Variable criteria sequential stopping rule: Validity and power with repeated measures ANOVA,multiple correlation,MANOVA and relation to Chi-square distribution

The variable criteria sequential stopping rule (vcSSR) is an efficient way to add sample size to planned ANOVA tests while holding the observed rate of Type I errors, α_o, constant. The only difference from regular null hypothesis testing is that criteria for stopping the experiment are obtained from a table based on the desired power, rate of Type I errors, and beginning sample size. The vcSSR was developed using between-subjects ANOVAs, but it should work with p values from any type of F test. In the present study, the α_o remained constant at the nominal level when using the previously published table of criteria with repeated measures designs with various numbers of treatments per subject, Type I error rates, values of ρ, and four different sample size models. New power curves allow researchers to select the optimal sample size model for a repeated measures experiment. The criteria held α_o constant either when used with a multiple correlation that varied the sample size model and the number of predictor variables, or when used with MANOVA with multiple groups and two levels of a within-subject variable at various levels of ρ. Although not recommended for use with χ² tests such as the Friedman rank ANOVA test, the vcSSR produces predictable results based on the relation between F and χ². Together, the data confirm the view that the vcSSR can be used to control Type I errors during sequential sampling with any t- or F-statistic rather than being restricted to certain ANOVA designs.     

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.     

Extending the CLAST sequential rule to one-way ANOVA under group sampling

Several studies have demonstrated that the fixed-sample stopping rule (FSR), in which the sample size is determined in advance, is less practical and efficient than are sequential-stopping rules. The composite limited adaptive sequential test (CLAST) is one such sequential-stopping rule. Previous research has shown that CLAST is more efficient in terms of sample size and power than are the FSR and other sequential rules and that it reflects more realistically the practice of experimental psychology researchers. The CLAST rule has been applied only to thet test of mean differences with two matched samples and to the chi-square independence test for twofold contingency tables. The present work extends previous research on the efficiency of CLAST to multiple group statistical tests. Simulation studies were conducted to test the efficiency of the CLAST rule for the one-way ANOVA for fixed effects models. The ANOVA general test and two linear contrasts of multiple comparisons among treatment means are considered. The article also introduces four rules for allocatingN observations toJ groups under the general null hypothesis and three allocation rules for the linear contrasts. Results show that the CLAST rule is generally more efficient than the FSR in terms of sample size and power for one-way ANOVA tests. However, the allocation rules vary in their optimality and have a differential impact on sample size and power. Thus, selecting an allocation rule depends on the cost of sampling and the intended precision.     

Approximate sample size formulas for the two‐sample trimmed mean test with unequal variances

Yuen's two‐sample trimmed mean test statistic is one of the most robust methods to apply when variances are heterogeneous. The present study develops formulas for the sample size required for the test. The formulas are applicable for the cases of unequal variances, non‐normality and unequal sample sizes. Given the specified α and the power (1?β), the minimum sample size needed by the proposed formulas under various conditions is less than is given by the conventional formulas. Moreover, given a specified size of sample calculated by the proposed formulas, simulation results show that Yuen's test can achieve statistical power which is generally superior to that of the approximate t test. A numerical example is provided.     

Optimization of sample size in controlled experiments: The CLAST rule

Sequential rules are explored in the context of null hypothesis significance testing. Several studies have demonstrated that the fixed-sample stopping rule, in which the sample size used by researchers is determined in advance, is less practical and less efficient than sequential stopping rules. It is proposed that a sequential stopping rule called CLAST (composite limited adaptive sequential test) is a superior variant of COAST (composite open adaptive sequential test), a sequential rule proposed by Frick (1998). Simulation studies are conducted to test the efficiency of the proposed rule in terms of sample size and power. Two statistical tests are used: the one-tailed t test of mean differences with two matched samples, and the chi-square independence test for twofold contingency tables. The results show that the CLAST rule is more efficient than the COAST rule and reflects more realistically the practice of experimental psychology researchers.     

