Rank Transformations and the Power of the Student t Test and Welch t' Test for Non-Normal Populations With Unequal Variances

• Published in: Canadian Journal of Experimental Psychology Vol. 47; no. 3; pp. 523 - 539
• Main Authors: ZIMMERMAN, DONALD W, ZUMBO, BRUNO D
• Format: Journal Article
• Language: English
• Published: Old Chelsea, PQ Canadian Psychological Association 01.09.1993; Educational Publishing Foundation
• Subjects: Design of experiments; Exact sciences and technology; Homogeneity of Variance; Human; Mathematics; Nonparametric inference; Probability and statistics; Psychology; Sciences and techniques of general use; Statistical Power; Statistics; T Test; Tests; Variance analysis; Non parametric method; Student test; Computer simulation; Rank test; Rank transformation
• ISSN: 1196-1961; 0008-4255; 1878-7290
• DOI: 10.1037/h0078850

Classical studies have disclosed that parametric significance tests such as t and F are robust under violation of homogeneity of variance, provided sample sizes are equal. But relatively little is known about effects of unequal variances on nonparametric counterparts of the tests or about non-normality combined with unequal variances. In the present computer simulation study, the Student t test and the Welch version of the t test (the t' test) were performed first on the initial sample values and then on ranks of the sample values. Unequal variances together with unequal N's markedly altered the probability of Type I and Type II errors for normal and for eight kinds of non-normal distributions, including mixed-normal, exponential, lognormal, and Cauchy distributions. Substitution of the Welch t' test for the Student t test eliminated effects of unequal variances, but not effects of non-normality. The t test on ranks, which is equivalent to the Mann-Whitney-Wilcoxon test, was more powerful than the Student t test for several non-normal distributions, but exhibited a substantial power loss when variances were unequal. The Welch t' test in conjunction with the rank transformation simultaneously counteracted effects of both non-normality and unequal variances.