Power comparisons of significance tests of location using scores,ranks, and modular ranks

The Type I error probability and the power of the independent samples t test, performed directly on the ranks of scores in combined samples in place of the original scores, are known to be the same as those of the non‐parametric Wilcoxon–Mann–Whitney (WMW) test. In the present study, simulations revealed that these probabilities remain essentially unchanged when the number of ranks is reduced by assigning the same rank to multiple ordered scores. For example, if 200 ranks are reduced to as few as 20, or 10, or 5 ranks by replacing sequences of consecutive ranks by a single number, the Type I error probability and power stay about the same. Significance tests performed on these modular ranks consistently reproduce familiar findings about the comparative power of the t test and the WMW tests for normal and various non‐normal distributions. Similar results are obtained for modular ranks used in comparing the one‐sample t test and the Wilcoxon signed ranks test.     

A comparison of the randomization test with the<Emphasis Type="Italic">F</Emphasis> test when error is skewed

I compared the randomization/permutation test and theF test for a two-cell comparative experiment. I varied (1) the number of observations per cell, (2) the size of the treatment effect, (3) the shape of the underlying distribution of error and, (4) for cases with skewed error, whether or not the skew was correlated with the treatment. With normal error, there was little difference between the tests. When error was skewed, by contrast, the randomization test was more sensitive than theF test, and if the amount of skew was correlated with the treatment, the advantage for the randomization test was both large and positively correlated with the treatment. I conclude that, because the randomization test was never less powerful than theF test, it should replace theF test in routine work.     

Replicability and randomization test logic in behavior analysis

Randomization tests are a class of nonparametric statistics that determine the significance of treatment effects. Unlike parametric statistics, randomization tests do not assume a random sample, or make any of the distributional assumptions that often preclude statistical inferences about single‐case data. A feature that randomization tests share with parametric statistics, however, is the derivation of a p‐value. P‐values are notoriously misinterpreted and are partly responsible for the putative “replication crisis.” Behavior analysts might question the utility of adding such a controversial index of statistical significance to their methods, so it is the aim of this paper to describe the randomization test logic and its potentially beneficial consequences. In doing so, this paper will: (1) address the replication crisis as a behavior analyst views it, (2) differentiate the problematic p‐values of parametric statistics from the, arguably, more useful p‐values of randomization tests, and (3) review the logic of randomization tests and their unique fit within the behavior analytic tradition of studying behavioral processes that cut across species.     

A Cathode Ray Oscillograph Apparatus for the Psychological Laboratory

A computer simulation which generated various sample distributions was performed. The equal interval assumption and degree of skewness were systematically manipulated to examine the sensitivity of the t and Mann-Whitney tests to varying degrees of violations of these t test assumptions. Results show that of the 348 sets (2436 t tests), 94 (27%) contained shifts from significant to nonsignificant t test values under mild violations of the equal-interval and normality assumption and would have led to decision errors. Of these, 55% would have been Type I and 45% would have been Type II. The implications of these results are discussed relative to Types I and II errors, to the use of parametric and nonparametric statistics, and to the likelihood of encountering such problems in sampled distributions.     

Bootstrap scree tests: A Monte Carlo simulation and applications to published data

A non‐parametric procedure for Cattell's scree test is proposed, using the bootstrap method. Bentler and Yuan developed parametric tests for the linear trend of scree eigenvalues in principal component analysis. The proposed method is for cases where parametric assumptions are not realistic. We define the break in the scree trend in several ways, based on linear slopes defined with two or three consecutive eigenvalues, or all eigenvalues after the k largest. The resulting scree test statistics are evaluated under various data conditions, among which Gorsuch and Nelson's bootstrap CNG performs best and is reasonably consistent and efficient under leptokurtic and skewed conditions. We also examine the bias‐corrected and accelerated bootstrap method for these statistics, and the bias correction is found to be too unstable to be useful. Using seven published data sets which Bentler and Yuan analysed, we compare the bootstrap approach to the scree test with the parametric linear trend test.     

