The Kruskal–Wallis test by ranks, Kruskal–Wallis H {\displaystyle H} test (named after William Kruskal and W. Allen Wallis), or one-way ANOVA on ranks is a non-parametric statistical test 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. It extends the Mann–Whitney U test, which is used for comparing only two groups. The parametric equivalent of the Kruskal–Wallis test is the one-way analysis of variance (ANOVA). A significant Kruskal–Wallis test indicates that at least one sample stochastically dominates one other sample. The test does not identify where this stochastic dominance occurs or for how many pairs of groups stochastic dominance obtains. For analyzing the specific sample pairs for stochastic dominance, Dunn's test, pairwise Mann–Whitney tests with Bonferroni correction, or the more powerful but less well known Conover–Iman test are sometimes used. It is supposed that the treatments significantly affect the response level and then there is an order among the treatments: one tends to give the lowest response, another gives the next lowest response is second, and so forth. Since it is a nonparametric method, the Kruskal–Wallis test does not assume a normal distribution of the residuals, unlike the analogous one-way analysis of variance. If the researcher can make the assumptions of an identically shaped and scaled distribution for all groups, except for any difference in medians, then the null hypothesis is that the medians of all groups are equal, and the alternative hypothesis is that at least one population median of one group is different from the population median of at least one other group. Otherwise, it is impossible to say, whether the rejection of the null hypothesis comes from the shift in locations or group dispersions. This is the same issue that happens also with the Mann-Whitney test. If the data contains potential outliers, if the population distributions have heavy tails, or if the population distributions are significantly skewed, the Kruskal–Wallis test is more powerful at detecting differences among treatments than ANOVA F-test. On the other hand, if the population distributions are normal or are light-tailed and symmetric, then ANOVA F-test will generally have greater power which is the probability of rejecting the null hypothesis when it indeed should be rejected.
Method
Rank all data from all groups together; i.e., rank the data from 1 to N ignoring group membership. Assign any tied values the average of the ranks they would have received had they not been tied. The test statistic is given by
H = ( N − 1 ) ∑ i = 1 g n i ( r ¯ i ⋅ − r ¯ ) 2 ∑ i = 1 g ∑ j = 1 n i ( r i j − r ¯ ) 2 , {\displaystyle \definecolor {Orange}{rgb}{1,0.5019607843137255,0}\definecolor {ChromeYellow}{rgb}{1,0.6549019607843137,0.011764705882352941}\definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}\definecolor {Blue}{rgb}{0,0,1}\definecolor {Purple}{rgb}{0.5019607843137255,0,0.5019607843137255}H=({\color {Red}N}-1){\frac {\sum _{i=1}^{\color {Orange}g}{\color {ChromeYellow}n_{i}}({\color {Blue}{\bar {r}}_{i\cdot }}-{\color {Purple}{\bar {r}}})^{2}}{\sum _{i=1}^{\color {Orange}g}\sum _{j=1}^{\color {ChromeYellow}n_{i}}({\color {Green}r_{ij}}-{\color {Purple}{\bar {r}}})^{2}}},} where
N {\textstyle \color {Red}N} is the total number of observations across all groups
g {\textstyle \definecolor {Orange}{rgb}{1,0.5019607843137255,0}\color {Orange}g} is the number of groups
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