A Note on Homogeneity of Variance of Scores and Ranks

Abstract

When any two or more sets of scores with unequal variances are combined and ranked together as one set, the corresponding sets of ranks inherit the unequal variances. This fact is well known in the theory of nonparametric statistics, but in practice researchers and applied statisticians frequently overlook its implications. Because of this property, familiar nonparametric rank tests cannot overcome effects of heterogeneous variances of treatment groups in statistical significance testing. A simulation study demonstrates explicitly that transformation of scores to ranks reduces variance heterogeneity, although not enough to prevent gross distortion of the probabilities of Type I and Type II errors of statistical significance tests, including the t test, the Wilcoxon–Mann–Whitney test, and the van der Waerden, or normal scores, test. The present note also focuses attention on an aspect of the problem that is neglected in the literature: The equivalence of various nonparametric tests and their parametric counterparts performed on ranks, or the rank transformation concept, provides a rationale for the influence of unequal variances on test statistics calculated from ranks.