Statistical significance levels of nonparametric tests biased by heterogeneous variances of treatment groups

The statistical significance levels of the Wilcoxon-Mann-Whitney test and the Kruskal-Wallis test are substantially biased by heterogeneous variances of treatment groups--even when sample sizes are equal. Under these conditions, the Type I error probabilities of the nonparametric tests, performed at the .01, .05, and .10 significance levels, increase by as much as 40%-50% in many cases and sometimes as much as 300%. The bias increases systematically as the ratio of standard deviations of treatment groups increases and remains fairly constant for various sample sizes. There is no indication that Type I error probabilities approach the significance level asymptotically as sample size increases.     

A simple and effective decision rule for choosing a significance test to protect against non-normality

There is no formal and generally accepted procedure for choosing an appropriate significance test for sample data when the assumption of normality is doubtful. Various tests of normality that have been proposed over the years have been found to have limited usefulness, and sometimes a preliminary test makes the situation worse. The present paper investigates a specific and easily applied rule for choosing between a parametric and non-parametric test, the Student t test and the Wilcoxon-Mann-Whitney test, that does not require a preliminary significance test of normality. Simulations reveal that the rule, which can be applied to sample data automatically by computer software, protects the Type I error rate and increases power for various sample sizes, significance levels, and non-normal distribution shapes. Limitations of the procedure in the case of heterogeneity of variance are discussed.     

Comparing one-step M-estimators of location when there are more than two groups

Methods for comparing means are known to be highly nonrobust in terms of Type II errors. The problem is that slight shifts from normal distributions toward heavy-tailed distributions inflate the standard error of the sample mean. In contrast, the standard error of various robust measures of location, such as the one-step M-estimator, are relatively unaffected by heavy tails. Wilcox recently examined a method of comparing the one-step M-estimators of location corresponding to two independent groups which provided good control over the probability of a Type I error even for unequal sample sizes, unequal variances, and different shaped distributions. There is a fairly obvious extension of this procedure to pairwise comparisons of more than two independent groups, but simulations reported here indicate that it is unsatisfactory. A slight modification of the procedure is found to give much better results, although some caution must be taken when there are unequal sample sizes and light-tailed distributions. An omnibus test is examined as well.     

A test of the hypothesis that Cronbach's alpha reliability coefficient is the same for two tests administered to the same sample

In measurement studies the researcher may wish to test the hypothesis that Cronbach's alpha reliability coefficient is the same for two measurement procedures. A statistical test exists for independent samples of subjects. In this paper three procedures are developed for the situation in which the coefficients are determined from the same sample. All three procedures are computationally simple and give tight control of Type I error when the sample size is 50 or greater.The author is indebted to Jerry S. Gilmer for development of the computer programs used in this study.     

Accounting for auto-dependency in binary dyadic time series data: A comparison of model- and permutation-based approaches for testing pairwise associations

Many theories have been put forward on how people become synchronized or co-regulate each other in daily interactions. These theories are often tested by observing a dyad and coding the presence of multiple target behaviours in small time intervals. The sequencing and co-occurrence of the partners’ behaviours across time are then quantified by means of association measures (e.g., kappa coefficient, Jaccard similarity index, proportion of agreement). We demonstrate that the association values obtained are not easy to interpret, because they depend on the marginal frequencies and the amount of auto-dependency in the data. Moreover, often no inferential framework is available to test the significance of the association. Even if a significance test exists (e.g., kappa coefficient) auto-dependencies are not taken into account, which, as we will show, can seriously inflate the Type I error rate. We compare the effectiveness of a model- and a permutation-based framework for significance testing. Results of two simulation studies show that within both frameworks test variants exist that successfully account for auto-dependency, as the Type I error rate is under control, while power is good.     

Tests for equality of several alpha coefficients when their sample estimates are dependent

In a variety of measurement situations, the researcher may wish to compare the reliabilities of several instruments administered to the same sample of subjects. This paper presents eleven statistical procedures which test the equality ofm coefficient alphas when the sample alpha coefficients are dependent. Several of the procedures are derived in detail, and numerical examples are given for two. Since all of the procedures depend on approximate asymptotic results, Monte Carlo methods are used to assess the accuracy of the procedures for sample sizes of 50, 100, and 200. Both control of Type I error and power are evaluated by computer simulation. Two of the procedures are unable to control Type I errors satisfactorily. The remaining nine procedures perform properly, but three are somewhat superior in power and Type I error control.A more detailed version of this paper is also available.     

