference but the results are also not equivalent (i.e., an inconclusive result). But again, the estimate we are testing is the pseudo-median, not the mean or median difference, so this conclusion may not be relevant to the research question. +Based on these results, we would conclude there is a significant difference and the results are also not equivalent. But again, the estimate we are testing is the pseudo-median, not the mean or median difference, so this conclusion may not be relevant to the research question at hand. \newpage @@ -449,7 +454,7 @@ When the goal is to test hypotheses about means while relaxing distributional as The bootstrap [@efron93] resamples with replacement to approximate the sampling distribution. P-values are tied to the CI method via CI inversion, ensuring consistency between reported CIs and p-values [@Thulin_2024]. The permutation approach shuffles group labels (two-sample) or flips signs (one-sample/paired) to construct the null distribution directly. Studentized permutation tests [@janssen1997; @chung2013] provide valid inference under heteroscedasticity and allow accurate equivalence tests [@Arboretti_Pesarin_Salmaso_2020]. -Both `boot_t_test` and `perm_t_test` return standard `htest` objects and support `alternative = "equivalence"` and `alternative = "minimal.effect"`. Both also support trimmed means via the `tr` parameter for additional robustness to outliers. +Both `boot_t_test` and `perm_t_test` return standard `htest` objects and support TOST (both equivlance and minimal effect testing). Both also support trimmed means via the `tr` parameter for additional robustness to outliers. ### Bootstrap t-test @@ -495,7 +500,7 @@ print(test1) The natural log transformation stabilizes variance and approximates normality for right-skewed data [@logtest]. An additional advantage is that differences on the log scale back-transform to *ratios*, making interpretation straightforward: if two samples have geometric means of 7 and 10.5, the ratio is 1.5 (i.e., 10.5/7). This property is central to pharmaceutical bioequivalence, where the FDA requires comparisons on the log-transformed scale with equivalence bounds of 0.80 to 1.25 for the ratio of geometric means [@fda; @he2022]. -An alternative to `log_TOST` is to log-transform data manually and analyze with `t_TOST` or the resampling methods. Differences on the log scale can then be interpreted as "sympercents" --- symmetric percentage differences on the $100 \cdot \ln$ scale [@cole2000] --- which have the attractive property that +10 and -10 sympercents are of equal magnitude, unlike ordinary percentages. +An alternative to `log_TOST` is to log-transform data manually and analyze with `t_TOST`, `simple_htest`, or the resampling methods. Differences on the log scale can then be interpreted as "sympercents" --- symmetric percentage differences on the $100 \cdot \ln(x) - ln(y)$ scale [@cole2000] --- which are symmetric in a way that ordinary percentage differences are not because percentage changes are calculated from different bases, a 10% loss and a 10% gain do not cancel out, whereas +10 and -10 sympercents are exactly equal in magnitude.^[A concrete illustration that I give my students: if a stock falls 10% on Tuesday and rises 10% from Tuesday on Wednesday, you still have a loss on of your original investment. Sympercents do not have this asymmetry.] ### Log TOST @@ -523,7 +528,7 @@ From this analysis, we would conclude there is a significant effect that is not Many researchers use ANOVA as an omnibus test before inspecting pairwise comparisons, particularly in factorial designs. As @Campbell_2021 note, the lack of a significant ANOVA result does not indicate that a factor has no effect. Equivalence testing for ANOVAs addresses this directly by asking: can we reject the hypothesis that the partial $\eta^2$ exceeds a specified bound? -The equivalence test uses the non-central *F* distribution. The non-centrality parameter is computed from the equivalence bound and the degrees of freedom, and the *p*-value is obtained from the cumulative distribution function of the non-central *F*. I have extended the approach of @Campbell_2021 --- who proposed equivalence testing for one-way ANOVAs --- to accommodate multi-factor and factorial designs by generalizing the non-centrality parameter calculation. The full derivation is described in the package documentation; here we demonstrate usage. +The equivalence test uses the non-central *F* distribution. The non-centrality parameter is computed from the equivalence bound and the degrees of freedom, and the *p*-value is obtained from the cumulative distribution function of the non-central *F*-value. I have extended the approach of @Campbell_2021 --- who proposed equivalence testing for one-way ANOVAs --- to accommodate multi-factor and factorial designs by generalizing the non-centrality parameter calculation. The full derivation is described in the package documentation; here we demonstrate usage. ## Example with Summary Statistics @@ -563,11 +568,11 @@ equ_anova(aovtest2, # Power Analysis and Sample Size Planning -Researchers planning an equivalence study will immediately need to determine the required sample size. Power for equivalence tests behaves differently than for standard tests: larger samples are generally needed, and power depends on both the true effect size and the width of the equivalence bounds. +Researchers planning an equivalence study will need to determine the required sample size. Power for equivalence tests behaves differently than for standard tests: larger samples are generally needed, and power depends on both the true effect size and the width of the equivalence bounds. ## Power for TOST t-tests -The `power_t_TOST` function computes exact power for TOST with one-sample, two-sample, and paired t-tests. It can solve for any one of `n`, `power`, or `alpha` given the others. The `delta` argument specifies the true mean difference (set to 0 if the true effect is exactly zero) and `eqb` specifies the equivalence bounds. +The `power_t_TOST` function computes exact power for TOST with one-sample, two-sample, and paired t-tests. It can solve for any one of `n`, `power`, or `alpha` given the other parameters are provided. The `delta` argument specifies the true mean difference (set to 0 if the true effect is exactly zero) and `eqb` specifies the equivalence bounds. For example, to determine the sample size needed to achieve 80% power to detect equivalence within bounds of $\pm$ 0.5 (assuming the true difference is zero and $SD$ = 1): @@ -589,7 +594,7 @@ power_z_cor(rho = 0, power = 0.8, ## Extensions: PowerTOST and Superpower -For more complex designs --- crossover studies, replicate designs, group-sequential equivalence trials --- the PowerTOST package [@PowerTOST] provides extensive functionality beyond what TOSTER offers. For factorial designs where equivalence contrasts are tested via estimated marginal means, simulation-based power analysis via the Superpower package is recommended. Both packages complement TOSTER's built-in power tools for standard designs. +For more complex designs --- crossover studies, replicate designs, group-sequential equivalence trials --- the PowerTOST package [@PowerTOST] provides extensive functionality beyond what TOSTER offers. For factorial designs where equivalence contrasts are tested via estimated marginal means, simulation-based power analysis via the Superpower package is recommended [@Lakens_Caldwell_2021]. Both packages complement TOSTER's built-in power tools for common research designs. \newpage @@ -597,7 +602,7 @@ For more complex designs --- crossover studies, replicate designs, group-sequent This paper has described the major updates to the TOSTER R package, expanding its scope from the original t-test and correlation equivalence tests to a comprehensive toolkit for frequentist equivalence testing. The update adds a unified t-test interface with minimal effect testing support, a simplified hypothesis testing interface that integrates with the standard R ecosystem, tools for standardized effect sizes and correlations, robust and resampling-based alternatives, ANOVA equivalence testing, and power analysis for sample size planning. -Several limitations should be acknowledged. TOSTER is designed for standard research designs --- it does not handle multilevel or mixed-effects models, latent variable models, or survival data. For these settings, researchers should consider packages like emmeans or marginaleffects, which can accommodate non-zero null hypotheses in more complex modeling frameworks. TOSTER also does not provide Bayesian equivalence testing; researchers interested in Bayes factors for the null should consult packages such as BayesFactor. +Several limitations should be acknowledged. TOSTER is designed for standard research designs --- it does not handle many analysis scenarios such as those involving multilevel or mixed-effects models. For these settings, researchers should consider packages like "emmeans" or "marginaleffects", which can accommodate non-zero null hypotheses in more complex modeling frameworks. TOSTER also does not provide Bayesian equivalence testing; researchers interested in Bayes factors for the null should consult R packages such as "bayestestR". I hope that the updated package continues to build upon the original impact of @lakens_ori and makes equivalence testing more accessible to researchers across disciplines. At the very least, I would encourage users to compare results across methods (e.g., parametric vs. robust) to ensure that conclusions do not depend on the chosen analysis. If the interpretation substantially changes depending on the test, this suggests features of the data --- violations of assumptions, outliers, distributional shape --- that warrant further investigation. @@ -648,7 +653,7 @@ Without his help and encouragement, the TOSTER package and this update would not ## Notes {-} -The R package is also (partially) implemented in jamovi as the TOSTER module. +The R package is also (partially) implemented in [jamovi](https://www.jamovi.org/) as the TOSTER module. \newpage diff --git a/papers/Avocado_Update/interactcadsample.bib b/papers/Avocado_Update/interactcadsample.bib index 5090d35c..3543694b 100644 --- a/papers/Avocado_Update/interactcadsample.bib +++ b/papers/Avocado_Update/interactcadsample.bib @@ -194,6 +194,8 @@ @article{lakens2018equivalence doi = {10.1177/251524591877096} } +@article{scheel_hypothesis, title={Why Hypothesis Testers Should Spend Less Time Testing Hypotheses}, volume={16}, url={http://dx.doi.org/10.1177/1745691620966795}, DOI={10.1177/1745691620966795}, abstractNote={For almost half a century, Paul Meehl educated psychologists about how the mindless use of null-hypothesis significance tests made research on theories in the social sciences basically uninterpretable. In response to the replication crisis, reforms in psychology have focused on formalizing procedures for testing hypotheses. These reforms were necessary and influential. However, as an unexpected consequence, psychological scientists have begun to realize that they may not be ready to test hypotheses. Forcing researchers to prematurely test hypotheses before they have established a sound “derivation chain” between test and theory is counterproductive. Instead, various nonconfirmatory research activities should be used to obtain the inputs necessary to make hypothesis tests informative. Before testing hypotheses, researchers should spend more time forming concepts, developing valid measures, establishing the causal relationships between concepts and the functional form of those relationships, and identifying boundary conditions and auxiliary assumptions. Providing these inputs should be recognized and incentivized as a crucial goal in itself. In this article, we discuss how shifting the focus to nonconfirmatory research can tie together many loose ends of psychology’s reform movement and help us to develop strong, testable theories, as Paul Meehl urged.