# Friedman test

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The Friedman test is a non-parametric statistical test developed by the U.S. economist Milton Friedman. The procedure involves ranking each row (or block) together, then considering the values of ranks by columns.

## MethodEdit

1. Given data $\{x_{ij}\}_{m\times n}$, that is, a tableau with $m$ rows (the blocks), $n$ columns (the treatments) and a single observation at the intersection of each block and treatment, calculate the ranks within each block. Replace the data with a new tableau $\{r_{ij}\}_{m \times n}$ where the entry $r_{ij}$ is the rank of $x_{ij}$ within block $i$.
2. Find the values:
• $SS_t = n\sum_{j=1}^n (\bar{r}_{j} - \bar{r})^2$,
• $SS_e = \frac{1}{m(n-1)} \sum_{i=1}^m \sum_{j=1}^n (r_{ij} - \bar{r})^2$
• $\bar{r}_{j} = \frac{1}{m} \sum_{i=1}^m {r_{ij}}$
• $\bar{r} = \frac{1}{mn}\sum_{i=1}^m \sum_{j=1}^n r_{ij}$
3. The test statistic is given by $Q = \frac{SS_t}{SS_e}$.
4. Finally, the p-value is given by $\mathbf{P}(\chi^2_{n-1} \ge Q)$.