5.6 Kruskal–Wallis Test

The Kruskal-Wallis H test8 measures the difference of a continuous or ordinal dependent variable between groups of a categorical independent variable. It is a rank-based nonparametric alternative to the one-way ANOVA test. Use Kruskal-Wallis if the dependent variable fails ANOVA’s normality or homogeneity conditions, or if it is ordinal.

How it Works

The Kruskal-Wallis H test ranks the dependent variable irrespective of its group. The test statistic is a function of the averaged square of the rank sum per group:

\[ H = \left[ \frac{12}{n(n+1)} \sum_{j} \frac{T_j^2}{n_j} \right] - 3(n + 1) \]

where \(T_j\) is the sum of the ranks of group j. The test statistic approximately follows a \(\chi^2\) distribution with k – 1 degrees of freedom, where k is the number of groups of the independent variable. The null hypothesis is that the rank means are equal. If you reject the null hypothesis, run a post hoc test to determine which groups differ.

Assumptions

Kruskal-Wallis has no assumptions per se, but the test interpretation depends on the distribution of the dependent variable. If its distribution has a similar shape across the groups of the categorical independent variable, then Kruskal-Wallis is a test of differences in their medians. Otherwise, Kruskal-Wallis is a test of differences in their distributions.

  1. The Kruskal-Wallis H test is also called the one-way ANOVA on ranks↩︎