A Note on Inferences About the Median of the Distribution of Difference Scores

Abstract

For two random variables, X and Y, let D = X - Y, and let θ _x , θ _y , and θ _d be the corresponding medians. It is known that the Wilcoxon-Mann-Whitney test and its modern extensions do not test H ₀: θ _x = θ _y , but rather, they test H ₀: θ _d = 0. The article deals with an extension of these methods to dependent groups, so the goal is not to make inferences about the median of the pairwise differences or even the median of the marginal distributions. Rather, if an individual is randomly sampled from the first group and another individual is randomly sampled from the second group, the goal is to make inferences about the median value of the typical difference. The goal is not to suggest that alternative methods be abandoned but rather to provide a perspective on how dependent groups compare that it is similar in spirit to the perspective provided by the Wilcoxon-Mann-Whitney test.