This is a non-parametric version of the F-Test (ANOVA). The test statistic is:
\[\begin{align} H = (\sum_{j=1}^g\sum_{i=1}^{n_j}w_{ij}-1)\frac{\sum_{j=1}^g\sum_{i=1}^{n_j} w_{ij} (\bar{r}_{\cdot j} - \bar{r})^2}{\sum_{j=1}^g\sum_{i=1}^{n_j} w_{ij} (r_{ij} - \bar{r})^2} \end{align}\]
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,
\(n = \sum^g_{j=1} n_j \),
\(w_{ij} \) is the Calibrated Weight,
\(\bar{r}_{\cdot j} = \frac{\sum_{i=1}^{n_j}{w_{ij} r_{ij}}}{{\sum_{i=1}^{n_j}w_{ij}}} \),
\(\bar{r} = \frac{\sum_{j=1}^g\sum_{i=1}^{n_j}w_{ij} r_{ij}}{\sum_{j=1}^g\sum_{i=1}^{n_j}w_{ij}} \),
\(p\approx \Pr(\chi^2_{g-1} \ge H) \)