Kruskal Wallis Test

TL;DR

• A non-parametric alternative to ANOVA that compares median ranks across multiple groups.
• Works without assuming a specific data distribution and can handle ordinal, interval, or some nominal data.
• Procedure: rank observations, sum ranks by group, compute the H statistic, and compare it to a critical value; results assume independent observations.

Definition

The Kruskal-Wallis test is a non-parametric statistical test used to determine if there are significant differences among the median ranks of several groups.

Explanation

• The test does not assume a normal distribution of the data and can be applied to ordinal, interval, or some nominal data.
• To perform the test, observations across all groups are ranked from lowest to highest. The sum of the ranks for each group is calculated.
• The Kruskal-Wallis statistic, H, is computed from those rank sums and group sizes using the formula below.
• If the calculated H exceeds the critical value from a table of critical values, the null hypothesis is rejected, indicating significant differences among the median ranks of the groups.
• A key assumption is independence of observations; lack of independence can bias the test results.
• The test compares median ranks only; if comparison of means or other measures is required, a different test such as ANOVA may be more appropriate.

Examples

Comparing three exercise groups (weight loss)

A researcher randomly assigns participants to one of three exercise groups and measures their weight at the beginning and end of the study. The researcher can use the Kruskal-Wallis test to determine if there are significant differences in the amount of weight loss among the three exercise groups.

Comparing four teaching methods (test scores)

A study randomly assigns students to one of four teaching methods and measures their test scores at the end of the semester. The Kruskal-Wallis test can be used to determine if there are significant differences in the median test scores among the four teaching methods.

Notes or pitfalls

• Independence among observations is required; related or clustered participants (for example, household members) may violate this assumption and bias results.
• The test assesses differences in median ranks only; it does not compare group means or other summary measures.

Related terms

• ANOVA

Formula

$$H = \frac{n_1 n_2 \dots n_r}{N(N+1)} \sum \frac{R_i^2}{n_i}$$

where n1, n2, …, nr are the number of observations in each group, N is the total number of observations, and Ri is the sum of the ranks for group i.