Friedman Test for Correlated Samples

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This is a non-parametric version of the Repeated Measures ANOVA. The test statistic is:

[math]\displaystyle{ Q =(g-1)\sum_{i=1}^n w_i \frac{ \sum_{j=1}^g\sum_{i=1}^{n} w_i (\bar{r}_{\cdot j} - \bar{r})^2}{\sum_{j=1}^g\sum_{i=1}^{n} w_i (r_{ij} - \bar{r})^2} }[/math]


[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 within observations (e.g., if [math]\displaystyle{ g=4 }[/math] then [math]\displaystyle{ r_{ij} \in {{1,2,3,4}} }[/math]), where 1 is assigned to the lowest value and the average rank is used for ties,
[math]\displaystyle{ n }[/math] is the number of matched samples,
[math]\displaystyle{ w_i }[/math] is the Calibrated Weight,
[math]\displaystyle{ \bar{r}_{\cdot j} = \frac{\sum_{i=1}^n{w_i r_{ij}}}{{\sum_{i=1}^n w_i}} }[/math],
[math]\displaystyle{ \bar{r} = \frac{g \sum_{i=1}^n w_i r_{ij}}{\sum_{j=1}^g\sum_{i=1}^n w_i} }[/math],
[math]\displaystyle{ p \approx \Pr(\chi^2_{g-1} \ge Q) }[/math]

See also