A Note on Homogeneity of Variance of Scores and Ranks

Abstract

When any two or more sets of scores with unequal variances are com bined 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 implica tions. Because of this property, familiar nonparametric rank tests cannot overcome effects of heterogeneous variances of treatment groups in statistical significance test ing. A simulation study demonstrates explicitly that transformation of scores to ranks reduces variance heterogeneity, although not enough to prevent gross distor tion 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 prob lem that is neglected in the literature: The equivalence of various nonparametric tests and their parametric counterparts performed on ranks, or the rank transfor mation concept, provides a rationale for the influence of unequal variances on test statistics calculated from ranks. MY PURPOSE in the present note is to call attention to a simple property of ranks that researchers and applied statisticians sometimes overlook when using statistical methods based on ranks. This property, noted by Pratt (1964) and Zaremba (1965), and later by many other authors, has implications for the prob lem of homogeneity of variance in statistical significance testing. It reveals that substitution of nonparametric methods for parametric tests such as t and F to overcome violation of the assumption, a procedure widely recommended in introductory textbooks, does not accomplish what is intended. Although some textbook authors have become aware of this problem in recent years, they have missed its connection with the rank transformation concept, or the equivalence of various nonparametric tests with parametric counterparts per formed on ranks replacing scores. In the present note, I emphasize that this con