Friedman test

The Friedman test is a non-parametric statistical test developed by the U.S. economist Milton Friedman. Similar to the parametric repeated measures ANOVA, it is used to detect differences in treatments across multiple test attempts. The procedure involves ranking each row (or block) together, then considering the values of ranks by columns.

Classic examples of use are:
 * n wine judges rate k different wines. Are the ratings consistent?
 * n welders use k welding torches, and the ensuing welds were rated on quality. Is there one torch that produced better welds than the others?

The Friedman test is used for two-way repeated measures analysis of variance by ranks. In its use of ranks it is similar to the Kruskal-Wallis one-way analysis of variance by ranks.

Method

 * 1) Given data $$\{x_{ij}\}_{n\times k}$$, that is, a tableau with $$n$$ rows (the blocks), $$k$$ columns (the treatments) and a single observation at the intersection of each block and treatment, calculate the ranks within each block. If there are tied values, assign to each tied value the average of the ranks that would have been assigned without ties. Replace the data with a new tableau $$\{r_{ij}\}_{n \times k}$$ where the entry $$r_{ij}$$ is the rank of $$x_{ij}$$ within block $$i$$.
 * 2) Find the values:
 * 3) *$$\bar{r}_{\cdot j} = \frac{1}{n} \sum_{i=1}^n {r_{ij}}$$
 * 4) *$$\bar{r} = \frac{1}{nk}\sum_{i=1}^n \sum_{j=1}^k r_{ij}$$
 * 5) *$$SS_t = n\sum_{j=1}^k (\bar{r}_{\cdot j} - \bar{r})^2$$,
 * 6) *$$SS_e = \frac{1}{n(k-1)} \sum_{i=1}^n \sum_{j=1}^k (r_{ij} - \bar{r})^2$$
 * 7) The test statistic is given by $$Q = \frac{SS_t}{SS_e}$$. Note that the value of Q as computed above does not need to be adjusted for tied values in the data.
 * 8) Finally, when n or k is large (i.e. n > 15 or k > 4), the probability distribution of Q can be approximated by that of a chi-square distribution. In this case the p-value is given by $$\mathbf{P}(\chi^2_{k-1} \ge Q)$$. If n or k is small, the approximation to chi-square becomes poor and the p-value should be obtained from tables of Q specially prepared for the Friedman test. If the p-value is significant, appropriate post-hoc multiple comparisons tests would be performed.