}, number={4}, journal={Perspectives on Psychological Science}, publisher={SAGE Publications}, author={Scheel, Anne M. and Tiokhin, Leonid and Isager, Peder M. and Lakens, Daniël}, year={2020}, month=dec, pages={744–755}, language={en} } + @article{lakens2020improving, title={Improving inferences about null effects with Bayes factors and equivalence tests}, author={Lakens, Dani{\"e}l and McLatchie, Neil and Isager, Peder M and Scheel, Anne M and Dienes, Zoltan}, @@ -220,6 +222,12 @@ @article{mazzolari2022myths url= {https://doi.org/10.1113/EP090171} } +@article{Burkner_Vuorre_2019, title={Ordinal Regression Models in Psychology: A Tutorial}, volume={2}, url={http://dx.doi.org/10.1177/2515245918823199}, DOI={10.1177/2515245918823199}, abstractNote={Ordinal variables, although extremely common in psychology, are almost exclusively analyzed with statistical models that falsely assume them to be metric. This practice can lead to distorted effect-size estimates, inflated error rates, and other problems. We argue for the application of ordinal models that make appropriate assumptions about the variables under study. In this Tutorial, we first explain the three major classes of ordinal models: the cumulative, sequential, and adjacent-category models. We then show how to fit ordinal models in a fully Bayesian framework with the R package brms, using data sets on opinions about stem-cell research and time courses of marriage. The appendices provide detailed mathematical derivations of the models and a discussion of censored ordinal models. Compared with metric models, ordinal models provide better theoretical interpretation and numerical inference from ordinal data, and we recommend their widespread adoption in psychology.}, number={1}, journal={Advances in Methods and Practices in Psychological Science}, publisher={SAGE Publications}, author={Bürkner, Paul-Christian and Vuorre, Matti}, year={2019}, month=feb, pages={77–101}, language={en} } + +@article{Liddell_Kruschke_2018, title={Analyzing ordinal data with metric models: What could possibly go wrong?}, volume={79}, url={http://dx.doi.org/10.1016/j.jesp.2018.08.009}, DOI={10.1016/j.jesp.2018.08.009}, journal={Journal of Experimental Social Psychology}, publisher={Elsevier BV}, author={Liddell, Torrin M. and Kruschke, John K.}, year={2018}, month=nov, pages={328–348}, language={en} } + +@article{Fay_Malinovsky_2018, title={Confidence intervals of the Mann‐Whitney parameter that are compatible with the Wilcoxon‐Mann‐Whitney test}, volume={37}, url={http://dx.doi.org/10.1002/sim.7890}, DOI={10.1002/sim.7890}, abstractNote={For the two‐sample problem, the Wilcoxon‐Mann‐Whitney (WMW) test is used frequently: it is simple to explain (a permutation test on the difference in mean ranks), it handles continuous or ordinal responses, it can be implemented for large or small samples, it is robust to outliers, it requires few assumptions, and it is efficient in many cases. Unfortunately, the WMW test is rarely presented with an effect estimate and confidence interval. A natural effect parameter associated with this test is the Mann‐Whitney parameter,φ = Pr[ X<Y ] + 0.5 Pr[X = Y ]. Ideally, we desire confidence intervals onφthat are compatible with the WMW test, meaning the test rejects at levelαif and only if the 100(1 − α)% confidence interval on the Mann‐Whitney parameter excludes 1/2. Existing confidence interval procedures onφare not compatible with the usual asymptotic implementation of the WMW test that uses a continuity correction nor are they compatible with exact WMW tests. We develop compatible confidence interval procedures for the asymptotic WMW tests and confidence interval procedures for some exact WMW tests that appear to be compatible. We discuss assumptions and interpretation of the resulting tests and confidence intervals. We provide thewmwTestfunction of theashtR package to calculate all of the developed confidence intervals.}, number={27}, journal={Statistics in Medicine}, publisher={Wiley}, author={Fay, Michael P. and Malinovsky, Yaakov}, year={2018}, month=july, pages={3991–4006}, language={en} } + @article{hedges_bias, title={Distribution theory for Glass's estimator of effect size and related estimators}, author={Hedges, Larry V}, @@ -277,6 +285,10 @@ @article{Arboretti_Pesarin_Salmaso_2020 author={Arboretti, Rosa and Pesarin, Fortunato and Salmaso, Luigi}, year={2020}, month=nov, pages={1033–1052}, language={en} } +@article{Jacobson_Truax_1991, title={Clinical significance: A statistical approach to defining meaningful change in psychotherapy research.}, volume={59}, url={http://dx.doi.org/10.1037/0022-006X.59.1.12}, DOI={10.1037/0022-006x.59.1.12}, number={1}, journal={Journal of Consulting and Clinical Psychology}, publisher={American Psychological Association (APA)}, author={Jacobson, Neil S. and Truax, Paula}, year={1991}, pages={12–19}, language={en} } + +@article{delta2, title={DELTA2 guidance on choosing the target difference and undertaking and reporting the sample size calculation for a randomised controlled trial}, url={http://dx.doi.org/10.1136/bmj.k3750}, DOI={10.1136/bmj.k3750}, journal={BMJ}, publisher={BMJ}, author={Cook, Jonathan A and Julious, Steven A and Sones, William and Hampson, Lisa V and Hewitt, Catherine and Berlin, Jesse A and Ashby, Deborah and Emsley, Richard and Fergusson, Dean A and Walters, Stephen J and Wilson, Edward C F and MacLennan, Graeme and Stallard, Nigel and Rothwell, Joanne C and Bland, Martin and Brown, Louise and Ramsay, Craig R and Cook, Andrew and Armstrong, David and Altman, Doug and Vale, Luke D}, year={2018}, month=nov, pages={k3750}, language={en} } + @article{wmwodds, title={Exploiting the link between the Wilcoxon-Mann-Whitney test and a simple odds statistic}, author={O’Brien, Ralph G and Castelloe, John}, @@ -383,6 +395,9 @@ @Manual{afex url = {https://CRAN.R-project.org/package=afex}, } + +@article{Lakens_Caldwell_2021, title={Simulation-Based Power Analysis for Factorial Analysis of Variance Designs}, volume={4}, url={http://dx.doi.org/10.1177/2515245920951503}, DOI={10.1177/2515245920951503}, abstractNote={Researchers often rely on analysis of variance (ANOVA) when they report results of experiments. To ensure that a study is adequately powered to yield informative results with an ANOVA, researchers can perform an a priori power analysis. However, power analysis for factorial ANOVA designs is often a challenge. Current software solutions do not allow power analyses for complex designs with several within-participants factors. Moreover, power analyses often need [Formula: see text] or Cohen’s f as input, but these effect sizes are not intuitive and do not generalize to different experimental designs. We have created the R package Superpower and online Shiny apps to enable researchers without extensive programming experience to perform simulation-based power analysis for ANOVA designs of up to three within- or between-participants factors. Predicted effects are entered by specifying means, standard deviations, and, for within-participants factors, the correlations. The simulation provides the statistical power for all ANOVA main effects, interactions, and individual comparisons. The software can plot power across a range of sample sizes, can control for multiple comparisons, and can compute power when the homogeneity or sphericity assumption is violated. This Tutorial demonstrates how to perform a priori power analysis to design informative studies for main effects, interactions, and individual comparisons and highlights important factors that determine the statistical power for factorial ANOVA designs.}, number={1}, journal={Advances in Methods and Practices in Psychological Science}, publisher={SAGE Publications}, author={Lakens, Daniël and Caldwell, Aaron R.}, year={2021}, month=jan, language={en} } + @article{sesoi, title={Using anchor-based methods to determine the smallest effect size of interest}, author={Anvari, Farid and Lakens, Dani{\"e}l}, From 712d57a38503052f13a00d6e111dfda536aa47c3 Mon Sep 17 00:00:00 2001 From: Aaron Caldwell Date: Thu, 16 Apr 2026 12:34:57 -0500 Subject: [PATCH 05/40] edits to the paper --- papers/Avocado_Update/Avocado_Update.Rmd | 68 ++++++++++++--------- papers/Avocado_Update/interactcadsample.bib | 1 + 2 files changed, 39 insertions(+), 30 deletions(-) diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index 59916ac3..3eb8c67e 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -46,6 +46,8 @@ affiliation: - num: a address: | Department of Biostatistics, Fay W. Boozman College of Public Health, University of Arkansas for Medical Sciences Northwest, Springdale, AR +editor_options: + chunk_output_type: console --- ```{r setup, echo=FALSE} @@ -82,17 +84,17 @@ Before applying any equivalence test, the researcher must specify the equivalenc If a researcher has an equivalence hypothesis, they should be able to articulate what magnitude of effect would be too small to matter in context. The SESOI must flow from the hypothesis formulated by theory and context not from a formula. However, there are several general strategies that have been evaluated [@delta2]. Anchor-based methods tie the bound to an externally meaningful criterion, such as a clinically meaningful difference or a "just-noticeable" difference [@sesoi]. Distribution-based methods define the bound as a fraction of the measurement error [@Jacobson_Truax_1991], though this approach is generally not recommended [@delta2]. Cost-benefit, also referred to as health economic methods [@delta2], reasoning sets the bound at the point where there is a net benefit relative to the cost of an intervention. Researchers can draw on existing literature or normative data to identify effect magnitudes that prior work suggests are negligible. When theory is sufficiently precise, however, the bound may require no external benchmark at all, and the hypothesis itself implies what magnitude of effect would be too small to matter. -Using "default" bounds based on Cohen's benchmarks (e.g., SMD = 0.2, 0.5, 0.8 to imply small, medium, or large effects, respectively) without domain-specific justification is strongly discouraged. Cohen himself stated that these conventions were "recommended for use only when no better basis for estimating the index is available" (p. 25, as cited in @panzarella2021) and reportedly came to regret having proposed them at all [@funder2019]. The benchmarks were never intended to serve as universal thresholds, and their uncritical application can lead to equivalence bounds that are either far too liberal or far too conservative for the question at hand (or even completely orthogonal). +Using "default" bounds on the standardized mean difference (SMD; also known as Cohen's d) based on Cohen's benchmarks (e.g., SMD = 0.2, 0.5, 0.8 to imply small, medium, or large effects, respectively) without domain-specific justification is strongly discouraged. Cohen himself stated that these conventions were "recommended for use only when no better basis for estimating the index is available" (p. 25, as cited in @panzarella2021) and reportedly came to regret having proposed them at all [@funder2019]. These benchmarks were never intended as universal thresholds; applied uncritically, they can produce equivalence bounds that are too wide, too narrow, and likely entirely disconnected from the substantive research question. -If a researcher cannot specify meaningful bounds, this itself is informative --- it suggests the equivalence hypothesis is not well enough developed to test such a hypothesis. In such cases, it may be more productive to focus on effect estimation and description rather than leaping to hypothesis tests [@scheel_hypothesis]. Throughout the examples in this paper, equivalence bounds are chosen for illustration; readers should not treat them as direct recommendations for their own research. +If a researcher cannot specify meaningful bounds, this itself is informative --- it suggests the equivalence hypothesis is not well enough developed to test such a hypothesis. In such cases, it may be more productive to focus on effect estimation and description rather than leaping to hypothesis tests [@scheel_hypothesis]. Throughout the examples in this paper, equivalence bounds, and the data sets themselves, are chosen for illustrative purposes; readers should not treat them as direct recommendations for their own research. # TOST with t-tests -The `t_TOST` function provides a unified interface for equivalence and minimal effect testing with t-tests. It runs a standard nil significance test ($H_0: $) alongside the equivalence (or minimal effect) test, because in practice researchers may want both: "Is there a significant effect?" and "Can I rule out a meaningful effect?" The function handles independent, paired, and one-sample designs through the `paired` argument and accepts data via either a formula interface or raw vectors, making it a direct analog to base R's `t.test`. For researchers who only want to test a single hypothesis at a time, the `simple_htest` function (described in the next section) provides a lighter alternative. +The `t_TOST` function provides a unified interface for equivalence and minimal effect testing with t-tests. It runs a standard nil significance test ($H_0: $) alongside the equivalence (or minimal effect) test, because in practice researchers may want both: "Is there a significant effect?" and "Can I rule out a meaningful effect?" The function handles independent, paired, and one-sample designs through the `paired` argument and accepts data via either a formula interface or raw vectors, making it a direct analog to base R's `t.test`. Additionally, the output includes an SMD calculation alongside the unstandarized mean difference with appropriate, albeit approximate, confidence intervals. For researchers who only want to test a single hypothesis at a time, the `simple_htest` function (described in the next section) provides a lighter alternative. Minimal effect testing (MET) is also supported. MET is useful when the hypothesis is that an effect *exceeds* a meaningful threshold --- the null hypothesis is equivalence, and rejection implies a practically meaningful effect (see Figure 1) [@mazzolari2022myths]. -```{r hypplot, fig.width=6, fig.height=2.75, echo=FALSE, message = FALSE, warning = FALSE, fig.show='hold', fig.cap = "Type of Hypothesis"} +```{r hypplot, fig.width=6, fig.height=2.75, echo=FALSE, message = FALSE, warning = FALSE, fig.show='hold', fig.cap = "Types of hypotheses tests supported by two one-sided tests (TOST) procedures."