Kruskal-Wallis one-way analysis of variance

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. Intuitively, 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

# 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: , 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
#*,
#* is the average of all the $r\_\{ij\}$.
#*:Notice that the denominator of the expression for $K$ is exactly $(N-1)N(N+1)/12$. Thus .
# 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.
# Finally, the p-value is approximated by $Pr(chi^2\_\{g-1\}\; ge\; K)$. If some n_i's 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, $chi^2\_\{alpha:\; g-1\}$, 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 $K\; ge\; chi^2\_\{alpha:\; g-1\}$. Appropriate multiple comparisons would then be performed on the group medians.

ee also

*Mann-Whitney U

References

* William H. Kruskal and W. Allen Wallis. Use of ranks in one-criterion variance analysis. "Journal of the American Statistical Association" 47 (260): 583–621, December 1952. [http://homepages.ucalgary.ca/~jefox/Kruskal%20and%20Wallis%201952.pdf]
* Sidney Siegel and N. John Castellan, Jr. (1988). "Nonparametric Statistics for the Behavioral Sciences" (second edition). New York: McGraw-Hill.