A Comparison of the Exact Kruskal-Wallis Distribution to Asymptotic Approximations for All Sample Sizes up to 105

Abstract

The authors generated exact probability distributions for sample sizes up to 35 in each of three groups (n ≤ 105) and up to 10 in each of four groups (n ≤ 40). They compared the exact distributions to the chi-square, gamma, and beta approximations. The beta approximation was best in terms of the root mean square error. At specific significance levels, either the gamma or beta approximation was best. These results suggest that the most common approximation, the chi-square approximation, is not a good choice. The authors demonstrate this point using an applied example. Critical value tables for the exact distribution are available online at http://faculty.virginia.edu/kruskal-wallis. The portion of these tables that provides critical values for equal sample sizes appears in this article. The authors recommend that researchers use critical values from the exact distribution whenever possible. If sample sizes exceed those included in the authors’ exact probability tables, they recommend using the beta approximation instead of the chi-square and gamma approximations.