Kruskal-Wallis Test

1. Overview

Kruskal-Wallis Test is a nonparametric test used to compare the medians of $k\ge 2$ independent population groups. It is a nonparametric alternative to the one-way ANOVA test.

In the Kruskal-Wallis Test:

• There is no normal distribution assumption for the data.
• It still requires the homoscedasticity assumption, i.e. equal variance.

2. Test Procedure

Assume there are $k$ independent population groups with sample sizes $n_1,n_2,n_3$. Given the null hypothesis $H_0:{\text{median}}_1={\text{median}}_2={\text{median}}_3$, and the alternative hypothesis $H_1:\text{medians are not all equal}$.

Compute the test statistic $H$ as follows:

$$\begin{array}{cccc}\text{Pop.Grp. }& \text{Rank}& \text{Sample Size}& \text{Rank Sum}\\ 1& r_{1,1},r_{1,2},\dots ,r_{1,n_1}& n_1& R_1={\sum }_{i=1}^{n_1}{r}_{1,i}\\ 2& r_{2,1},r_{2,2},\dots ,r_{2,n_2}& n_2& R_2={\sum }_{i=1}^{n_2}{r}_{2,i}\\ ⋮& ⋮& ⋮& ⋮\\ k& r_{k,1},r_{k,2},\dots ,r_{k,n_k}& n_k& R_k={\sum }_{i=1}^{n_k}{r}_{k,i}\end{array}$$

Define the test statistic $H$ as

$$H=\frac{12}{n(n+1)}\sum _{j=1}^k\frac{R_j^2}{n_j}-3(n+1)\stackrel{H_0}{⟶}{\chi }^2(k-1).$$

As a chi-square test, the rejection region is always right-tailed:

$$\mathcal{RR}=\{H\ge {\chi }_{\alpha }^2(k-1)\}.$$

Related Notes

• Wilcoxon Rank Sum Test
• Mann-Whitney U Test
• Nonparametric Tests Overview