Kruskal-Wallis Test
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This is a non-parametric version of the F-Test (ANOVA). The test statistic is:
[math]\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} }[/math]
where:
- [math]\displaystyle{ n_j }[/math] is the number of observations in group [math]\displaystyle{ j }[/math] of [math]\displaystyle{ g }[/math]groups,
- [math]\displaystyle{ r_{ij} }[/math] is the rank of the [math]\displaystyle{ i }[/math]th observation from group [math]\displaystyle{ j }[/math] 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,
- [math]\displaystyle{ n = \sum^g_{j=1} n_j }[/math],
- [math]\displaystyle{ w_{ij} }[/math] is the Calibrated Weight,
- [math]\displaystyle{ \bar{r}_{\cdot j} = \frac{\sum_{i=1}^{n_j}{w_{ij} r_{ij}}}{{\sum_{i=1}^{n_j}w_{ij}}} }[/math],
- [math]\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}} }[/math],
- [math]\displaystyle{ p\approx \Pr(\chi^2_{g-1} \ge H) }[/math]