} p1 = ggplot() + geom_vline(aes(xintercept = -.5), @@ -230,11 +232,13 @@ library(jmv) data('bugs') ``` -For the first example, we use the `sleep` data with equivalence bounds of $\pm$ 0.5 units. The formula interface works identically to `t.test`: +For the first example, we use the `sleep` data with equivalence bounds (set by the `eqb` argument) of $\pm$ 0.5 units. The `smd_ci` argument can also be set to use one of four possible calculations for the SMD confidence interval. In the example below, I have opted to go with the normal approximation for computational ease; such approximations tend provide adequate coverage [@Viechtbauer_2007]. The formula interface works almost identically to `t.test`: ```{r} -res1 = t_TOST(formula = extra ~ group, data = sleep, - eqb = .5, smd_ci = "t") +res1 = t_TOST(formula = extra ~ group, # difference in "group" of outcome "extra" + data = sleep, # data + eqb = .5, # equivalence bound + smd_ci = "z") # sets type of SMD confidence interval calculation ``` The `print` method provides a verbose summary including the two-tailed test, both one-sided equivalence tests, and the standardized mean difference (Hedges' $g$ by default): @@ -245,9 +249,16 @@ print(res1) \newpage -The generic `plot` method produces a consonance density plot, inspired by the concurve R package [@rafi2020], that shows the estimate, confidence intervals, and equivalence bounds simultaneously: +The default visualization provided by the generic `plot` method for `t_TOST` +provides a simple visualization of both effect sizes and their respective confidence intervals alongside the equivalence bounds. Test information is also printed at the top of the plot. -```{r cdplot,fig.width=6, fig.height=5,fig.cap="Consonance density plot for the independent-samples equivalence test."} +```{r genericplot,fig.width=6, fig.height=5,fig.cap="Results of the independent-samples equivalence test visualized with the default plot method. The top panel displays the raw mean difference with its 90% confidence interval, and the bottom panel displays the standardized effect size (Hedges's g~av~) with its 90% confidence interval. In both panels, the dashed vertical lines indicate the equivalence bounds (raw: -0.5 to 0.5; standardized: approximately -0.3 to 0.3), and the point estimate is marked by a filled circle. A 90% confidence interval is used because it corresponds to the two one-sided tests (TOST) procedure at an alpha level of 0.05."} +plot(res1) +``` + +Also when the generic `plot` method has the `type` argument set to "cd" this produces a consonance density plot, inspired by the "concurve" R package [@rafi2020], that shows the estimate, confidence intervals, and equivalence bounds simultaneously: + +```{r cdplot,fig.width=6, fig.height=5,fig.cap="Consonance density plot for the independent-samples equivalence test. The top panel displays the standardized effect size (Hedges's g~av~) and the bottom panel displays the raw mean difference. In each panel, the consonance density curve represents the distribution of confidence intervals across all levels, with color-coded regions corresponding to the 68%, 90%, 95%, and 99.9% confidence intervals. The point estimate is marked by a filled circle along the x-axis. The dashed vertical lines indicate the equivalence bounds (standardized: -0.26 to 0.26; raw: -0.5 to 0.5). This visualization conveys the full range of effect sizes compatible with the data at varying confidence levels, rather than relying on a single confidence interval. Wider, less certain intervals (e.g., 99.9%) extend further from the point estimate, while narrower intervals (e.g., 68%) cluster near it."} plot(res1, type = "cd") ``` @@ -257,13 +268,13 @@ Other plot types are available: `type = "c"` produces a consonance plot showing ## One-Sample t-test with Asymmetric Bounds -When the equivalence bounds are not symmetric around the null, two values can be supplied to `eqb`. This is common when the research question implies asymmetric consequences. Here we test whether the mean of `LDHF` falls within the bounds of 5.5 and 8.5 around a hypothesized value of 7.5: +When the equivalence bounds are not symmetric around the null, two values can be supplied to `eqb`. This is common when the research question implies asymmetric consequences. Similarly, we may want to test the mean itself rather than a mean difference. This harkens back to the original use of the t-test by Gosset at Guinness Brewery, where the concern was whether a single sample mean fell within acceptable limits. Here we test whether the mean of LDHF falls within the bounds of 5.5 and 8.5 around a hypothesized value of 7.5: ```{r} -res4 = t_TOST(x = bugs$LDHF, - hypothesis = "EQU", - mu = 7.5, - eqb = c(5.5,8.5)) +res4 = t_TOST(x = bugs$LDHF, # simple sample, vector from which to estimate mean + mu = 7.5, # set nil significance test to 7.5 + eqb = c(5.5,8.5) # set TOST bounds to 5.5 and 8.5 + ) res4 ``` @@ -289,7 +300,7 @@ res_tsum # Simplified Hypothesis Testing -The `t_TOST` function is designed to be comprehensive in that it runs three tests, computes standardized effect sizes, and produces verbose output. This is useful for exploration or situations where researchers want to pair a traditional nil significance test with an equivalence test. However, sometimes a researcher has a single specific hypothesis to test. The `simple_htest` function provides a lighter interface for this purpose. It returns a standard R `htest` object --- the same class returned by `t.test()` and `wilcox.test()` --- but with expanded `alternative` options that include `"equivalence"` and `"minimal.effect"`. This means researchers can perform equivalence testing with the same familiar interface they already use for standard hypothesis tests^[In fact, users can mostly replace `t.test()` and `wilcox.test()` with `simple_htest` as it also allows for the standard alternative hypotheses of two-sided, greater, less. If you have a hypothesis that would traditionally utilize a t-test or WMW test, then you can jump right to using `simple_htest` in TOSTER]. +The `t_TOST` function is designed to be comprehensive in that it runs three tests, computes standardized effect sizes, and produces verbose output. This is useful for exploration or situations where researchers *always* want to pair a traditional nil significance test with an equivalence test. However, sometimes a researcher has a single specific hypothesis to test, or simply wants less verbose output (e.g., SMD calculation is not wanted). The `simple_htest` function provides a lighter interface for this purpose. It returns a standard R `htest` object --- the same class returned by `t.test()` and `wilcox.test()` --- but with expanded `alternative` options that include `"equivalence"` and `"minimal.effect"`. This means researchers can perform equivalence testing with the same familiar interface they already use for standard hypothesis tests^[In fact, users can mostly replace `t.test()` and `wilcox.test()` with `simple_htest` as it also allows for the standard alternative hypotheses of two-sided, greater, less. If you have a hypothesis that would traditionally utilize a t-test or Wilcoxon-Mann-Whitney (WMW) test, then you can jump right to using `simple_htest` in TOSTER]. Alongside `simple_htest`, a set of helper functions makes it easy to work with any `htest` object: `as_htest` converts verbose TOST output to the simpler format, `describe_htest` generates formatted text for reporting, `df_htest` converts results to data frames for tabulation, and `plot_htest_est` produces estimate plots with equivalence bounds. @@ -298,8 +309,10 @@ Alongside `simple_htest`, a set of helper functions makes it easy to work with a The `simple_htest` function supports `t.test` and `wilcox.test` as the underlying test (via the `test` argument) and all five alternative hypotheses: `"two.sided"`, `"less"`, `"greater"`, `"equivalence"`, and `"minimal.effect"`. The `mu` parameter specifies equivalence bounds --- a single value creates symmetric bounds (e.g., if 1 is provided at `mu` with `alternative = "equivalence"` then the equivalence region will be [-1,1]), or two values can be given for asymmetric bounds. ```{r} -res_sh1 <- simple_htest(extra ~ group, data = sleep, - alternative = "equivalence", mu = 0.5) +res_sh1 <- simple_htest(extra ~ group, + data = sleep, + alternative = "equivalence", + mu = 0.5) res_sh1 ``` @@ -310,7 +323,7 @@ Notice how the output is a single clean `htest` object. This is much more concis For writing up results, `describe_htest` generates a formatted text string following typical scientific reporting conventions: ```{r} -describe_htest(simple_htest(extra ~ group, data = sleep)) +describe_htest(res_sh1) ``` @@ -326,13 +339,13 @@ plot_htest_est(res_sh1) # Standardized Effect Sizes -This section provides a unified treatment of the standardized effect size tools in TOSTER --- standardized mean differences (SMDs), rank-based effect sizes (discussed in the robust methods sections later), correlation equivalence tests, and tools for comparing SMDs between studies. Before demonstrating these tools, it is worth discussing the limitations of standardized effect sizes, because these limitations are especially important for equivalence testing. +This section provides a unified treatment of the standardized effect size tools in TOSTER --- standardized mean differences (SMDs), correlation coefficients, and tools for comparing both between studies. Before demonstrating these tools, it is worth discussing the limitations of standardized effect sizes, because these limitations are especially important for equivalence testing. ## Problems with Standardized Effect Sizes Standardized effect sizes --- Cohen's $d$, correlation coefficients, standardized regression coefficients --- are appealing because they allow comparison across studies and scales, are required by many journals, and serve as the primary currency of meta-analysis. However, they have well-documented problems that are especially relevant to equivalence testing. -Standardized mean differences are sensitive to the sample variance. The "same" real-world effect produces different $d$ values across populations and study designs. The ubiquitously reported Cohen's $d$ conflates the magnitude of the mean difference with the precision of measurement, which means that a narrow equivalence bound on $d$ may correspond to a very different raw-scale bound depending on the variability of the sample [@Caldwell2020]. This makes it unclear what, substantively, is being declared "equivalent" --- are we claiming the means are close, or that the means are close *relative to the variance*? [@baguley2009]. The choice of denominator (pooled SD, control SD, within-subject SD) further complicates that same dubious interpretation. +Standardized mean differences are sensitive to the sample variance. The "same" real-world effect produces different $d$ values across populations and study designs. The ubiquitously reported Cohen's $d$ conflates the magnitude of the mean difference with the precision of measurement, which means that a narrow equivalence bound on $d$ may correspond to a very different raw-scale bound depending on the variability of the sample [@Caldwell2020; @baguley2009]. This makes it unclear what, substantively, is being declared "equivalent" --- are we claiming the means are close, or that the means are close *relative to the variance*? The choice of denominator (pooled SD, control SD, within-subject SD) further complicates that same dubious interpretation. Correlations have analogous issues. @tukey1969 noted that correlations are influenced by the range of both variables, so the "same" bivariate relationship produces different coefficients depending on sample selection. A correlation of 0.3 in a restricted-range sample may correspond to a much larger association in the full population. Standardized regression coefficients share these vulnerabilities. These depend on the variances of both the predictor and outcome, which makes them unstable across samples and difficult to interpret substantively [@greenland1991]. @@ -385,14 +398,9 @@ compare_smd(smd1 = 0.95, n1 = 25, smd2 = 0.23, n2 = 50, - paired = TRUE) -``` - -Setting `TOST = TRUE` with an equivalence bound via the `null` argument tests whether the two SMDs are practically equivalent: - -```{r} -compare_smd(smd1 = 0.95, n1 = 25, smd2 = 0.23, n2 = 50, - paired = TRUE, TOST = TRUE, null = .25) + paired = TRUE, + null = .25, + alternative = "equivalence") ``` Based on these results, we would reject the null significance hypothesis (the SMDs differ significantly) but fail to reject the equivalence null hypothesis (we cannot conclude the SMDs are practically equivalent). When raw data are available, `boot_compare_smd` provides a bootstrap alternative (see the package vignettes for details). @@ -414,7 +422,7 @@ These tests are *not* tests of means or medians [@median_test]; they target diff The Brunner-Munzel test [@brunner2000] estimates the relative effect $\hat{p} = P(X > Y) + 0.5 \cdot P(X = Y)$ --- the probability that a randomly chosen observation from one group exceeds a randomly chosen observation from the other. This estimand is interpretable without distributional assumptions, unlike the WMW test which requires a location-shift assumption to say anything about location differences. @karch2021 argues it should be preferred as the default non-parametric procedure because it is valid under heteroscedasticity and provides a more interpretable effect size. -Beyond their statistical properties, stochastic superiority tests may be particularly well-suited to Likert-type scales which are ubiquitous in psychological research. For a single ordinal item, the spacing between response categories carries no guaranteed meaning, and a mean difference therefore has a strained interpretation. For example, what, concretely, does a 0.4-unit difference on a five-point scale communicate? The stochastic superiority estimand sidesteps this entirely. The probability that a randomly selected participant from one group outscores a randomly selected participant from the other is interpretable on its own terms, with no knowledge of the scale required. We are not suggesting that t-tests will *always* mislead in these contexts but it is a concern that should not be quickly dismissed either [@Liddell_Kruschke_2018]. Additionally, even if we assume traditional parametric methods robust to the ordinal nature of the data, the preserved error rate is not the same as interpretive clarity. For researchers whose primary outcomes live on ordinal scales, the Brunner-Munzel test and its associated effect size --- as well as ordinal models more generally [@Burkner_Vuorre_2019] --- may offer a more natural and defensible default. +Stochastic superiority tests may be particularly well-suited to Likert-type scales which are ubiquitous in psychological research. For a single ordinal item, the spacing between response categories carries no guaranteed meaning, and a mean difference therefore has a strained interpretation. For example, what, concretely, does a 0.4-unit mean difference on a five-point scale communicate? The stochastic superiority estimand sidesteps this entirely. The probability that a randomly selected participant from one group outscores a randomly selected participant from the other is interpretable on its own terms, with no knowledge of the scale required. We are not suggesting that t-tests will *always* mislead in these contexts but it is a concern that should not be quickly dismissed either [@Liddell_Kruschke_2018]. Additionally, even if we assume traditional parametric methods are robust to the ordinal nature of the data, the preserved error rate is not the same as interpretive clarity. For researchers whose primary outcomes live on ordinal scales, the Brunner-Munzel test and its associated effect size --- as well as ordinal models more generally [@Burkner_Vuorre_2019] --- may offer a more natural and defensible default. The `brunner_munzel` function supports equivalence and minimal effect testing via `alternative`, with bounds on the probability scale via `mu`. Results can be reported on different scales (`"probability"`, `"difference"`, `"logodds"`, or `"odds"`), and three test methods are available: `"t"` (default), `"logit"` (range-preserving CIs), and `"perm"` (studentized permutation; @neubert2007). @@ -428,7 +436,7 @@ print(bm_test) \newpage -### Wilcoxon-Mann-Whitney TOST +### Wilcoxon-Mann-Whitney Tests The WMW group of tests provide non-parametric tests based on ranks. Within the TOST framework, two directional location-shift tests determine if the shift is within (equivalence) or outside (minimal effect) the equivalence bounds. diff --git a/papers/Avocado_Update/interactcadsample.bib b/papers/Avocado_Update/interactcadsample.bib index 3543694b..3bb5a55f 100644 --- a/papers/Avocado_Update/interactcadsample.bib +++ b/papers/Avocado_Update/interactcadsample.bib @@ -408,6 +408,7 @@ @article{sesoi publisher={Elsevier} } +@article{Viechtbauer_2007, title={Approximate Confidence Intervals for Standardized Effect Sizes in the Two-Independent and Two-Dependent Samples Design}, volume={32}, url={http://dx.doi.org/10.3102/1076998606298034}, DOI={10.3102/1076998606298034}, abstractNote={Standardized effect sizes and confidence intervals thereof are extremely useful devices for comparing results across different studies using scales with incommensurable units. However, exact confidence intervals for standardized effect sizes can usually be obtained only via iterative estimation procedures. The present article summarizes several closed-form approximations to the exact confidence interval bounds in the two-independent and two-dependent samples design. Monte Carlo simulations were conducted to determine the accuracy of the various approximations under a wide variety of conditions. All methods except one provided accurate results for moderately large sample sizes and converged to the exact confidence interval bounds as sample size increased.}, number={1}, journal={Journal of Educational and Behavioral Statistics}, publisher={American Educational Research Association (AERA)}, author={Viechtbauer, Wolfgang}, year={2007}, month=mar, pages={39–60}, language={en} } @article{rafi2020, title={Semantic and cognitive tools to aid statistical science: replace confidence and significance by compatibility and surprise}, From 408a4f7867597705892d18c2d9603a43719ec397 Mon Sep 17 00:00:00 2001 From: Aaron Caldwell Date: Thu, 16 Apr 2026 13:01:17 -0500 Subject: [PATCH 06/40] Update Avocado_Update.Rmd Add more for SMD/correlation section --- papers/Avocado_Update/Avocado_Update.Rmd | 61 +++++++++++++++++++++--- 1 file changed, 54 insertions(+), 7 deletions(-) diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index 3eb8c67e..1d0e35d1 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -357,26 +357,62 @@ With these caveats in mind, the following subsections describe the tools TOSTER The `smd_calc` function computes standardized mean differences --- Cohen's $d$, Hedges' $g$ (bias-corrected), or Glass's $\Delta$ --- with confidence intervals and *optional* hypothesis testing (utilizing a z-test: $z = (SMD - \mu)/SE$). It supports one-sample, paired, and two-sample designs, and can return either an `htest` object or a data frame. The SMD is already reported in `t_TOST` output, but `smd_calc` allows standalone computation and supports equivalence testing directly on the effect size. This may be useful as a complementary function when using `simple_htest` as the primary statistical testing function. +When estimation rather than testing is the goal, `smd_calc` can be called with `alternative = "none"` (the default), which returns only the point estimate and confidence interval without performing a hypothesis test. This is the recommended usage when the primary objective is to report the magnitude of the effect. + +```{r} +smd_calc(formula = extra ~ group, + data = sleep, + paired = TRUE, + bias_correction = TRUE, + rm_correction = TRUE, + output = "htest") +``` + +When hypothesis testing on the SMD is desired, `smd_calc` supports equivalence and minimal-effect alternatives directly: + ```{r} -smd_calc(formula = extra ~ group, +smd_calc(formula = extra ~ group, data = sleep, - paired = TRUE, + paired = TRUE, bias_correction = TRUE, rm_correction = TRUE, - alternative = "equivalence", + alternative = "equivalence", null.value = c(-0.5,0.5), output = "htest") ``` +However, the parametric z-test used by `smd_calc` relies on distributional assumptions about the sampling distribution of the SMD that may not hold in practice. When a hypothesis test on the SMD is the goal, bootstrapping via `boot_smd_calc` is recommended because it provides more robust confidence intervals and p-values without requiring these assumptions. + +```{r} +boot_smd_calc(formula = extra ~ group, + data = sleep, + paired = TRUE, + bias_correction = TRUE, + rm_correction = TRUE, + alternative = "equivalence", + null.value = c(-0.5,0.5)) +``` + ## Equivalence Tests for Correlations -Researchers sometimes need to test that a correlation is practically zero --- for example, when evaluating discriminant validity or confirming that a potential confounder is unrelated to an outcome. The `boot_cor_test` function supports provides robust support for testing equivalence on correlation coefficients. Bounds are specified via the `null` argument: +Researchers sometimes need to test that a correlation is practically zero --- for example, when evaluating discriminant validity or confirming that a potential confounder is unrelated to an outcome. The `z_cor_test` function provides a parametric approach using the Fisher z-transformation. However, this is only an approximate method and is generally *not recommended* as the primary inferential tool for equivalence testing on correlations. ```{r} X = rnorm(30) Y = rnorm(30) +z_cor_test( + x = X, + y = Y, + method = "pearson", + alternative = "equivalence", + null = 0.4) +``` + +Because the z-test relies on distributional approximations that can be inaccurate --- particularly with small samples or non-normal data --- bootstrapping via `boot_cor_test` is recommended for more robust inference. The `boot_cor_test` function supports all three major correlation methods (Pearson, Kendall, and Spearman) as well as robust alternatives (winsorized and percentage-bend correlations), and returns standard `htest` objects that integrate with the helper functions described above. + +```{r} boot_cor_test( x = X, y = Y, @@ -385,9 +421,9 @@ boot_cor_test( null = 0.4) ``` -This extends the original TOSTER package's correlation coverage by supporting all three major correlation methods and returning standard `htest` objects that integrate with the helper functions described above. If summary statistics are only available, an approximate method is provided through `corsum_test`, which uses the Fisher z-transformation to compute confidence intervals and p-values. +If only summary statistics are available, an approximate method is provided through `corsum_test`, which uses the Fisher z-transformation to compute confidence intervals and p-values. -## Comparing SMDs Between Studies +## Comparing Standardized Effects Between Studies When evaluating whether a replication study produced results consistent with the original, researchers need a way to formally compare effect sizes. The `compare_smd` function tests whether two SMD estimates --- from an original study and a replication, for example --- differ more than would be expected by sampling variability alone. @@ -398,13 +434,24 @@ compare_smd(smd1 = 0.95, n1 = 25, smd2 = 0.23, n2 = 50, - paired = TRUE, + paired = TRUE, null = .25, alternative = "equivalence") ``` Based on these results, we would reject the null significance hypothesis (the SMDs differ significantly) but fail to reject the equivalence null hypothesis (we cannot conclude the SMDs are practically equivalent). When raw data are available, `boot_compare_smd` provides a bootstrap alternative (see the package vignettes for details). +For comparing correlation coefficients between independent studies, the `compare_cor` function provides an analogous test. It uses either the Fisher z-transformation (default) or the Kraatz (Anderson-Hauck) method to compare two correlation coefficients. + +```{r} +compare_cor(r1 = 0.45, + df1 = 48, + r2 = 0.25, + df2 = 78, + alternative = "equivalence", + null = 0.25) +``` + \newpage **A note on argument names for equivalence bounds.