Rank transformations and the power of the Student t test and Welch t' test for non-normal populations with unequal variances.

Abstract Classical studies have disclosed that parametric significance tests such as t and F are robust under violation of of 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. Resume Des etudes classiques ont revele que des tests d'hypothese parametriques comme les tests t et F sont rigoureux dans les cas ou l'homogeneite de la variance est perturbee, pourvu que les echantillons aient la me@me taille. Mais on en sait relativement peu au sujet des effets des variances inegales sur les versions non parametriques des tests ou au sujet de la non - normalite combinee a des variances inegales. Dans la presente etude de simulation par ordinateur, le test de Student et la version Welch du test t ont ete appliques d'abord aux valeurs initiales de l'echantillon, puis aux rangs des valeurs. Les variances inegales jointes aux N inegaux modifiaient nettement la probabilite des erreurs de type I et de type II dans le cas des distributions normales et de huit genres de distributions non normales, dont les distributions mixtes - normales, exponentielles, normales logarithmiques, et des distributions de Cauchy. Le remplacement du test de Student par le test t de Welch a elimine les effets des variances inegales, mais non ceux de la non - normalite. Le test t effectue sur les rangs, qui equivaut au test de Mann - Whitney - Wilcoxon, etait plus rigoureux que le test de Student pour plusieurs distributions non normales, mais il perdait considerablement de pouvoir lorsque les variances etaient inegales. Le test t de Welch joint a la transformation en rangs neutralise simultanement les effets de la non - normalite et des variances inegales.It is well known that parametric significance tests such as t and F are based on an assumption of equality of variances in treatment groups, or homogeneity of variance, as it is known. For a long time, researchers have been concerned about how violation of this assumption affects statistical tests (see, for example, Box, 1953, Glass, Peckham, & Sanders, 1972, Scheffe, 1959). As a result of numerous simulation studies, as well as theoretical investigations, there is now general agreement that the t and F tests are robust under violation of of provided sample sizes are equal, although some exceptions have been found recently (Tomarkin & Serlin, 1986).When sample sizes are unequal, the Type I error probabilities of the tests are decidedly influenced by unequal variances (see, for example, Boneau, 1960, Box, 1953, Games & Howell, 1976, Hsu, 1938, Kohr & Games, 1977, Ramsey, 1980, Rogan & Keselman, 1977, Scheffe, 1959). It has been found that, when the larger variance is associated with the larger sample size, there is a depression of the Type I error probability, and when the larger variance is associated with the smaller sample size, there is a spurious elevation of that probability. …