Improved stopping rules for the design of efficient small-sample experiments in biomedical and biobehavioral research

Sequential stopping rules (SSRs) should augment traditional hypothesis tests in many planned experiments, because they can provide the same statistical power with up to 30% fewer subjects without additional education or software. This article includes new Monte-Carlo-generated power curves and tables of stopping criteria based on the p values from simulated t tests and one-way ANOVAs. The tables improve existing SSR techniques by holding alpha very close to a target value when 1–10 subjects are added at each iteration. The emphasis is on small sample sizes (3–40 subjects per group) and large standardized effect sizes (0.8–2.0). The generality of the tables for dependent samples and one-tailed tests is discussed. SSR methods should be of interest to ethics bodies governing research when it is desirable to limit the number of subjects tested, such as in studies of pain, experimental disease, or surgery with animal or human subjects.     

The frequentist implications of optional stopping on Bayesian hypothesis tests

Null hypothesis significance testing (NHST) is the most commonly used statistical methodology in psychology. The probability of achieving a value as extreme or more extreme than the statistic obtained from the data is evaluated, and if it is low enough, the null hypothesis is rejected. However, because common experimental practice often clashes with the assumptions underlying NHST, these calculated probabilities are often incorrect. Most commonly, experimenters use tests that assume that sample sizes are fixed in advance of data collection but then use the data to determine when to stop; in the limit, experimenters can use data monitoring to guarantee that the null hypothesis will be rejected. Bayesian hypothesis testing (BHT) provides a solution to these ills because the stopping rule used is irrelevant to the calculation of a Bayes factor. In addition, there are strong mathematical guarantees on the frequentist properties of BHT that are comforting for researchers concerned that stopping rules could influence the Bayes factors produced. Here, we show that these guaranteed bounds have limited scope and often do not apply in psychological research. Specifically, we quantitatively demonstrate the impact of optional stopping on the resulting Bayes factors in two common situations: (1) when the truth is a combination of the hypotheses, such as in a heterogeneous population, and (2) when a hypothesis is composite—taking multiple parameter values—such as the alternative hypothesis in a t-test. We found that, for these situations, while the Bayesian interpretation remains correct regardless of the stopping rule used, the choice of stopping rule can, in some situations, greatly increase the chance of experimenters finding evidence in the direction they desire. We suggest ways to control these frequentist implications of stopping rules on BHT.     

ANOVA and the variance homogeneity assumption: Exploring a better gatekeeper

Valid use of the traditional independent samples ANOVA procedure requires that the population variances are equal. Previous research has investigated whether variance homogeneity tests, such as Levene's test, are satisfactory as gatekeepers for identifying when to use or not to use the ANOVA procedure. This research focuses on a novel homogeneity of variance test that incorporates an equivalence testing approach. Instead of testing the null hypothesis that the variances are equal against an alternative hypothesis that the variances are not equal, the equivalence-based test evaluates the null hypothesis that the difference in the variances falls outside or on the border of a predetermined interval against an alternative hypothesis that the difference in the variances falls within the predetermined interval. Thus, with the equivalence-based procedure, the alternative hypothesis is aligned with the research hypothesis (variance equality). A simulation study demonstrated that the equivalence-based test of population variance homogeneity is a better gatekeeper for the ANOVA than traditional homogeneity of variance tests.     

A note on preliminary tests of equality of variances

Preliminary tests of equality of variances used before a test of location are no longer widely recommended by statisticians, although they persist in some textbooks and software packages. The present study extends the findings of previous studies and provides further reasons for discontinuing the use of preliminary tests. The study found Type I error rates of a two‐stage procedure, consisting of a preliminary Levene test on samples of different sizes with unequal variances, followed by either a Student pooled‐variances t test or a Welch separate‐variances t test. Simulations disclosed that the twostage procedure fails to protect the significance level and usually makes the situation worse. Earlier studies have shown that preliminary tests often adversely affect the size of the test, and also that the Welch test is superior to the t test when variances are unequal. The present simulations reveal that changes in Type I error rates are greater when sample sizes are smaller, when the difference in variances is slight rather than extreme, and when the significance level is more stringent. Furthermore, the validity of the Welch test deteriorates if it is used only on those occasions where a preliminary test indicates it is needed. Optimum protection is assured by using a separate‐variances test unconditionally whenever sample sizes are unequal.     

Some Thoughts Regarding the Sensitivity of t to Error in Estimated Within-Group Variance

A. J. Riopelle (2003) has eloquently demonstrated that the null hypothesis assessed by the t test involves not only mean differences but also error in the estimation of the within-group standard deviation, s. He is correct in his conclusion that the precision of the interpretation of a significant t and the null hypothesis tested is complex, particularly when sample sizes are small. In this article, the author expands on Riopelle's thoughts by comparing t with some equivalent or closely related tests that make the reliance of t on the accurate estimation of error perhaps more salient and by providing a simulation that may address more directly the magnitude of the interpretational problem.     