A note on consistency of non-parametric rank tests and related rank transformations

The extent to which rank transformations result in the same statistical decisions as their non‐parametric counterparts is investigated. Simulations are presented using the Wilcoxon–Mann–Whitney test, the Wilcoxon signed‐rank test and the Kruskal–Wallis test, together with the rank transformations and t and F tests corresponding to each of those non‐parametric methods. In addition to Type I errors and power over all simulations, the study examines the consistency of the outcomes of the two methods on each individual sample. The results show how acceptance or rejection of the null hypothesis and differences in p‐values of the test statistics depend in a regular and predictable way on sample size, significance level, and differences between means, for normal and various non‐normal distributions.     

Sample size determination for at test given at value from a previous study: A FORTRAN 77 program

When uncertain about the magnitude of an effect, researchers commonly substitute in the standard sample-size-determination formula an estimate of effect size derived from a previous experiment. A problem with this approach is that the traditional sample-size-determination formula was not designed to deal with the uncertainty inherent in an effect-size estimate. Consequently, estimate-substitution in the traditional sample-size-determination formula can lead to a substantial loss of power. A method of sample-size determination designed to handle uncertainty in effect-size estimates is described. The procedure uses thet value and sample size from a previous study, which might be a pilot study or a related study in the same area, to establish a distribution of probable effect sizes. The sample size to be employed in the new study is that which supplies an expected power of the desired amount over the distribution of probable effect sizes. A FORTRAN 77 program is presented that permits swift calculation of sample size for a variety oft tests, including independentt tests, relatedt tests,t tests of correlation coefficients, andt tests of multiple regressionb coefficients.     

A note on statistical power in multi‐site randomized trials with multiple treatments at each site

We derive the statistical power functions in multi‐site randomized trials with multiple treatments at each site, using multi‐level modelling. An F statistic is used to test multiple parameters in the multi‐level model instead of the Wald chi square test as suggested in the current literature. The F statistic is shown to be more conservative than the Wald statistic in testing any overall treatment effect among the multiple study conditions. In addition, we improvise an easy way to estimate the non‐centrality parameters for the means comparison t‐tests and the F test, using Helmert contrast coding in the multi‐level model. The variance of treatment means, which is difficult to fathom but necessary for power analysis, is decomposed into intuitive simple effect sizes in the contrast tests. The method is exemplified by a multi‐site evaluation study of the behavioural interventions for cannabis dependence.     

SAS procedures for approximate randomization tests

Approximate randomization tests are alternatives to conventional parametric statistical methods used when the normality and homoscedasticity assumptions are violated This article presents an SAS program that tests the equality of two means using an approximate randomization test This program can serve as a template for testing other hypotheses, which is illustrated by modifications to test the significance of a correlation coefficient or the equality of more than two means.     

Testing overall and moderator effects in random effects meta‐regression

Random effects meta‐regression is a technique to synthesize results of multiple studies. It allows for a test of an overall effect, as well as for tests of effects of study characteristics, that is, (discrete or continuous) moderator effects. We describe various procedures to test moderator effects: the z, t, likelihood ratio (LR), Bartlett‐corrected LR (BcLR), and resampling tests. We compare the Type I error of these tests, and conclude that the common z test, and to a lesser extent the LR test, do not perform well since they may yield Type I error rates appreciably larger than the chosen alpha. The error rate of the resampling test is accurate, closely followed by the BcLR test. The error rate of the t test is less accurate but arguably tolerable. With respect to statistical power, the BcLR and t tests slightly outperform the resampling test. Therefore, our recommendation is to use either the resampling or the BcLR test. If these statistics are unavailable, then the t test should be used since it is certainly superior to the z test.     

