Kruskal–Wallis one-way analysis of variance

1. 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.
2. The test statistic is given by

   ${\displaystyle H=(N-1){\frac {\sum _{i=1}^{g}n_{i}({\bar {r}}_{i\cdot }-{\bar {r}})^{2}}{\sum _{i=1}^{g}\sum _{j=1}^{n_{i}}(r_{ij}-{\bar {r}})^{2}}},}$ where

   • ${\displaystyle N}$ is the total number of observations across all groups
   • ${\displaystyle g}$ is the number of groups
   • ${\displaystyle n_{i}}$ is the number of observations in group ${\displaystyle i}$
   • ${\displaystyle r_{ij}}$ is the rank (among all observations) of observation ${\displaystyle j}$ from group ${\displaystyle i}$
   • ${\displaystyle {\bar {r}}_{i\cdot }={\frac {\sum _{j=1}^{n_{i}}{r_{ij}}}{n_{i}}}}$ is the average rank of all observations in group ${\displaystyle i}$
   • ${\displaystyle {\bar {r}}={\tfrac {1}{2}}(N+1)}$ is the average of all the ${\displaystyle r_{ij}}$.
3. If the data contain no ties the denominator of the expression for ${\displaystyle H}$ is exactly ${\displaystyle (N-1)N(N+1)/12}$ and ${\displaystyle {\bar {r}}={\tfrac {N+1}{2}}}$. Thus

   ${\displaystyle {\begin{aligned}H&={\frac {12}{N(N+1)}}\sum _{i=1}^{g}n_{i}\left({\bar {r}}_{i\cdot }-{\frac {N+1}{2}}\right)^{2}\\&={\frac {12}{N(N+1)}}\sum _{i=1}^{g}n_{i}{\bar {r}}_{i\cdot }^{2}-\ 3(N+1)\end{aligned}}}$
   The last formula only contains the squares of the average ranks.

4. A correction for ties if using the short-cut formula described in the previous point can be made by dividing ${\displaystyle H}$ by ${\displaystyle 1-{\frac {\sum _{i=1}^{G}(t_{i}^{3}-t_{i})}{N^{3}-N}}}$, where G is the number of groupings of different tied ranks, and t_i is the number of tied values within group i that are tied at a particular value. This correction usually makes little difference in the value of H unless there are a large number of ties.
5. Finally, the decision to reject or not the null hypothesis is made by comparing ${\displaystyle H}$ to a critical value ${\displaystyle H_{c}}$ obtained from a table or a software for a given significance or alpha level. If ${\displaystyle H}$ is bigger than ${\displaystyle H_{c}}$, the null hypothesis is rejected. If possible (no ties, sample not too big) one should compare ${\displaystyle H}$ to the critical value obtained from the exact distribution of ${\displaystyle H}$. Otherwise, the distribution of H can be approximated by a chi-squared distribution with g-1 degrees of freedom. If some ${\displaystyle n_{i}}$ values are small (i.e., less than 5) the exact probability distribution of ${\displaystyle H}$ can be quite different from this chi-squared distribution. If a table of the chi-squared probability distribution is available, the critical value of chi-squared, ${\displaystyle \chi _{\alpha :g-1}^{2}}$, can be found by entering the table at g − 1 degrees of freedom and looking under the desired significance or alpha level.
6. If the statistic is not significant, then there is no evidence of stochastic dominance between the samples. However, if the test is significant then at least one sample stochastically dominates another sample. Therefore, a researcher might use sample contrasts between individual sample pairs, or post hoc tests using Dunn's test, which (1) properly employs the same rankings as the Kruskal–Wallis test, and (2) properly employs the pooled variance implied by the null hypothesis of the Kruskal–Wallis test in order to determine which of the sample pairs are significantly different.[5] When performing multiple sample contrasts or tests, the Type I error rate tends to become inflated, raising concerns about multiple comparisons.

A large amount of computing resources is required to compute exact probabilities for the Kruskal–Wallis test. Existing software only provides exact probabilities for sample sizes of less than about 30 participants. These software programs rely on the asymptotic approximation for larger sample sizes.

Exact probability values for larger sample sizes are available. Spurrier (2003) published exact probability tables for samples as large as 45 participants.[10] Meyer and Seaman (2006) produced exact probability distributions for samples as large as 105 participants.[11]

Choi et al.[12] made a review of two methods that had been developed to compute the exact distribution of ${\displaystyle H}$, proposed a new one, and compared the exact distribution to its chi-squared approximation.

• Friedman test
• Jonckheere's trend test

• Daniel, Wayne W. (1990). "Kruskal–Wallis one-way analysis of variance by ranks". Applied Nonparametric Statistics (2nd ed.). Boston: PWS-Kent. pp. 226–234. ISBN 0-534-91976-6.

• An online version of the test