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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.
Anna E. Bargagliotti & Raymond N. Greenwell (2015) Combinatorics and Statistical Issues Related to the Kruskal–Wallis Statistic, Communications in Statistics - Simulation and Computation, 44:2, 533-550, DOI: 10.1080/03610918.2013.786781