** The equivalence bound argument is named differently across TOSTER functions for historical and compatibility reasons: `eqb` in `t_TOST`, `wilcox_TOST`, `equ_ftest`, and `equ_anova`; `mu` in `simple_htest`, `brunner_munzel`, `boot_t_test`, and `perm_t_test` (for compatibility with base R's `t.test`); and `null` in `compare_smd` and `z_cor_test`. Despite the different names, the interpretation is the same: the bound(s) defining the region of practical equivalence. From abdd345dba964f876e22d305b1657a870273b5e9 Mon Sep 17 00:00:00 2001 From: Aaron Caldwell Date: Thu, 16 Apr 2026 15:43:19 -0500 Subject: [PATCH 07/40] updated manuscript --- R/boot_cor_test.R | 6 +- man/boot_cor_test.Rd | 4 +- papers/Avocado_Update/Avocado_Update.R | 149 +++++------------- papers/Avocado_Update/Avocado_Update.Rmd | 118 ++++++++------ .../figure-docx/cdplot-1.pdf | Bin 18730 -> 18842 bytes .../figure-docx/cdplot-1.png | Bin 27237 -> 27099 bytes .../figure-docx/genericplot-1.pdf | Bin 0 -> 5027 bytes .../figure-docx/genericplot-1.png | Bin 0 -> 32209 bytes .../figure-docx/hypplot-1.pdf | Bin 4622 -> 4622 bytes .../figure-docx/shplot1-1.pdf | Bin 4660 -> 4660 bytes papers/Avocado_Update/interactcadsample.bib | 3 + 11 files changed, 115 insertions(+), 165 deletions(-) create mode 100644 papers/Avocado_Update/Avocado_Update_files/figure-docx/genericplot-1.pdf create mode 100644 papers/Avocado_Update/Avocado_Update_files/figure-docx/genericplot-1.png diff --git a/R/boot_cor_test.R b/R/boot_cor_test.R index 078c15d1..bcd6a481 100644 --- a/R/boot_cor_test.R +++ b/R/boot_cor_test.R @@ -127,8 +127,10 @@ #' null = c(-0.1, 0.4), R = 999) #' #' @section References: -#' Wilcox, R.R. (2009) Comparing Pearson Correlations: Dealing with Heteroscedasticity and Nonnormality. -#' Communications in Statistics - Simulation and Computation, 38, 2220–2234. +#' +#' Wilcox, R. R. (1994). The Percentage Bend Correlation Coefficient. Psychometrika, 59(4), 601–616. https://doi.org/10.1007/bf02294395 +#' +#' Wilcox, R. R. (1993). Some results on a Winsorized correlation coefficient. British Journal of Mathematical and Statistical Psychology, 46(2), 339–349. https://doi.org/10.1111/j.2044-8317.1993.tb01020.x #' #' Wilcox, R.R. (2017) Introduction to Robust Estimation and Hypothesis Testing, 4th edition. Academic Press. #' diff --git a/man/boot_cor_test.Rd b/man/boot_cor_test.Rd index f1c7273c..f75a004f 100644 --- a/man/boot_cor_test.Rd +++ b/man/boot_cor_test.Rd @@ -152,8 +152,8 @@ See \code{vignette("correlations")} for more details. } \section{References}{ -Wilcox, R.R. (2009) Comparing Pearson Correlations: Dealing with Heteroscedasticity and Nonnormality. -Communications in Statistics - Simulation and Computation, 38, 2220–2234. + +Wilcox, R. R. (1994). The Percentage Bend Correlation Coefficient. Psychometrika, 59(4), 601–616. https://doi.org/10.1007/bf02294395 Wilcox, R.R. (2017) Introduction to Robust Estimation and Hypothesis Testing, 4th edition. Academic Press. } diff --git a/papers/Avocado_Update/Avocado_Update.R b/papers/Avocado_Update/Avocado_Update.R index c1b0c2d8..28f12184 100644 --- a/papers/Avocado_Update/Avocado_Update.R +++ b/papers/Avocado_Update/Avocado_Update.R @@ -11,7 +11,7 @@ library(ggplot2) library(ggdist) library(patchwork) -## ----hypplot, fig.width=6, fig.height=2.75, echo=FALSE, message = FALSE, warning = FALSE, fig.show='hold', fig.cap = "Type of Hypothesis"---- +## ----hypplot, fig.width=6, fig.height=2.75, echo=FALSE, message = FALSE, warning = FALSE, fig.show='hold', fig.cap = "Types of hypotheses tests supported by two one-sided tests (TOST) procedures."---- p1 = ggplot() + geom_vline(aes(xintercept = -.5), @@ -142,144 +142,75 @@ library(jmv) data('bugs') ## ----------------------------------------------------------------------------- -res1 = t_TOST(formula = extra ~ group, data = sleep, - eqb = .5, smd_ci = "t") +res1 = t_TOST(formula = extra ~ group, # difference in "group" of outcome "extra" + data = sleep, # data + eqb = .5, # equivalence bound + smd_ci = "z") # sets type of SMD confidence interval calculation ## ----------------------------------------------------------------------------- print(res1) -## ----cdplot,fig.width=6, fig.height=5,fig.cap="Consonance density plot for the independent-samples equivalence test."---- +## ----genericplot,fig.width=6, fig.height=5,fig.cap="Results of the independent-samples equivalence test visualized with the default plot method. The top panel displays the raw mean difference with its 90% confidence interval, and the bottom panel displays the standardized effect size (Hedges's g~av~) with its 90% confidence interval. In both panels, the dashed vertical lines indicate the equivalence bounds (raw: -0.5 to 0.5; standardized: approximately -0.3 to 0.3), and the point estimate is marked by a filled circle. A 90% confidence interval is used because it corresponds to the two one-sided tests (TOST) procedure at an alpha level of 0.05."---- +plot(res1) + +## ----cdplot,fig.width=6, fig.height=5,fig.cap="Consonance density plot for the independent-samples equivalence test. The top panel displays the standardized effect size (Hedges's g~av~) and the bottom panel displays the raw mean difference. In each panel, the consonance density curve represents the distribution of confidence intervals across all levels, with color-coded regions corresponding to the 68%, 90%, 95%, and 99.9% confidence intervals. The point estimate is marked by a filled circle along the x-axis. The dashed vertical lines indicate the equivalence bounds (standardized: -0.26 to 0.26; raw: -0.5 to 0.5). This visualization conveys the full range of effect sizes compatible with the data at varying confidence levels, rather than relying on a single confidence interval. Wider, less certain intervals (e.g., 99.9%) extend further from the point estimate, while narrower intervals (e.g., 68%) cluster near it."---- plot(res1, type = "cd") ## ----------------------------------------------------------------------------- -res4 = t_TOST(x = bugs$LDHF, - hypothesis = "EQU", - mu = 7.5, - eqb = c(5.5,8.5)) +res4 = t_TOST(x = bugs$LDHF, # simple sample, vector from which to estimate mean + mu = 7.5, # set nil significance test to 7.5 + eqb = c(5.5,8.5) # set TOST bounds to 5.5 and 8.5 + ) res4 ## ----------------------------------------------------------------------------- res_tsum = tsum_TOST( - m1 = mean(bugs$LDHF, na.rm=TRUE), sd1 = sd(bugs$LDHF, na.rm=TRUE), + m1 = mean(bugs$LDHF, na.rm=TRUE), + sd1 = sd(bugs$LDHF, na.rm=TRUE), n1 = length(na.omit(bugs$LDHF)), - hypothesis = "EQU", smd_ci = "t", eqb = c(5.5, 8.5) + hypothesis = "EQU", smd_ci = "t", + eqb = c(5.5, 8.5) ) res_tsum ## ----------------------------------------------------------------------------- -res_sh1 <- simple_htest(extra ~ group, data = sleep, - alternative = "equivalence", mu = 0.5) +res_sh1 <- simple_htest(extra ~ group, + data = sleep, + alternative = "equivalence", + mu = 0.5) res_sh1 ## ----------------------------------------------------------------------------- -describe_htest(simple_htest(extra ~ group, data = sleep)) +describe_htest(res_sh1) ## ----shplot1, fig.width=6, fig.height=3.5, fig.cap = "Estimate plot for an equivalence test via simple_htest."---- plot_htest_est(res_sh1) ## ----------------------------------------------------------------------------- -smd_calc(formula = extra ~ group, +smd_calc(formula = extra ~ group, data = sleep, - paired = FALSE, + paired = TRUE, bias_correction = TRUE, - alternative = "equivalence", - null.value = c(-0.5,0.5), + rm_correction = TRUE, output = "htest") ## ----------------------------------------------------------------------------- -X = rnorm(30) -Y = rnorm(30) - -boot_cor_test( - x = X, - y = Y, - method = "pearson", - alternative = "equivalence", - null = 0.4) - -## ----------------------------------------------------------------------------- -compare_smd(smd1 = 0.95, - n1 = 25, - smd2 = 0.23, - n2 = 50, - paired = TRUE) - -## ----------------------------------------------------------------------------- -compare_smd(smd1 = 0.95, n1 = 25, smd2 = 0.23, n2 = 50, - paired = TRUE, TOST = TRUE, null = .25) - -## ----------------------------------------------------------------------------- -bm_test = brunner_munzel(formula = extra ~ group, - data = sleep, - alternative = "equivalence", - mu = c(0.3, 0.7)) -print(bm_test) - -## ----------------------------------------------------------------------------- -test1 = wilcox_TOST(formula = extra ~ group, - data = sleep, - paired = FALSE, - eqb = .5) -print(test1) - -## ----------------------------------------------------------------------------- -set.seed(4522) -boot_t_test(formula = extra ~ group, - data = sleep, - alternative = "equivalence", - mu = c(-0.5, 0.5), - R = 999) - -## ----------------------------------------------------------------------------- -set.seed(8812) -perm_t_test(formula = extra ~ group, - data = sleep, - alternative = "equivalence", - mu = c(-0.5, 0.5), - R = 999) - -## ----------------------------------------------------------------------------- -set.seed(891111) -test1 = boot_t_TOST(formula = extra ~ group, - data = sleep, - paired = TRUE, - eqb = .5, - R = 999) -print(test1) - -## ----error=FALSE-------------------------------------------------------------- -log_TOST(mpg ~ am, data = mtcars) - -## ----error=FALSE-------------------------------------------------------------- -boot_log_TOST(mpg ~ am, data = mtcars, R=999) - -## ----warning=FALSE, message=FALSE--------------------------------------------- -data("InsectSprays") -aovtest = aov(count ~ spray, data = InsectSprays) -anova(aovtest) - - -## ----------------------------------------------------------------------------- -equ_ftest(Fstat = 34.70228, df1 = 5, df2 = 66, eqb = 0.35) - -## ----------------------------------------------------------------------------- -# Example using a purely within-subjects design -# (Maxwell & Delaney, 2004, Chapter 12, Table 12.5, p. 578): -library(afex) -data(md_12.1) -aovtest2 = aov_ez("id", "rt", md_12.1, within = c("angle", "noise"), - anova_table=list(correction = "none", es = "none")) -equ_anova(aovtest2, - eqb = 0.35) - -## ----------------------------------------------------------------------------- -power_t_TOST(delta = 0, sd = 1, eqb = 0.5, - alpha = 0.05, power = 0.8, - type = "two.sample") +smd_calc(formula = extra ~ group, + data = sleep, + paired = TRUE, + bias_correction = TRUE, + rm_correction = TRUE, + alternative = "equivalence", + null.value = c(-0.5,0.5), + output = "htest") ## ----------------------------------------------------------------------------- -power_z_cor(rho = 0, power = 0.8, - null = 0.3, alpha = 0.05, - alternative = "equivalence") +boot_smd_calc(formula = extra ~ group, + data = sleep, + paired = TRUE, + bias_correction = TRUE, + rm_correction = TRUE, + alternative = "equivalence", + null.value = c(-0.5,0.5)) diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index 1d0e35d1..fb2944fe 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -396,34 +396,35 @@ boot_smd_calc(formula = extra ~ group, ## Equivalence Tests for Correlations -Researchers sometimes need to test that a correlation is practically zero --- for example, when evaluating discriminant validity or confirming that a potential confounder is unrelated to an outcome. The `z_cor_test` function provides a parametric approach using the Fisher