Functional Anatomy of the Null Hypothesis and of Tests of It

The author compared simulations of the “true” null hypothesis (z) test, in which ò was known and fixed, with the t test, in which s, an estimate of ò, was calculated from the sample because the t test was used to emulate the “true” test. The true null hypothesis test bears exclusively on calculating the probability that a sample distance (mean) is larger than a specified value. The results showed that the value of t was sensitive to sampling fluctuations in both distance and standard error. Large values of t reflect small standard errors when n is small. The value of t achieves sensitivity primarily to distance only when the sample sizes are large. One cannot make a definitive statement about the probability or “significance” of a distance solely on the basis of the value of t.     

Effect of non-normality on test statistics for one-way independent groups designs

The data obtained from one‐way independent groups designs is typically non‐normal in form and rarely equally variable across treatment populations (i.e. population variances are heterogeneous). Consequently, the classical test statistic that is used to assess statistical significance (i.e. the analysis of variance F test) typically provides invalid results (e.g. too many Type I errors, reduced power). For this reason, there has been considerable interest in finding a test statistic that is appropriate under conditions of non‐normality and variance heterogeneity. Previously recommended procedures for analysing such data include the James test, the Welch test applied either to the usual least squares estimators of central tendency and variability, or the Welch test with robust estimators (i.e. trimmed means and Winsorized variances). A new statistic proposed by Krishnamoorthy, Lu, and Mathew, intended to deal with heterogeneous variances, though not non‐normality, uses a parametric bootstrap procedure. In their investigation of the parametric bootstrap test, the authors examined its operating characteristics under limited conditions and did not compare it to the Welch test based on robust estimators. Thus, we investigated how the parametric bootstrap procedure and a modified parametric bootstrap procedure based on trimmed means perform relative to previously recommended procedures when data are non‐normal and heterogeneous. The results indicated that the tests based on trimmed means offer the best Type I error control and power when variances are unequal and at least some of the distribution shapes are non‐normal.     

Robust tests of equivalence for k independent groups

A common question of interest to researchers in psychology is the equivalence of two or more groups. Failure to reject the null hypothesis of traditional hypothesis tests such as the ANOVA F‐test (i.e., H₀: μ₁ = … = μ_k) does not imply the equivalence of the population means. Researchers interested in determining the equivalence of k independent groups should apply a one‐way test of equivalence (e.g., Wellek, 2003). The goals of this study were to investigate the robustness of the one‐way Wellek test of equivalence to violations of homogeneity of variance assumption, and compare the Type I error rates and power of the Wellek test with a heteroscedastic version which was based on the logic of the one‐way Welch (1951) F‐test. The results indicate that the proposed Wellek–Welch test was insensitive to violations of the homogeneity of variance assumption, whereas the original Wellek test was not appropriate when the population variances were not equal.     

Stopping Rules for Computer Adaptive Testing When Item Banks Have Nonuniform Information

The standard error (SE) stopping rule, which terminates a computer adaptive test (CAT) when the SE is less than a threshold, is effective when there are informative questions for all trait levels. However, in domains such as patient-reported outcomes, the items in a bank might all target one end of the trait continuum (e.g., negative symptoms), and the bank may lack depth for many individuals. In such cases, the predicted standard error reduction (PSER) stopping rule will stop the CAT even if the SE threshold has not been reached and can avoid administering excessive questions that provide little additional information. By tuning the parameters of the PSER algorithm, a practitioner can specify a desired tradeoff between accuracy and efficiency. Using simulated data for the Patient-Reported Outcomes Measurement Information System Anxiety and Physical Function banks, we demonstrate that these parameters can substantially impact CAT performance. When the parameters were optimally tuned, the PSER stopping rule was found to outperform the SE stopping rule overall, particularly for individuals not targeted by the bank, and presented roughly the same number of items across the trait continuum. Therefore, the PSER stopping rule provides an effective method for balancing the precision and efficiency of a CAT.     

