KruskalWallisin

KruskalWallisin, often written as the Kruskal-Wallis test, refers to a non-parametric method for comparing more than two independent groups. It generalizes the Mann-Whitney U test to k groups and is used when the assumptions of one-way ANOVA are not met or when data are ordinal.

The procedure involves ranking all observations together across all groups. For each group i, with sample size n_i, calculate the sum of ranks R_i. Let N be the total sample size. The test statistic is H = (12 / (N(N+1))) * sum over groups of (R_i^2 / n_i) - 3(N+1). If ties are present, a correction factor is applied. This statistic is compared to a chi-square distribution with k-1 degrees of freedom under the null hypothesis that the populations have the same distribution.

A significant H indicates that at least one group tends to differ from the others, but it does not specify which groups are different. Post hoc procedures, such as Dunn’s test with a Bonferroni or Holm correction, are used to identify specific group differences.

Assumptions include independent observations and at least ordinal data. The test does not assume normality but does assume similarly shaped distributions under the null, aside from possible differences in central tendency. The Kruskal-Wallis test is named after William Kruskal and W. Allen Wallis, who introduced it in 1952. It is a common non-parametric alternative to one-way ANOVA when parametric assumptions are violated.