13-5 Kruskal-Wallis Test for Three or More Samples 671 Key Concept This section describes the Kruskal-Wallis test, which uses ranks of data from three or more independent simple random samples to test the null hypothesis that the samples come from populations with the same median. Section 12-1 described one-way analysis of variance (ANOVA) as a method for testing the null hypothesis that three or more populations have the same mean, but that ANOVA procedure requires that all of the involved populations have normal distributions. The Kruskal-Wallis test for equal medians does not require normal distributions, so it is a distribution-free or nonparametric test. 13-5 Kruskal-Wallis Test for Three or More Samples DEFINITION The Kruskal-Wallis test (also called the H test) is a nonparametric test that uses ranks of combined simple random samples from three or more independent populations to test the null hypothesis that the populations have the same median. (The alternative hypothesis is the claim that the populations have medians that are not all equal.) In applying the Kruskal-Wallis test, we compute the test statistic H, which has a distribution that can be approximated by the chi-square distribution provided that each sample has at least five observations. (For a quick review of the key features of the chi-square distribution, see Section 7-3.) The H test statistic measures the variance of the rank sums R1, R2, c,Rk from the different samples. If the ranks are distributed evenly among the sample groups, then H should be a relatively small number. If the samples are very different, then the ranks will be excessively low in some groups and high in others, with the net effect that H will be large. Consequently, only large values of H lead to rejection of the null hypothesis that the samples come from identical populations. The Kruskal-Wallis test is therefore a right-tailed test. Kruskal-Wallis Test Objective Use the Kruskal-Wallis test with simple random samples from three or more independent populations for the following null and alternative hypotheses: H0: The samples come from populations with the same median. H1: The samples come from populations with medians that are not all equal. Notation N = total number of observations in all samples combined k = number of different samples R1 = sum of ranks for Sample 1 n1 = number of observations in Sample 1 For Sample 2, the sum of ranks is R2 and the number of observations is n2, and similar notation is used for the other samples. Requirements 1. We have at least three independent simple random samples. 2. Each sample has at least five observations. (If samples have fewer than five observations, refer to special tables of critical values, such as CRC Standard Probability and Statistics Tables and Formulae, published by CRC Press.) Note: There is no requirement that the populations have a normal distribution or any other particular distribution. KEY ELEMENTS continued
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