Combinatorics and Statistical Issues Related to the Kruskal–Wallis Statistic

Abstract

We explore criteria that data must meet in order for the Kruskal–Wallis test to reject the null hypothesis by computing the number of unique ranked datasets in the balanced case where each of the m alternatives has n observations. We show that the Kruskal–Wallis test tends to be conservative in rejecting the null hypothesis, and we offer a correction that improves its performance. We then compute the number of possible datasets producing unique rank-sums. The most commonly occurring data lead to an uncommonly small set of possible rank-sums. We extend prior findings about row- and column-ordered data structures.

Keywords:

• Continuity correction
• Kruskal–Wallis
• Rank-sums
• Row-ordered data
• Tournaments
• Young tableaux

Mathematics Subject Classification:

• 62
• 5