Consequences of choosing samples in hypothesis testing to ensure homogeneity of variance

The two‐sample Student t test of location was performed on random samples of scores and on rank‐transformed scores from normal and non‐normal population distributions with unequal variances. The same test also was performed on scores that had been explicitly selected to have nearly equal sample variances. The desired homogeneity of variance was brought about by repeatedly rejecting pairs of samples having a ratio of standard deviations that exceeded a predetermined cut‐off value of 1.1, 1.2, or 1.3, while retaining pairs with ratios less than the cut‐off value. Despite this forced conformity with the assumption of equal variances, the tests on the selected samples were no more robust than tests on unselected samples, and in most cases substantially less robust. Under conditions where sample sizes were unequal, so that Type I error rates were inflated and power curves were atypical, the selection procedure produced still greater inflation and distortion of the power curves.     

Further evaluating the conditional decision rule for comparing two independent means

Many books on statistical methods advocate a ‘conditional decision rule’ when comparing two independent group means. This rule states that the decision as to whether to use a ‘pooled variance’ test that assumes equality of variance or a ‘separate variance’ Welch t test that does not should be based on the outcome of a variance equality test. In this paper, we empirically examine the Type I error rate of the conditional decision rule using four variance equality tests and compare this error rate to the unconditional use of either of the t tests (i.e. irrespective of the outcome of a variance homogeneity test) as well as several resampling‐based alternatives when sampling from 49 distributions varying in skewness and kurtosis. Several unconditional tests including the separate variance test performed as well as or better than the conditional decision rule across situations. These results extend and generalize the findings of previous researchers who have argued that the conditional decision rule should be abandoned.     

Are Effect Sizes and Confidence Levels Problems For or Solutions To the Null Hypothesis Test?

Some have proposed that the null hypothesis significance test, as usually conducted using the t test of the difference between means, is an impediment to progress in psychology. To improve its prospects, using Neyman-Pearson confidence intervals and Cohen's standardized effect sizes, d, is recommended. The purpose of these approaches is to enable us to understand what can appropriately be said about the distances between the means and their reliability. Others have written extensively that these recommended strategies are highly interrelated and use identical information. This essay was written to remind us that the t test, based on the sample—not the true—standard deviation, does not apply solely to distance between means. The t test pertains to a much more ambiguous specification: the difference between samples, including sampling variations of the standard deviation.     

New heterogeneous test statistics for the unbalanced fixed‐effect nested design

When the underlying variances are unknown or/and unequal, using the conventional F test is problematic in the two‐factor hierarchical data structure. Prompted by the approximate test statistics (Welch and Alexander–Govern methods), the authors develop four new heterogeneous test statistics to test factor A and factor B nested within A for the unbalanced fixed‐effect two‐stage nested design under variance heterogeneity. The actual significance levels and statistical power of the test statistics were compared in a simulation study. The results show that the proposed procedures maintain better Type I error rate control and have greater statistical power than those obtained by the conventional F test in various conditions. Therefore, the proposed test statistics are recommended in terms of robustness and easy implementation.     

How and how not to make predictions with temporal Copernicanism

Gott (Nature 363:315–319, 1993) considers the problem of obtaining a probabilistic prediction for the duration of a process, given the observation that the process is currently underway and began a time t ago. He uses a temporal Copernican principle according to which the observation time can be treated as a random variable with uniform probability density. A simple rule follows: with a 95% probability,

Sample size determinations for Welch's test in one‐way heteroscedastic ANOVA

For one‐way fixed effects ANOVA, it is well known that the conventional F test of the equality of means is not robust to unequal variances, and numerous methods have been proposed for dealing with heteroscedasticity. On the basis of extensive empirical evidence of Type I error control and power performance, Welch's procedure is frequently recommended as the major alternative to the ANOVA F test under variance heterogeneity. To enhance its practical usefulness, this paper considers an important aspect of Welch's method in determining the sample size necessary to achieve a given power. Simulation studies are conducted to compare two approximate power functions of Welch's test for their accuracy in sample size calculations over a wide variety of model configurations with heteroscedastic structures. The numerical investigations show that Levy's (1978a) approach is clearly more accurate than the formula of Luh and Guo (2011) for the range of model specifications considered here. Accordingly, computer programs are provided to implement the technique recommended by Levy for power calculation and sample size determination within the context of the one‐way heteroscedastic ANOVA model.