# Kruskal-Wallis Test

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

$\displaystyle{ H = (\sum_{j=1}^g\sum_{i=1}^{n_j}w_{ij}-1)\frac{\sum_{j=1}^g\sum_{i=1}^{n_j} w_{ij} (\bar{r}_{\cdot j} - \bar{r})^2}{\sum_{j=1}^g\sum_{i=1}^{n_j} w_{ij} (r_{ij} - \bar{r})^2} }$

where:

$\displaystyle{ n_j }$ is the number of observations in group $\displaystyle{ j }$ of $\displaystyle{ g }$groups,
$\displaystyle{ r_{ij} }$ is the rank of the $\displaystyle{ i }$th observation from group $\displaystyle{ j }$ 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,
$\displaystyle{ n = \sum^g_{j=1} n_j }$,
$\displaystyle{ w_{ij} }$ is the Calibrated Weight,
$\displaystyle{ \bar{r}_{\cdot j} = \frac{\sum_{i=1}^{n_j}{w_{ij} r_{ij}}}{{\sum_{i=1}^{n_j}w_{ij}}} }$,
$\displaystyle{ \bar{r} = \frac{\sum_{j=1}^g\sum_{i=1}^{n_j}w_{ij} r_{ij}}{\sum_{j=1}^g\sum_{i=1}^{n_j}w_{ij}} }$,
$\displaystyle{ p\approx \Pr(\chi^2_{g-1} \ge H) }$