The new and improved two-sample T test

Abstract This article considers the problem of comparing two independent groups in terms of some measure of location. It is well known that with Student's two-independent-sample t test, the actual level of significance can be well above or below the nominal level, confidence intervals can have inaccurate probability coverage, and power can be low relative to other methods. A solution to deal with heterogeneity is Welch's (1938) test. Welch's test deals with heteroscedasticity but can have poor power under arbitrarily small departures from normality. Yuen (1974) generalized Welch's test to trimmed means; her method provides improved control over the probability of a Type I error, but problems remain. Transformations for skewness improve matters, but the probability of a Type I error remains unsatisfactory in some situations. We find that a transformation for skewness combined with a bootstrap method improves Type I error control and probability coverage even if sample sizes are small.     

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.     

Randomization test for coupled data

Coupled data arise in perceptual research when subjects are contributing two scores to the data pool. These two scores, it can be reasonably argued, cannot be assumed to be independent of one another; therefore, special treatment is needed when performing statistical inference. This paper shows how the Type I error rate of randomization-based inference is affected by coupled data. It is demonstrated through Monte Carlo simulation that a randomization test behaves much like its parametric counterpart except that, for the randomization test, a negative correlation results in an inflation in the Type I error rate. A new randomization test, the couplet-referenced randomization test, is developed and shown to work for sample sizes of 8 or more observations. An example is presented to demonstrate the computation and interpretation of the new randomization test.     

Evaluating the robustness of repeated measures analyses: The case of small sample sizes and nonnormal data

Repeated measures analyses of variance are the method of choice in many studies from experimental psychology and the neurosciences. Data from these fields are often characterized by small sample sizes, high numbers of factor levels of the within-subjects factor(s), and nonnormally distributed response variables such as response times. For a design with a single within-subjects factor, we investigated Type I error control in univariate tests with corrected degrees of freedom, the multivariate approach, and a mixed-model (multilevel) approach (SAS PROC MIXED) with Kenward–Roger’s adjusted degrees of freedom. We simulated multivariate normal and nonnormal distributions with varied population variance–covariance structures (spherical and nonspherical), sample sizes (N), and numbers of factor levels (K). For normally distributed data, as expected, the univariate approach with Huynh–Feldt correction controlled the Type I error rate with only very few exceptions, even if samples sizes as low as three were combined with high numbers of factor levels. The multivariate approach also controlled the Type I error rate, but it requires N ≥ K. PROC MIXED often showed acceptable control of the Type I error rate for normal data, but it also produced several liberal or conservative results. For nonnormal data, all of the procedures showed clear deviations from the nominal Type I error rate in many conditions, even for sample sizes greater than 50. Thus, none of these approaches can be considered robust if the response variable is nonnormally distributed. The results indicate that both the variance heterogeneity and covariance heterogeneity of the population covariance matrices affect the error rates.     

Comparison of a two-stage and three-stage interim-analysis procedure.

A statistical model for combining p values from multiple tests of significance is used to define rejection and acceptance regions for two-stage and three-stage sampling plans. Type I error rates, power, frequencies of early termination decisions, and expected sample sizes are compared. Both the two-stage and three-stage procedures provide appropriate protection against Type I errors. The two-stage sampling plan with its single interim analysis entails minimal loss in power and provides substantial reduction in expected sample size as compared with a conventional single end-of-study test of significance for which power is in the adequate range. The three-stage sampling plan with its two interim analyses introduces somewhat greater reduction in power, but it compensates with greater reduction in expected sample size. Either interim-analysis strategy is more efficient than a single end-of-study analysis in terms of power per unit of sample size.     