On sample size calculation for 2×2 fixed‐effect ANOVA when variances are unknown and possibly unequal

The factorial 2 × 2 fixed‐effect ANOVA is a procedure used frequently in scientific research to test mean differences between‐subjects in all of the groups. But if the assumption of homogeneity is violated, the test for the row, column, and the interaction effect might be invalid or less powerful. Therefore, for planning research in the case of unknown and possibly unequal variances, it is worth developing a sample size formula to obtain the desired power. This article suggests a simple formula to determine the sample size for 2 × 2 fixed‐effect ANOVA for heterogeneous variances across groups. We use the approximate Welch t test and consider the variance ratio to derive the formula. The sample size determination requires two‐step iterations but the approximate sample sizes needed for the main effect and the interaction effect can be determined separately with the specified power. The present study also provides an example and a SAS program to facilitate the calculation process.     

Randomization tests as alternative analysis methods for behavior‐analytic data

Randomization statistics offer alternatives to many of the statistical methods commonly used in behavior analysis and the psychological sciences, more generally. These methods are more flexible than conventional parametric and nonparametric statistical techniques in that they make no assumptions about the underlying distribution of outcome variables, are relatively robust when applied to small‐n data sets, and are generally applicable to between‐groups, within‐subjects, mixed, and single‐case research designs. In the present article, we first will provide a historical overview of randomization methods. Next, we will discuss the properties of randomization statistics that may make them particularly well suited for analysis of behavior‐analytic data. We will introduce readers to the major assumptions that undergird randomization methods, as well as some practical and computational considerations for their application. Finally, we will demonstrate how randomization statistics may be calculated for mixed and single‐case research designs. Throughout, we will direct readers toward resources that they may find useful in developing randomization tests for their own data.     

A nonparametric method to analyze interactions: The adjusted rank transform test

Experimental social psychologists routinely rely on ANOVA to study interactions between factors even when the assumptions underlying the use of parametric tests are not met. Alternative nonparametric methods are often relatively difficult to conduct, have seldom been presented into detail in regular curriculum and have the reputation - sometimes incorrectly - of being less powerful than parametric tests. This article presents the adjusted rank transform test (ART); a nonparametric test, easy to conduct, having the advantage of being much more powerful than parametric tests when certain assumptions underlying the use of these tests are violated. To specify the conditions under which the adjusted rank transform test is superior to the usual parametric tests, results of a Monte Carlo simulation are presented.     

Evaluation of global testing procedures for item fit to the Rasch model

Two types of global testing procedures for item fit to the Rasch model were evaluated using simulation studies. The first type incorporates three tests based on first‐order statistics: van den Wollenberg's Q₁ test, Glas's R₁ test, and Andersen's LR test. The second type incorporates three tests based on second‐order statistics: van den Wollenberg's Q₂ test, Glas's R₂ test, and a non‐parametric test proposed by Ponocny. The Type I error rates and the power against the violation of parallel item response curves, unidimensionality and local independence were analysed in relation to sample size and test length. In general, the outcomes indicate a satisfactory performance of all tests, except the Q₂ test which exhibits an inflated Type I error rate. Further, it was found that both types of tests have power against all three types of model violation. A possible explanation is the interdependencies among the assumptions underlying the model.     

G*Power 3: A flexible statistical power analysis program for the social,behavioral, and biomedical sciences

G*Power (Erdfelder, Faul, & Buchner, 1996) was designed as a general stand-alone power analysis program for statistical tests commonly used in social and behavioral research. G*Power 3 is a major extension of, and improvement over, the previous versions. It runs on widely used computer platforms (i.e., Windows XP, Windows Vista, and Mac OS X 10.4) and covers many different statistical tests of thet, F, and χ₂ test families. In addition, it includes power analyses forz tests and some exact tests. G*Power 3 provides improved effect size calculators and graphic options, supports both distribution-based and design-based input modes, and offers all types of power analyses in which users might be interested. Like its predecessors, G*Power 3 is free.     

SPSS procedures for approximate randomization tests

Randomization tests are valid alternatives to parametric tests like the t test and analysis of variance when the normality or random sampling assumptions of these tests are violated. Three SPSS programs are listed and described that will conduct approximate randomization tests for testing the null hypotheses that two or more means or distributions are the same or that two variables are independent (i.e., uncorrelated or “randomly associated”). The programs will work on both desktop and mainframe versions of SPSS. Although the SPSS programs are slower on desktop machines than software designed explicitly for randomization tests, these programs bring randomization tests into the reach of researchers who prefer the SPSS computing environment for data analysis.     

