Friedman Test for Correlated Samples

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

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

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 within observations (e.g., if $g=4$ then $r_{ij}\in 1,2,3,4$), where 1 is assigned to the lowest value and the average rank is used for ties,

$n$ is the number of matched samples,

$w_i$ is the Calibrated Weight,

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

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

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

See also

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