Kruskal–Wallis test

In statistics, the Kruskal–Wallis one-way analysis of variance by ranks (named after William Kruskal and W. Allen Wallis) is a non-parametric method for testing equality of population medians among groups. It is identical to a one-way analysis of variance with the data replaced by their ranks. It is an extension of the Mann–Whitney U test to 3 or more groups.

Since it is a non-parametric method, the Kruskal–Wallis test does not assume a normal population, unlike the analogous one-way analysis of variance. However, the test does assume an identically-shaped and scaled distribution for each group, except for any difference in medians.

Method

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:

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

   • $n_{i}$ is the number of observations in group $i$
   • $r_{ij}$ is the rank (among all observations) of observation $j$ from group $i$
   • $N$ is the total number of observations across all groups
   • ${\bar {r}}_{i\cdot }={\frac {\sum _{j=1}^{n_{i}}{r_{ij}}}{n_{i}}}$,
   • ${\bar {r}}={\tfrac {1}{2}}(N+1)$ is the average of all the $r_{ij}$.
3. Notice that the denominator of the expression for $K$ is exactly $(N-1)N(N+1)/12$ and ${\bar {r}}={\tfrac {N+1}{2}}$. Thus

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

4. A correction for ties can be made by dividing $K$ by $1-{\frac {\sum _{i=1}^{G}(t_{i}^{3}-t_{i})}{N^{3}-N}}$, 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 K unless there are a large number of ties.
5. Finally, the p-value is approximated by $\Pr(\chi _{g-1}^{2}\geq K)$. If some $n_{i}$ values are small (i.e., less than 5) the probability distribution of K can be quite different from this chi-square distribution. If a table of the chi-square probability distribution is available, the critical value of chi-square, $\chi _{\alpha :g-1}^{2}$, can be found by entering the table at g − 1 degrees of freedom and looking under the desired significance or alpha level. The null hypothesis of equal population medians would then be rejected if $K\geq \chi _{\alpha :g-1}^{2}$. Appropriate multiple comparisons would then be performed on the group medians.

See also

• Mann–Whitney U
• Friedman test

References

• Gwet, Kilem Li (2011). "The Practical Guide to Statistis: Applications with Excel, R, and Calc" (section 10.3, page 364). Advanced Analytics, LLC ISBN: 978-0970806291.
• Corder and Foreman. (2009). "Nonparametric Statistics for Non-Statisticians: A Step-by-Step Approach". New York: Wiley.
• Kruskal and Wallis. Use of ranks in one-criterion variance analysis. Journal of the American Statistical Association 47 (260): 583–621, December 1952.
• Siegel and Castellan. (1988). "Nonparametric Statistics for the Behavioral Sciences" (second edition). New York: McGraw–Hill.

External links

• An online version of the test