认知诊断测验中基于信息矩阵的多群组DIF检验

基于改进的Wald统计量,将适用于两群组的DIF检测方法拓展至多群组的项目功能差异（DIF）检验;改进的Wald统计量将分别通过计算观察信息矩阵（Obs）和经验交叉相乘信息矩阵（XPD）而得到。模拟研究探讨了此二者与传统计算方法在多个群组下的DIF检验情况,结果表明：（1）Obs和XPD的一类错误率明显低于传统方法,DINA模型估计下Obs和XPD的一类错误率接近理论水平;（2）样本量和DIF量较大时,Obs和XPD具有与传统Wald统计量大体相同的统计检验力。     

A warning about statistical significance tests performed on large samples of nonindependent observations

When sample observations are not independent, the variance estimate in the denominator of the Student t statistic is altered, inflating the value of the test statistic and resulting in far too many Type I errors. Furthermore, how much the Type I error probability exceeds the nominal significance level is an increasing function of sample size. If N is quite large, in the range of 100 to 200 or larger, small apparently inconsequential correlations that are unknown to a researcher, such as .01 or .02, can have substantial effects and lead to false reports of statistical significance when effect size is zero.     

Out with .05, in with Replication and Measurement: Isolating and Working with the Particular Effect Sizes that are Troublesome for Inferential Statistics

It is difficult to obtain adequate power to test a small effect size with a set criterion alpha of 0.05. Probably an inferential test will indicate non-statistical significance and not be published. Rarely, statistical significance will be obtained, and an exaggerated effect size calculated and reported. Accepting all inferential probabilities and associated effect sizes could solve exaggeration problems. Graphs, generated through Monte Carlo methods, are presented to illustrate this. The first graph presents effect sizes (Cohen's d) as lines from 1 to 0 with probabilities on the Y axis and the number of measures on the X axis. This graph shows effect sizes of .5 or less should yield non-significance with sample sizes below 120 measures. The other graphs show results with as many as 10 small sample size replications. There is a convergence of means with the effect size as sample size increases and measurement accuracy emerges.     

Non‐parametric three‐way mixed ANOVA with aligned rank tests

Research problems that require a non‐parametric analysis of multifactor designs with repeated measures arise in the behavioural sciences. There is, however, a lack of available procedures in commonly used statistical packages. In the present study, a generalization of the aligned rank test for the two‐way interaction is proposed for the analysis of the typical sources of variation in a three‐way analysis of variance (ANOVA) with repeated measures. It can be implemented in the usual statistical packages. Its statistical properties are tested by using simulation methods with two sample sizes (n = 30 and n = 10) and three distributions (normal, exponential and double exponential). Results indicate substantial increases in power for non‐normal distributions in comparison with the usual parametric tests. Similar levels of Type I error for both parametric and aligned rank ANOVA were obtained with non‐normal distributions and large sample sizes. Degrees‐of‐freedom adjustments for Type I error control in small samples are proposed. The procedure is applied to a case study with 30 participants per group where it detects gender differences in linguistic abilities in blind children not shown previously by other methods.     

A Monte Carlo evaluation of tests for comparing dependent correlations

The authors conducted a Monte Carlo simulation of 8 statistical tests for comparing dependent zero-order correlations. In particular, they evaluated the Type I error rates and power of a number of test statistics for sample sizes (Ns) of 20, 50, 100, and 300 under 3 different population distributions (normal, uniform, and exponential). For the Type I error rate analyses, the authors evaluated 3 different magnitudes of the predictor-criterion correlations (rho(y,x1) = rho(y,x2) = .1, .4, and .7). For the power analyses, they examined 3 different effect sizes or magnitudes of discrepancy between rho(y,x1) and rho(y,x2) (values of .1, .3, and .6). They conducted all of the simulations at 3 different levels of predictor intercorrelation (rho(x1,x2) = .1, .3, and .6). The results indicated that both Type I error rate and power depend not only on sample size and population distribution, but also on (a) the predictor intercorrelation and (b) the effect size (for power) or the magnitude of the predictor-criterion correlations (for Type I error rate). When the authors considered Type I error rate and power simultaneously, the findings suggested that O. J. Dunn and V. A. Clark's (1969) z and E. J. Williams's (1959) t have the best overall statistical properties. The findings extend and refine previous simulation research and as such, should have greater utility for applied researchers.     

