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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) $.
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