Kruskal–Wallis test

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== Kruskal–Wallis test ==

The Kruskal–Wallis test is a non-parametric method for testing whether samples originate from the same distribution. It is used for comparing two or more independent samples of equal or different sample sizes. The test extends the Mann–Whitney U test, which is used for comparing only two groups. It is named after William Kruskal and W. Allen Wallis.

Overview

The Kruskal–Wallis test is used when the assumptions of ANOVA (analysis of variance) are not met, particularly the assumption of normality. It is an alternative to the one-way ANOVA and is used when the data is ordinal or when the assumptions of ANOVA are not satisfied.

Assumptions

The Kruskal–Wallis test makes the following assumptions:

• The samples are independent.
• The dependent variable is measured at least at the ordinal level.
• The distributions of the groups are similar in shape.

Procedure

1. Rank all the data from all groups together. 2. Calculate the sum of ranks for each group. 3. Compute the test statistic H using the formula:

 \[
 H = \left( \frac{12}{N(N+1)} \sum \frac{R_i^2}{n_i} \right) - 3(N+1)
 \]
 where \( N \) is the total number of observations, \( R_i \) is the sum of ranks for the i-th group, and \( n_i \) is the number of observations in the i-th group.

4. Compare the test statistic H to the chi-square distribution with \( k-1 \) degrees of freedom, where \( k \) is the number of groups.

Interpretation

If the calculated H value is greater than the critical value from the chi-square distribution table, the null hypothesis that all groups have the same distribution is rejected. This indicates that at least one group distribution is different from the others.

Applications

The Kruskal–Wallis test is widely used in various fields such as psychology, medicine, education, and agriculture to compare the effectiveness of different treatments or conditions.

Related Tests

• Mann–Whitney U test
• Friedman test
• Wilcoxon signed-rank test
• ANOVA

See also

• Non-parametric statistics
• Chi-square test
• Statistical hypothesis testing