The Wilcoxon–Mann–Whitney Procedure Fails as a Test of Medians

To illustrate and document the tenuous connection between the Wilcoxon–Mann–Whitney (WMW) procedure and medians, its relationship to mean ranks is first contrasted with the relationship of a t-test to means. The quantity actually tested: Pr ^ (X1<X2) + Pr ^ (X1=X2) /2 is then described and recommended as the basis for an alternative summary statistic that can be employed instead of medians. In order to graphically represent an estimate of the quantity: Pr (X1 < X2) + Pr (X1 = X2) /2, use of a bubble plot, an ROC curve and a dominance diagram are illustrated. Several counter-examples (real and constructed) are presented, all demonstrating that the WMW procedure fails to be a test of medians. The discussion also addresses another, less common and perhaps less clear cut, but potentially even more important misconception: that the WMW procedure requires continuous data in order to be valid. Discussion of other issues surrounding the question of the WMW procedure and medians is presented, along with the authors' teaching experience with the topic. SAS code used for the examples is included as supplementary material.