A General Family of Limited Information Goodness-of-Fit Statistics for Multinomial Data

Maydeu-Olivares and Joe (J. Am. Stat. Assoc. 100:1009–1020, 2005; Psychometrika 71:713–732, 2006) introduced classes of chi-square tests for (sparse) multidimensional multinomial data based on low-order marginal proportions. Our extension provides general conditions under which quadratic forms in linear functions of cell residuals are asymptotically chi-square. The new statistics need not be based on margins, and can be used for one-dimensional multinomials. We also provide theory that explains why limited information statistics have good power, regardless of sparseness. We show how quadratic-form statistics can be constructed that are more powerful than X ² and yet, have approximate chi-square null distribution in finite samples with large models. Examples with models for truncated count data and binary item response data are used to illustrate the theory.     

Comparing score tests and other local dependence diagnostics for the graded response model

Score tests for identifying locally dependent item pairs have been proposed for binary item response models. In this article, both the bifactor and the threshold shift score tests are generalized to the graded response model. For the bifactor test, the generalization is straightforward; it adds one secondary dimension associated only with one pair of items. For the threshold shift test, however, multiple generalizations are possible: in particular, conditional, uniform, and linear shift tests are discussed in this article. Simulation studies show that all of the score tests have accurate Type I error rates given large enough samples, although their small‐sample behaviour is not as good as that of Pearson's Χ² and M₂ as proposed in other studies for the purpose of local dependence (LD) detection. All score tests have the highest power to detect the LD which is consistent with their parametric form, and in this case they are uniformly more powerful than Χ² and M₂; even wrongly specified score tests are more powerful than Χ² and M₂ in most conditions. An example using empirical data is provided for illustration.     

Properties of bootstrap tests for N‐of‐1 studies

N‐of‐1 study designs involve the collection and analysis of repeated measures data from an individual not using an intervention and using an intervention. This study explores the use of semi‐parametric and parametric bootstrap tests in the analysis of N‐of‐1 studies under a single time series framework in the presence of autocorrelation. When the Type I error rates of bootstrap tests are compared to Wald tests, our results show that the bootstrap tests have more desirable properties. We compare the results for normally distributed errors with those for contaminated normally distributed errors and find that, except when there is relatively large autocorrelation, there is little difference between the power of the parametric and semi‐parametric bootstrap tests. We also experiment with two intervention designs: ABAB and AB, and show the ABAB design has more power. The results provide guidelines for designing N‐of‐1 studies, in the sense of how many observations and how many intervention changes are needed to achieve a certain level of power and which test should be performed.     

Nonparametric tests for equality of psychometric functions

Many empirical studies measure psychometric functions (curves describing how observers’ performance varies with stimulus magnitude) because these functions capture the effects of experimental conditions. To assess these effects, parametric curves are often fitted to the data and comparisons are carried out by testing for equality of mean parameter estimates across conditions. This approach is parametric and, thus, vulnerable to violations of the implied assumptions. Furthermore, testing for equality of means of parameters may be misleading: Psychometric functions may vary meaningfully across conditions on an observer-by-observer basis with no effect on the mean values of the estimated parameters. Alternative approaches to assess equality of psychometric functions per se are thus needed. This paper compares three nonparametric tests that are applicable in all situations of interest: The existing generalized Mantel–Haenszel test, a generalization of the Berry–Mielke test that was developed here, and a split variant of the generalized Mantel–Haenszel test also developed here. Their statistical properties (accuracy and power) are studied via simulation and the results show that all tests are indistinguishable as to accuracy but they differ non-uniformly as to power. Empirical use of the tests is illustrated via analyses of published data sets and practical recommendations are given. The computer code in matlab and R to conduct these tests is available as Electronic Supplemental Material.