z-transformation. However, this is only an approximate method and is generally *not recommended* as the primary inferential tool for equivalence testing on correlations. +Researchers sometimes need to test that a correlation is practically zero. The `z_cor_test` function provides a parametric approach using the Fisher z-transformation. However, this is only an approximate method and is generally *not recommended* as the primary inferential tool for equivalence testing on correlations. ```{r} -X = rnorm(30) -Y = rnorm(30) z_cor_test( - x = X, - y = Y, + x = bugs$LDLF, + y = bugs$LDHF, method = "pearson", - alternative = "equivalence", - null = 0.4) + alternative = "t") ``` -Because the z-test relies on distributional approximations that can be inaccurate --- particularly with small samples or non-normal data --- bootstrapping via `boot_cor_test` is recommended for more robust inference. The `boot_cor_test` function supports all three major correlation methods (Pearson, Kendall, and Spearman) as well as robust alternatives (winsorized and percentage-bend correlations), and returns standard `htest` objects that integrate with the helper functions described above. +Because the z-test relies on distributional approximations that can be inaccurate --- particularly with small samples --- bootstrapping via `boot_cor_test` is recommended for more robust inference. The `boot_cor_test` function supports all three major correlation methods (Pearson, Kendall, and Spearman) as well as the Winsorized [@Wilcox_1993] and percentage bend [@Wilcox_1994] correlation coefficients as robust alternatives, and returns standard `htest` objects that integrate with the helper functions described above. ```{r} -boot_cor_test( - x = X, - y = Y, +res_boot_cor = boot_cor_test( + x = bugs$LDLF, + y = bugs$LDHF, method = "pearson", alternative = "equivalence", null = 0.4) + +print(res_boot_cor) + +describe_htest(res_boot_cor) ``` If only summary statistics are available, an approximate method is provided through `corsum_test`, which uses the Fisher z-transformation to compute confidence intervals and p-values. -## Comparing Standardized Effects Between Studies +## Comparing Standardized Effect Sizes Between Studies When evaluating whether a replication study produced results consistent with the original, researchers need a way to formally compare effect sizes. The `compare_smd` function tests whether two SMD estimates --- from an original study and a replication, for example --- differ more than would be expected by sampling variability alone. @@ -458,36 +459,17 @@ compare_cor(r1 = 0.45, # Robust Methods for Equivalence Testing -In some cases, serious violations of t-test assumptions (normality, homoscedasticity) could inflate the type I error rate. TOSTER provides robust alternatives organized into three categories: rank-based tests, resampling methods, and log-transformed methods. +When t-test assumptions (normality, homoscedasticity) are seriously violated, or when the mean difference is a poor summary of the effect (e.g., with heavy skew or ordinal psychometric scales) alternative approaches are worth considering. TOSTER organizes these into three categories: rank-based tests, resampling methods, and log-transformed methods. ## Rank-Based Tests -Rank-based tests avoid distributional assumptions by operating on the ranks of the data rather than the raw values. -These tests are *not* tests of means or medians [@median_test]; they target different estimands (i.e., the quantity a study is attempting to estimate). The choice between them should be driven by the estimand of interest. - -### Brunner-Munzel Test - -The Brunner-Munzel test [@brunner2000] estimates the relative effect $\hat{p} = P(X > Y) + 0.5 \cdot P(X = Y)$ --- the probability that a randomly chosen observation from one group exceeds a randomly chosen observation from the other. This estimand is interpretable without distributional assumptions, unlike the WMW test which requires a location-shift assumption to say anything about location differences. @karch2021 argues it should be preferred as the default non-parametric procedure because it is valid under heteroscedasticity and provides a more interpretable effect size. - -Stochastic superiority tests may be particularly well-suited to Likert-type scales which are ubiquitous in psychological research. For a single ordinal item, the spacing between response categories carries no guaranteed meaning, and a mean difference therefore has a strained interpretation. For example, what, concretely, does a 0.4-unit mean difference on a five-point scale communicate? The stochastic superiority estimand sidesteps this entirely. The probability that a randomly selected participant from one group outscores a randomly selected participant from the other is interpretable on its own terms, with no knowledge of the scale required. We are not suggesting that t-tests will *always* mislead in these contexts but it is a concern that should not be quickly dismissed either [@Liddell_Kruschke_2018]. Additionally, even if we assume traditional parametric methods are robust to the ordinal nature of the data, the preserved error rate is not the same as interpretive clarity. For researchers whose primary outcomes live on ordinal scales, the Brunner-Munzel test and its associated effect size --- as well as ordinal models more generally [@Burkner_Vuorre_2019] --- may offer a more natural and defensible default. - -The `brunner_munzel` function supports equivalence and minimal effect testing via `alternative`, with bounds on the probability scale via `mu`. Results can be reported on different scales (`"probability"`, `"difference"`, `"logodds"`, or `"odds"`), and three test methods are available: `"t"` (default), `"logit"` (range-preserving CIs), and `"perm"` (studentized permutation; @neubert2007). - -```{r} -bm_test = brunner_munzel(formula = extra ~ group, - data = sleep, - alternative = "equivalence", - mu = c(0.3, 0.7)) -print(bm_test) -``` - -\newpage +Rank-based tests operate on the ranks of the data rather than the raw values, which is why they are typically described as non-parametric. It is worth emphasizing that this label applies to the test, not the data. The underlying observations may still come from any distribution, and rank-based methods should not be treated as automatic substitutes for a t-test when normality is in doubt. These tests are not tests of means or medians [@median_test; @karch2021]; they target different estimands (i.e., the quantity a study is attempting to estimate), and the choice between a rank-based test and a t-test should be driven by which estimand is of scientific interest. ### Wilcoxon-Mann-Whitney Tests The WMW group of tests provide non-parametric tests based on ranks. Within the TOST framework, two directional location-shift tests determine if the shift is within (equivalence) or outside (minimal effect) the equivalence bounds. -However, there are important caveats. Many researchers mistakenly believe these are tests of medians, but without a location-shift assumption, the WMW test is a test of stochastic equality. As @Fay_Malinovsky_2018 demonstrate, the effect size can even be presented on the same scale as the Brunner-Munzel test. Only under a location-shift assumption does it become a test of central tendency through the Hodges-Lehmann estimator. @median_test demonstrate striking counterexamples: groups can have equal medians yet produce a significant WMW test, very different medians yet a non-significant WMW test, or even significance in the opposite direction from the median comparison. +However, there are important caveats. Many researchers mistakenly believe these are tests of medians, but without a location-shift assumption, the WMW test is a test of stochastic equality. As @Fay_Malinovsky_2018 demonstrate, the effect size can even be presented as stochastic superiority/dominance. Only under a location-shift assumption does it become a test of central tendency through the Hodges-Lehmann estimator. @median_test demonstrate striking counterexamples: groups can have equal medians yet produce a significant WMW test, very different medians yet a non-significant WMW test, or even significance in the opposite direction from the median comparison. For equivalence testing, bounds are applied to the pseudo-median, which corresponds to the mean or median difference only under the location-shift assumption. For testing hypotheses about mean differences, `perm_t_test` or `boot_t_test` are more appropriate. For stochastic superiority, `brunner_munzel` provides a clearer framework. @@ -501,53 +483,81 @@ print(test1) Based on these results, we would conclude there is a significant difference and the results are also not equivalent. But again, the estimate we are testing is the pseudo-median, not the mean or median difference, so this conclusion may not be relevant to the research question at hand. +### Brunner-Munzel Test + +The Brunner-Munzel test [@brunner2000] estimates the relative effect $\hat{p} = P(X > Y) + 0.5 \cdot P(X = Y)$, sometimes referred to as the probability of superiority --- the probability that a randomly chosen observation from one group exceeds a randomly chosen observation from the other, with ties split evenly. This estimand is interpretable without distributional assumptions, unlike the WMW test which requires a location-shift assumption to say anything about location differences. @karch2021 argues it should be preferred as the default non-parametric procedure because it is valid under heteroscedasticity and provides an interpretable effect size. + +Stochastic superiority tests may be particularly well-suited to Likert-type scales which are ubiquitous in psychological research. For a single ordinal item, the spacing between response categories carries no guaranteed meaning, and a mean difference therefore has a strained interpretation. For example, what, concretely, does a 0.4-unit mean difference on a five-point scale communicate? The stochastic superiority estimand sidesteps this entirely. The probability that a randomly selected participant from one group outscores a randomly selected participant from the other is interpretable on its own terms, with no knowledge of the scale required. We are not suggesting that t-tests will *always* mislead in these contexts but it is a concern that should not be quickly dismissed either [@Liddell_Kruschke_2018]. Additionally, even if we assume traditional parametric methods are robust to the ordinal nature of the data, the preserved error rate is not the same as interpretive clarity. For researchers whose primary outcomes live on ordinal scales, the Brunner-Munzel test and its associated effect size --- as well as ordinal models more generally [@Burkner_Vuorre_2019] --- may offer a more natural and defensible default. + +The `brunner_munzel` function supports equivalence and minimal effect testing via `alternative`, with bounds on the probability scale via `mu`. Results can be reported on different scales (`"probability"`, `"difference"`, `"logodds"`, or `"odds"`), and three test methods are available: `"t"` (default), `"logit"` (range-preserving CIs), and `"perm"` (studentized permutation; @neubert2007). + +```{r} +bm_test = brunner_munzel(formula = extra ~ group, + data = sleep, + alternative = "equivalence", + mu = c(0.3, 0.7)) +print(bm_test) +``` + + \newpage ## Resampling Methods -When the goal is to test hypotheses about means while relaxing distributional assumptions, resampling methods provide powerful tools. TOSTER provides bootstrap and permutation approaches, each with distinct inferential logic. +When the goal is to test hypotheses about means while relaxing distributional assumptions, resampling methods provide powerful tools. TOSTER implements both bootstrap and permutation approaches through `boot_t_test` and `perm_t_test`, each with distinct inferential logic but a shared motivation to retain the mean as the estimand while obtaining reference distributions empirically --- through resampling --- rather than relying on the normal-theory derivations (i.e., Student's t-test). + +Both approaches are natural extensions of the traditional t-test rather than alternatives to it. The test statistic itself is unchanged; what changes is how the null distribution is constructed. In the parametric t-test, the null distribution of the standardized mean difference is derived analytically under the assumption that the data are drawn from a population with a normal distribution (and, for Student's version, with equal variances). The bootstrap approximates the sampling distribution of the same statistic by resampling from the observed data with replacement, whereas the permutation approach constructs an exact or near-exact null distribution by shuffling group labels (two-sample) or flipping signs (one-sample and paired). In both cases, the researcher continues to ask the same question about means; only the calibration of the reference distribution differs. This is important for researchers who may view resampling methods as a departure from familiar testing logic: the estimand, the null hypothesis, and the test statistic remain the same. -The bootstrap [@efron93] resamples with replacement to approximate the sampling distribution. P-values are tied to the CI method via CI inversion, ensuring consistency between reported CIs and p-values [@Thulin_2024]. The permutation approach shuffles group labels (two-sample) or flips signs (one-sample/paired) to construct the null distribution directly. Studentized permutation tests [@janssen1997; @chung2013] provide valid inference under heteroscedasticity and allow accurate equivalence tests [@Arboretti_Pesarin_Salmaso_2020]. +Because the reference distribution is generated from the data rather than assumed, distributional requirements are relaxed, though not eliminated. The bootstrap relies on asymptotic arguments and assumes that the empirical distribution is a reasonable stand-in for the population distribution; it performs well with moderate-to-large samples and skewed or heavy-tailed data, but can struggle with very small samples or extreme outliers. The permutation approach is exact under the null hypothesis of exchangeability, meaning group labels carry no information about outcomes. In its basic form this exactness requires identical distributions under the null, which implicitly assumes equal variances. The studentized permutation test [@janssen1997; @chung2013] addresses this limitation by permuting a scale-invariant statistic (essentially the Welch t-statistic), yielding asymptotically valid inference even under heteroscedasticity. @Arboretti_Pesarin_Salmaso_2020 extended this logic to the equivalence testing setting, showing that studentized permutation TOST maintains nominal Type I error rates when variances differ across groups. In short, these methods are assumption-lighter rather than assumption-free, and users should still consider whether independence and exchangeability are reasonable for their design. -Both `boot_t_test` and `perm_t_test` return standard `htest` objects and support TOST (both equivlance and minimal effect testing). Both also support trimmed means via the `tr` parameter for additional robustness to outliers. +A common reflex when normality seems implausible is to switch to the WMW or other rank-based tests. However, these tests answer a different question. As discussed earlier in this manuscript, rank-based procedures test hypotheses about stochastic ordering or pseudo-medians rather than means, and under heteroscedasticity they can reject the null even when the groups have identical means. When the scientific question is genuinely about means --— as it often is in applied research where effect sizes and interventions are naturally expressed on the original scale —-- bootstrap and permutation t-tests should generally be preferred over rank-based methods for handling such violations of assumptions. The rank-based tests remain useful, but as answers to different questions, not as robust substitutes for tests of means. + +Choosing between the bootstrap and permutation approaches involves tradeoffs rather than a clear winner in most situations. The studentized permutation test has the strongest theoretical footing for two-sample mean comparisons under heteroscedasticity, and is particularly attractive for equivalence testing given the @Arboretti_Pesarin_Salmaso_2020 results. The bootstrap offers greater flexibility because it extends naturally to statistics beyond the mean difference, provides readily interpretable confidence intervals via CI inversion [@Thulin_2024], and ensures consistency between reported intervals and p-values. In small samples, permutation approaches tend to have better finite-sample calibration because the null distribution is constructed exactly (or nearly so) from the data at hand, whereas the bootstrap's asymptotic justification can produce liberal tests with very small sample sizes. When outliers are a concern, both methods accept the `tr` argument for trimmed means, which reduces the influence of extreme observations while preserving the mean-based interpretation. As a rough heuristic, researchers comparing two independent groups with suspected unequal variances may find the studentized permutation test most defensible, while those wanting a single framework that yields coherent intervals and p-values across a range of designs may prefer the bootstrap. Both are substantial improvements over ignoring assumption violations, and both return standard `htest` objects that integrate with the TOST framework for equivalence and minimal effect testing. ### Bootstrap t-test +The bootstrap t-test in TOSTER defaults to the studentized bootstrap (`boot_ci = "stud"`) because this matches the original bootstrap t-test proposed by @efron93 (Chapter 16), which is the procedure `boot_t_TOST` implements. In this approach, the t-statistic itself is resampled rather than the raw mean difference, and confidence intervals are constructed from the bootstrap distribution of these pivotal quantities. Other CI methods (percentile, basic, and BCa) remain available via the `boot_ci` argument, but the studentized default preserves consistency with the canonical Efron and Tibshirani formulation. It also ensures that the reported p-value and confidence interval are derived from the same bootstrap distribution, so the two will always agree; previous versions of the package used the studentized p-value regardless of the selected CI method, which could produce disagreement between reported intervals and p-values. + ```{r} set.seed(4522) boot_t_test(formula = extra ~ group, data = sleep, + paired = TRUE, alternative = "equivalence", mu = c(-0.5, 0.5), R = 999) ``` +The output follows the familiar `htest` format. The reported p-value is the larger of the two one-sided tests against the equivalence bounds, and the confidence interval is the studentized bootstrap CI for the mean difference. Because the p-value is tied to this interval via CI inversion, a researcher reporting both will find the two tell a consistent story. The interval will fall within the equivalence bounds if and only if the equivalence test rejects. +For situations where both a nil hypothesis test and an equivalence test are desired simultaneously with bootstrapping, `boot_t_TOST` returns a `TOSTt` object with bootstrapped SMD estimates [@Kirby2013]: + +```{r} +set.seed(891111) +test1 = boot_t_TOST(formula = extra ~ group, + data = sleep, + paired = TRUE, + eqb = .5, + R = 999) +print(test1) +``` + +The bootstrapped SMDs are a useful feature in their own right. Analytic confidence intervals for standardized effect sizes rely on approximations, which assume normality and can behave poorly with small or skewed samples. Bootstrap intervals for SMDs sidestep these assumptions by resampling the full estimation pipeline, yielding intervals that inherit the robustness properties of the bootstrap itself. + ### Permutation t-test -When `R = NULL`, the function enumerates all possible permutations for an exact test; otherwise, it draws `R` random permutations with the "+1" correction [@phipson2010]. By default the test is studentized (`perm_se = TRUE`). +When `R = NULL`, the function enumerates all possible permutations for an exact test; otherwise, it draws R random permutations with the "+1" correction [@phipson2010]. By default the test is studentized (`perm_se = TRUE`), consistent with the recommendations discussed in the methods section above. The `perm_se = FALSE` option is available primarily to match the behavior of other permutation test packages that do not studentize by default. ```{r} set.seed(8812) perm_t_test(formula = extra ~ group, data = sleep, + paired = TRUE, alternative = "equivalence", mu = c(-0.5, 0.5), R = 999) ``` -### Bootstrap TOST - -For situations where both a nil hypothesis test and an equivalence test are desired simultaneously with bootstrapping, `boot_t_TOST` returns a `TOSTt` object with bootstrapped SMD estimates [@Kirby2013]: - -```{r} -set.seed(891111) -test1 = boot_t_TOST(formula = extra ~ group, - data = sleep, - paired = TRUE, - eqb = .5, - R = 999) -print(test1) -``` +Running the bootstrap and permutation equivalence tests on the same paired sleep data using matching equivalence bounds produces broadly similar conclusions, as would be expected. The two approaches can, however, diverge in small samples, under strong heteroscedasticity, or in the presence of influential outliers, which is where the choice between them becomes substantively important. Users interested in robustness to outliers can additionally supply the `tr` argument to either function to use trimmed means, preserving the mean-based interpretation while downweighting extreme observations. \newpage @@ -657,7 +667,7 @@ For more complex designs --- crossover studies, replicate designs, group-sequent This paper has described the major updates to the TOSTER R package, expanding its scope from the original t-test and correlation equivalence tests to a comprehensive toolkit for frequentist equivalence testing. The update adds a unified t-test interface with minimal effect testing support, a simplified hypothesis testing interface that integrates with the standard R ecosystem, tools for standardized effect sizes and correlations, robust and resampling-based alternatives, ANOVA equivalence testing, and power analysis for sample size planning. -Several limitations should be acknowledged. TOSTER is designed for standard research designs --- it does not handle many analysis scenarios such as those involving multilevel or mixed-effects models. For these settings, researchers should consider packages like "emmeans" or "marginaleffects", which can accommodate non-zero null hypotheses in more complex modeling frameworks. TOSTER also does not provide Bayesian equivalence testing; researchers interested in Bayes factors for the null should consult R packages such as "bayestestR". +Several limitations should be acknowledged. TOSTER only covers a few standard research designs --- it does not handle many analysis scenarios such as those involving multilevel or mixed-effects models. For these settings, researchers should consider packages like "emmeans" or "marginaleffects", which can accommodate non-zero null hypotheses in more complex modeling frameworks. TOSTER also does not provide Bayesian equivalence testing; researchers interested in Bayesian methods for the null should consult other R packages such as "bayestestR". I hope that the updated package continues to build upon the original impact of @lakens_ori and makes equivalence testing more accessible to researchers across disciplines. At the very least, I would encourage users to compare results across methods (e.g., parametric vs. robust) to ensure that conclusions do not depend on the chosen analysis. If the interpretation substantially changes depending on the test, this suggests features of the data --- violations of assumptions, outliers, distributional shape --- that warrant further investigation. @@ -684,6 +694,10 @@ I would like to thank everyone from the Lakens laboratory group for their input The author of this manuscript is the author of the TOSTER package. Citations of this manuscript will benefit his citation count. +# AI Use Disclosure + +During the preparation of the TOSTER package updates and this manuscript, the author used Claude Opus 4.5-4.7 (Anthropic) to assist with editing R code and drafting unit tests. For the manuscript text, Claude Opus was used only for targeted copy-editing — suggesting revisions to specific passages, catching typographical and grammatical errors, and identifying inconsistencies across sections. The manuscript was written by the author, and all AI-assisted output was reviewed, revised, and verified by the author, who takes full responsibility for the content of the package and the manuscript. + ## Funding {-} No funding was provided for this work. diff --git a/papers/Avocado_Update/Avocado_Update_files/figure-docx/cdplot-1.pdf b/papers/Avocado_Update/Avocado_Update_files/figure-docx/cdplot-1.pdf index f2926e36bdf16219dcedf30ed85889a4adbb917a..acb08f8a3b91c1f2fe2e0a9092d600fd2bf972d1 100644 GIT binary patch delta 15215 zcmZX)2RNKt6exN^h~A?IQ6ofWbU}1Nh!