KruskalWallisTests

Kruskal-Wallis tests, commonly referred to as the Kruskal-Wallis one-way analysis of ranks, are non-parametric methods for comparing more than two independent samples. They provide an alternative to one-way ANOVA when data do not meet normality or equal-variance assumptions or when the measurements are ordinal. The test assesses whether the populations from which the samples are drawn have the same distribution, focusing on differences in central tendency as reflected by ranks rather than raw values.

To perform the test, all observations are pooled and ranked, from smallest to largest. For each group i, with n_i observations and rank sum R_i, the test statistic is H = (12 / [N(N+1)]) * sum_i (R_i^2 / n_i) - 3(N+1), where N is the total number of observations. When ties occur, a correction factor is applied to adjust H. Under the null hypothesis that the groups come from identical distributions, H approximately follows a chi-square distribution with k-1 degrees of freedom, where k is the number of groups; exact p-values may be obtained via permutation methods for small samples.

Interpretation centers on whether the observed rank differences among groups are larger than would be expected by chance. A significant result suggests that at least one group tends to have higher or lower values, but it does not specify which groups differ. If significant, post hoc pairwise comparisons, such as Dunn’s test with p-value adjustments (e.g., Bonferroni or Holm), are commonly used to identify specific group differences.

Assumptions include independent samples, measurements at least on an ordinal scale, and similar distribution shapes across groups. The test is robust to non-normality and is widely used in non-parametric statistics and applied research.