Kruskal-Wallis Test

This is a non-parametric version of the F-Test (ANOVA). The test statistic is:

$H=(\sum _{j=1}^g\sum _{i=1}^{n_j}{w}_{ij}-1)\frac{\sum _{j=1}^g\sum _{i=1}^{n_j}{w}_{ij}({\overline{r}}_{\cdot j}-\overline{r}{)}^2}{\sum _{j=1}^g\sum _{i=1}^{n_j}{w}_{ij}(r_{ij}-\overline{r}{)}^2}$

where:

$n_j$ is the number of observations in group $j$ of $g$groups,

$r_{ij}$ is the rank of the $i$th observation from group $j$ where the ranking is computed across all the groups with a 1 assigned to the lowest value and the average rank is used for ties,

$n=\sum _{j=1}^g{n}_j$,

$w_{ij}$ is the Calibrated Weight,

${\overline{r}}_{\cdot j}=\frac{\sum _{i=1}^{n_j}{w}_{ij}{r}_{ij}}{\sum _{i=1}^{n_j}{w}_{ij}}$,

$\overline{r}=\frac{\sum _{j=1}^g\sum _{i=1}^{n_j}{w}_{ij}{r}_{ij}}{\sum _{j=1}^g\sum _{i=1}^{n_j}{w}_{ij}}$,

$p\approx \mathrm{Pr}({\chi }_{g-1}^2\ge H)$

See also

• ANOVA-Type Tests - Comparing Three or More Groups
• Planned ANOVA-Type Tests