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Kruskal-Wallis ANOVAOn this page: Principles of Kruskal-Wallis one-way, non-parametric, ANOVA Procedure Assumptions
PrinciplesThe Kruskal-Wallis one-way ANOVA is a non-parametric method for comparing k independent samples. It is roughly equivalent to a parametric one way ANOVA with the data replaced by their ranks. Since ranking is conditional upon your observed values, so is this test.
The null hypothesis is that the k groups were randomly assigned from the same group of ranks - in which case each group is equally likely to obtain values above and below that common mean rank. The alternative hypothesis is that, in addition to this random assignment, two or more groups also differ in their mean rank - in which case, like ANOVA, this test assumes the only difference between samples is their mean rank, and any other differences are due to simple chance.
Provided the original observations are identically distributed this can be interpreted as testing for a difference between medians. But when observations represent very different distributions Kruskal-Wallis is a test of dominance, much as the Wilcoxon-Mann-Whitney test is a test of dominance comparing just two samples. The test statistic is in fact identical to the Wilcoxon-Mann-Whitney statistic in the two-sample case.
Kruskal-Wallis is commonly used as a test of equality of medians or even means. In the latter case, in addition to the distributional assumptions mentioned above, observations are also assumed to be distributed symmetrically.
At the asymptote the null distribution of Kruskal-Wallis statistic approximates to the χ2 distribution with k-1 degrees of freedom. The χ2 distribution generally furnishes a conservative test. For (k =) 3 groups and sample size (ni) less than or equal to 5, the exact distribution should be used. For smallish sets of tied data a randomization approach may be preferable.
Kruskal-Wallis is given in various different forms. We give it in its simplified form for no ties. A correction factor is then applied when ties present.
If each ni is at least 5, the statistic approximates a chi square distribution with k-1 degrees of freedom. For smaller sample sizes exact critical values are available in Table A8 in Conover
The assumptions of the Kruskal-Wallis test are similar to those for the Wilcoxon-Mann-Whitney test.