KruskalWallis

KruskalWallis, commonly referred to as the Kruskal-Wallis test, is a nonparametric method for testing whether samples originate from the same distribution. It extends the Mann-Whitney U test to more than two groups and serves as an alternative to one-way analysis of variance (ANOVA) when the assumption of normally distributed populations is questionable or when data are ordinal.

Procedure: All observations are pooled and ranked from smallest to largest. For each group, sum of ranks R_i is computed, and with sample sizes n_i, the test statistic H is defined as H = (12 / (N(N+1))) * sum_i (R_i^2 / n_i) - 3(N+1), where N is total observations. H approximately follows a chi-square distribution with k-1 degrees of freedom under the null hypothesis that the distributions are the same. When ties are present, a correction factor is applied to adjust H.

Assumptions: observations are independent within and across groups, data are at least ordinal, and the groups have similar shaped distributions; the test detects differences in central tendency but can be influenced by distribution shapes.

Interpretation: A significant result indicates at least one group differs from the others, but it does not specify which groups. Post hoc procedures, such as pairwise Mann-Whitney tests with multiple testing correction or Dunn's test, can identify specific differences.

History and usage: Developed by Kruskal and Wallis in 1952, it is widely used in fields where nonparametric comparisons across multiple groups are needed.