Combinatorics and Statistical Issues Related to the Kruskal-Wallis Statistic

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 1 data sets 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 data sets 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.