(w*V2Cyt{v%PMw`kFVM2Rv=5`-~&Z;?cg zHhLd@m>F-Jd+&Sq-8mue+)OENXC=C-t$v|PI4EnYzHmsj7j(v1!CT}vxMssM!k z-hSyEEzPaK+l2OyrzH~AweC&dJ3McZSR5W4u6C+T7520s911z9h6nC_`?4@NsAn3E zy~wabK(W7K5E#t0?Us`+to7ph#iOL~)3qI#on1`49$I#z@&sM?-dyh#xeJp~9xZ0N z-hy3kj{)HqmoqSxRlgWNf%7rD>)jpdN5JqJG#se9I*s*_?tv+PJ*uPb%_Q0Bt7n&a z`MxhFBBvJc{Ih(K9ojA(5#<5Y3nu18ZZLVpAe~Y*MV2zHfS)A_U}veA4W^>h{DME|>qyNr3Tt=g 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B4axuj delta 53 zcmeBE=~J0trfO_tW@u_)YG|&>rSF@c;*waBs-WRwWn^GvW&oF)7_G->xUs865CCum B4afih diff --git a/papers/Avocado_Update/Avocado_Update_files/figure-docx/shplot1-1.pdf b/papers/Avocado_Update/Avocado_Update_files/figure-docx/shplot1-1.pdf index 5a206f11be338ee5516a5e84d468b40ec6a66a28..3bb33295f14bd76d6ff9bc887df147abdf40f924 100644 GIT binary patch delta 53 zcmdm@vPETrnW~AQnW3qPfuWHmm%eX)ic4Zis)B}#m63swnE_mGVzeHk@y4!If&hN8 B4ln=! delta 53 zcmdm@vPETrnX0jonW3qHsgbEBm%eX)ic4Zis)B}#m63swnE_mGVzeHk;l{32f&hP$ B4mbb+ diff --git a/papers/Avocado_Update/interactcadsample.bib b/papers/Avocado_Update/interactcadsample.bib index 3bb5a55f..e46ea717 100644 --- a/papers/Avocado_Update/interactcadsample.bib +++ b/papers/Avocado_Update/interactcadsample.bib @@ -410,6 +410,9 @@ @article{sesoi @article{Viechtbauer_2007, title={Approximate Confidence Intervals for Standardized Effect Sizes in the Two-Independent and Two-Dependent Samples Design}, volume={32}, url={http://dx.doi.org/10.3102/1076998606298034}, DOI={10.3102/1076998606298034}, abstractNote={Standardized effect sizes and confidence intervals thereof are extremely useful devices for comparing results across different studies using scales with incommensurable units. However, exact confidence intervals for standardized effect sizes can usually be obtained only via iterative estimation procedures. The present article summarizes several closed-form approximations to the exact confidence interval bounds in the two-independent and two-dependent samples design. Monte Carlo simulations were conducted to determine the accuracy of the various approximations under a wide variety of conditions. All methods except one provided accurate results for moderately large sample sizes and converged to the exact confidence interval bounds as sample size increased.}, number={1}, journal={Journal of Educational and Behavioral Statistics}, publisher={American Educational Research Association (AERA)}, author={Viechtbauer, Wolfgang}, year={2007}, month=mar, pages={39–60}, language={en} } +@article{Wilcox_1994, title={The Percentage Bend Correlation Coefficient}, volume={59}, url={http://dx.doi.org/10.1007/BF02294395}, DOI={10.1007/bf02294395}, abstractNote={A well-known result is that the usual correlation coefficient, ρ, is highly nonrobust: very slight changes in only one of the marginal distributions can alter ρ by a substantial amount. There are a variety of methods for correcting this problem. This paper identifies one particular method which is useful in psychometrics and provides a simple test for independence. It is not recommended that the new test replace the usual test of H0: ρ = 0, but the new test has important advantages over the usual test in terms of both Type I errors and power.}, number={4}, journal={Psychometrika}, publisher={Cambridge University Press (CUP)}, author={Wilcox, Rand R.}, year={1994}, month=dec, pages={601–616}, language={en} } +@article{Wilcox_1993, title={Some results on a Winsorized correlation coefficient}, volume={46}, url={http://dx.doi.org/10.1111/j.2044-8317.1993.tb01020.x}, DOI={10.1111/j.2044-8317.1993.tb01020.x}, abstractNote={Recent investigations indicate that psychometric measures often have distributions with very heavy tails, and that outliers are quite common. It has long been known that in terms of both power and Type I errors, even slight departures from normality can have serious consequences, and it is fairly evident that there are problems when using conventional measures of effect size, as is briefly illustrated in this paper. Similar problems arise when dealing with the usual correlation coefficient. Robust correlation coefficients have already been proposed that reflect the linear relationship between two random variables, but many have the unfortunate property of not always being equal to 0 under independence. One exception is the Winsorized correlation. The primary goal in this paper is to suggest a simple method for testing the hypothesis that the Winsorized correlation is equal to zero. A minor goal is to describe a formal definition of the Winsorized correlation coefficient in terms of Winsorized expected values.}, number={2}, journal={British Journal of Mathematical and Statistical Psychology}, publisher={Wiley}, author={Wilcox, Rand R.}, year={1993}, month=nov, pages={339–349}, language={en} } + @article{rafi2020, title={Semantic and cognitive tools to aid statistical science: replace confidence and significance by compatibility and surprise}, author={Rafi, Zad and Greenland, Sander}, From 2ae7c12a56b12df678ecdae4d4a6d3e0479e338f Mon Sep 17 00:00:00 2001 From: Aaron Caldwell Date: Fri, 17 Apr 2026 09:57:42 -0500 Subject: [PATCH 08/40] Adding assumption checks --- papers/Avocado_Update/Avocado_Update.Rmd | 11 +- .../toster_assumption_checks_supplement.pdf | Bin 0 -> 855277 bytes .../toster_assumption_checks_supplement.qmd | 810 ++++++++++++++++++ 3 files changed, 819 insertions(+), 2 deletions(-) create mode 100644 papers/Avocado_Update/toster_assumption_checks_supplement.pdf create mode 100644 papers/Avocado_Update/toster_assumption_checks_supplement.qmd diff --git a/papers/Avocado_Update/Avocado_Update.Rmd b/papers/Avocado_Update/Avocado_Update.Rmd index fb2944fe..186ca5b7 100644 --- a/papers/Avocado_Update/Avocado_Update.Rmd +++ b/papers/Avocado_Update/Avocado_Update.Rmd @@ -280,6 +280,10 @@ res4 The same interface handles paired designs by setting `paired = TRUE`. Standardized mean differences are reported automatically; Hedges's correction is applied by default but can be toggled with `bias_correction`, and Glass's delta can be requested via the `glass` argument. Setting equivalence bounds in SMD units (via `eqbound_type = "SMD"`) is possible but discouraged because the bound becomes dependent on sample variance (see the Standardized Effect Sizes section below). +## Checking Assumptions + +The parametric t-tests implemented in `t_TOST` assume independent observations, approximately normal residuals (or differences, in the paired and one-sample cases), and, for Student's form, equal variances between groups; Welch's correction relaxes the equal-variance assumption and is the default. The residual normality assumption matters most in small samples, where the central limit theorem provides limited protection. Visual diagnostics are generally more informative than formal tests: QQ plots of residuals or differences for normality, side-by-side boxplots or density overlays for variance comparison, and inspection of residual plots or boxplots for extreme or influential observations. A worked diagnostic workflow for each design is provided in the online supplement. When assumptions are seriously in doubt, the bootstrap and permutation alternatives discussed in later sections provide valid inference under relaxed --— though not eliminated —-- distributional requirements. + \newpage ## Using Summary Statistics @@ -302,7 +306,7 @@ res_tsum The `t_TOST` function is designed to be comprehensive in that it runs three tests, computes standardized effect sizes, and produces verbose output. This is useful for exploration or situations where researchers *always* want to pair a traditional nil significance test with an equivalence test. However, sometimes a researcher has a single specific hypothesis to test, or simply wants less verbose output (e.g., SMD calculation is not wanted). The `simple_htest` function provides a lighter interface for this purpose. It returns a standard R `htest` object --- the same class returned by `t.test()` and `wilcox.test()` --- but with expanded `alternative` options that include `"equivalence"` and `"minimal.effect"`. This means researchers can perform equivalence testing with the same familiar interface they already use for standard hypothesis tests^[In fact, users can mostly replace `t.test()` and `wilcox.test()` with `simple_htest` as it also allows for the standard alternative hypotheses of two-sided, greater, less. If you have a hypothesis that would traditionally utilize a t-test or Wilcoxon-Mann-Whitney (WMW) test, then you can jump right to using `simple_htest` in TOSTER]. -Alongside `simple_htest`, a set of helper functions makes it easy to work with any `htest` object: `as_htest` converts verbose TOST output to the simpler format, `describe_htest` generates formatted text for reporting, `df_htest` converts results to data frames for tabulation, and `plot_htest_est` produces estimate plots with equivalence bounds. +Alongside `simple_htest`, a set of helper functions makes it easy to work with any `htest` object: `as_htest` converts verbose TOST output to the simpler format, `describe_htest` generates formatted text for reporting, `df_htest` converts results to data frames for tabulation, and `plot_htest_est` produces estimate plots with equivalence bounds. Checking assumptions would follow the same workflow as `t_TOST`, as both are using t-tests, as discussed in the online supplement. ## The simplified function @@ -399,7 +403,6 @@ boot_smd_calc(formula = extra ~ group, Researchers sometimes need to test that a correlation is practically zero. The `z_cor_test` function provides a parametric approach using the Fisher z-transformation. However, this is only an approximate method and is generally *not recommended* as the primary inferential tool for equivalence testing on correlations. ```{r} - z_cor_test( x = bugs$LDLF, y = bugs$LDHF, @@ -424,6 +427,10 @@ describe_htest(res_boot_cor) If only summary statistics are available, an approximate method is provided through `corsum_test`, which uses the Fisher z-transformation to compute confidence intervals and p-values. +### Checking Assumptions + +Correlation-based inference in `z_cor_test` and `boot_cor_test` assumes independent paired observations and, for Pearson's coefficient, a roughly linear relationship between variables with approximately symmetric marginal distributions; Spearman and Kendall relax the linearity requirement to monotonicity, making them appropriate when the relationship is non-linear but directionally consistent. The bootstrap version in `boot_cor_test` further relaxes distributional assumptions about the marginals but remains sensitive to observations that exert disproportionate influence on the estimate. Visual diagnostics are the primary tools: scatterplots with a loess smoother to assess linearity or monotonicity, marginal density plots or QQ plots to check distributional shape, and jackknife-after-bootstrap plots to identify individual observations driving bootstrap variability. A worked diagnostic workflow is provided in the online supplement. + ## Comparing Standardized Effect Sizes Between Studies When evaluating whether a replication study produced results consistent with the original, researchers need a way to formally compare effect sizes. The `compare_smd` function tests whether two SMD estimates --- from an original study and a replication, for example --- differ more than would be expected by sampling variability alone. diff --git a/papers/Avocado_Update/toster_assumption_checks_supplement.pdf b/papers/Avocado_Update/toster_assumption_checks_supplement.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4ce84a4aaf7509af6ba44bd7c8e66e2755cac629 GIT binary patch literal 855277 zcma&NWmII%vL*~oqm4U_yA5gG~%G7$!CXdIB4La~N)J0y+^hOGhL7 zzpbU7qmhu2fsLUN0iBeQwTYuC0RtNc8v!pb!GC>V931V9^sHc9S1&Y1toKL!EK}p)VCe#s?;}gUF6=cd(l_# zKHe>F-m&lBGtA&%Waz%up?yNK`^BL~GUCX60&=H|lD}kpHXPumG9d{}{fa5>6^( zUW60S$`rpOl4VzgM%)7N`{Lgd=`EV`TLudCqG+9DI?bT8P=*!X z{d^;hK11(+_|s6fZGu1^1CZFzgPUax+0FmpvGygSn^fb>qEC|bww!*FN3Nv)@Gl`k zH8$J^l^P`0jCW4(rxV|SR8e5GiU9d;>B>$}!ArxB2rZqvni^O>Ccecs(w(~ZX_FvC za5wv#WHja!6=7^MG8VnCN5RB8j%;7q0bU{G=d$k-^$&jzO}Bz9in2 zq+xX0urqGHBTdBaC5|X@VWZ%&F43BtYm~b}muQ}dY?1N)l`={nMmp1tK)+Yi)>r6f 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