Variable Error

The degree to which blocked (VE) data satisfies the assumptions of compound symmetry required for a repeated measures ANOVA was studied. Monte Carlo procedures were used to study the effect of violation of this assumption, under varying block sizes, on the Type I error rate. Populations of 10,000 subjects for each of two groups, the underlying variance-covariance matrices reflecting a specific condition of violation of the homogeneity of covariance assumptions, were generated based on each of three actual experimental data sets. The data were blocked in various ways, VE calculated, and subsequently analyzed by a repeated measures ANOVA. The complete process was replicated for four covariance homogeneity conditions for each of the three data sets, resulting in a total of 22,000 simulated experiments. Results indicated that the Type I error rate increases as the degree of heterogeneity within the variance-covariance matrices increases when raw (unblocked) data are analyzed. With VE, the effects of within-matrix heterogeneity on the Type I error rate are inconclusive. However, block size does seem to affect the probability of obtaining a significant interaction, but the nature of this relationship is not clear as there does not appear to be any consistent relationship between the size of the block and the probability of obtaining significance. For both raw and VE data there was no inflation in the number of Type I errors when the covariances within a given matrix were homogeneous, regardless of the differences between the group variance-covariance matrices.     

Hypothesis tests for population heterogeneity in meta‐analysis

Choice of the appropriate model in meta‐analysis is often treated as an empirical question which is answered by examining the amount of variability in the effect sizes. When all of the observed variability in the effect sizes can be accounted for based on sampling error alone, a set of effect sizes is said to be homogeneous and a fixed‐effects model is typically adopted. Whether a set of effect sizes is homogeneous or not is usually tested with the so‐called Q test. In this paper, a variety of alternative homogeneity tests – the likelihood ratio, Wald and score tests – are compared with the Q test in terms of their Type I error rate and power for four different effect size measures. Monte Carlo simulations show that the Q test kept the tightest control of the Type I error rate, although the results emphasize the importance of large sample sizes within the set of studies. The results also suggest under what conditions the power of the tests can be considered adequate.     

Testing the significance of a correlation with nonnormal data: Comparison of Pearson, Spearman, transformation, and resampling approaches

It is well known that when data are nonnormally distributed, a test of the significance of Pearson's r may inflate Type I error rates and reduce power. Statistics textbooks and the simulation literature provide several alternatives to Pearson's correlation. However, the relative performance of these alternatives has been unclear. Two simulation studies were conducted to compare 12 methods, including Pearson, Spearman's rank-order, transformation, and resampling approaches. With most sample sizes (n ≥ 20), Type I and Type II error rates were minimized by transforming the data to a normal shape prior to assessing the Pearson correlation. Among transformation approaches, a general purpose rank-based inverse normal transformation (i.e., transformation to rankit scores) was most beneficial. However, when samples were both small (n ≤ 10) and extremely nonnormal, the permutation test often outperformed other alternatives, including various bootstrap tests. (PsycINFO Database Record (c) 2012 APA, all rights reserved).     

Heterogeneous heterogeneity by default: Testing categorical moderators in mixed-effects meta-analysis

Categorical moderators are often included in mixed-effects meta-analysis to explain heterogeneity in effect sizes. An assumption in tests of categorical moderator effects is that of a constant between-study variance across all levels of the moderator. Although it rarely receives serious thought, there can be statistical ramifications to upholding this assumption. We propose that researchers should instead default to assuming unequal between-study variances when analysing categorical moderators. To achieve this, we suggest using a mixed-effects location-scale model (MELSM) to allow group-specific estimates for the between-study variance. In two extensive simulation studies, we show that in terms of Type I error and statistical power, little is lost by using the MELSM for moderator tests, but there can be serious costs when an equal variance mixed-effects model (MEM) is used. Most notably, in scenarios with balanced sample sizes or equal between-study variance, the Type I error and power rates are nearly identical between the MEM and the MELSM. On the other hand, with imbalanced sample sizes and unequal variances, the Type I error rate under the MEM can be grossly inflated or overly conservative, whereas the MELSM does comparatively well in controlling the Type I error across the majority of cases. A notable exception where the MELSM did not clearly outperform the MEM was in the case of few studies (e.g., 5). With respect to power, the MELSM had similar or higher power than the MEM in conditions where the latter produced non-inflated Type 1 error rates. Together, our results support the idea that assuming unequal between-study variances is preferred as a default strategy when testing categorical moderators.