Kruskal-Wallis one-way analysis of variance

In statistics, the Kruskal-Wallis one-way analysis of variance by ranks (named after William Kruskal and Allen Wallis) is a non-parametric method. Intuitively, it is identical to a one-way analysis of variance, with the data replaced by their ranks.

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.

Method

 * 1) Rank all data from all groups together.
 * 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:
 * 3) *$$n_g$$ is the number of observations in group $$g$$
 * 4) *$$r_{ij}$$ is observation $$j$$ from group $$i$$
 * 5) *$$N$$ is the total number of observations across all groups
 * 6) *$$\bar{r}_{i\cdot} = \frac{\sum_{j=1}^{n_i}{r_{ij}}}{n_i}$$,
 * 7) *$$\bar{r}$$ is the average of all the $$r_{ij}$$, equal to $$(N+1)/2$$.
 * Notice that the denominator of the expression for $$K$$ is exactly $$(N-1)N(N+1)/12$$.
 * 1) Finally, the p-value is approximated by $$\mathbf{P}(\chi^2_{N-g} \ge K)$$. If some ni's are small the distribution of K can be quite different from this.