A multivariate projection-type analogue of the Wilcoxon-Mann-Whitney test

Abstract

The problem of comparing two independent groups based on mulitivariate data is considered. Many such methods have been proposed, but it is difficult to gain a perspective on the extent to which the groups differ. The basic strategy here is to determine a robust measure of location for each group, project the data onto the line connecting these measures of location, and then compare the groups based on the ordering of the projected points. In the univariate case the method uses the same measure of effect size employed by the Wilcoxon-Mann-Whitney test. Under general conditions, the projected points are dependent, causing difficulties when testing hypotheses. Two methods are found to be effective when trying to avoid Type 1 error probabilities above the nominal level. The relative merits of the two methods are discussed. The projected data provide not only a useful (numerical) measure of effect size, but also a graphical indication of the extent to which groups differ.