Friedman Test

Analysis of Variance by Ranks

In 1937, an American statistician and economist, Milton Friedman, suggested that the assumption of normality in the parametric analysis of variance could be circumvented by converting the data in question into ranks. If the data table contains k columns and n rows, the entries in each row are replaced by the numbers from 1 to k, where 1 refers to the smallest observation and k refers to the largest observation across the k conditions. (Friedman noted that it was immaterial whether the ranking was from the lowest to the highest or from the highest to the lowest.) Suppose that Ri is the sum of the ranks in the ith condition and that R is the sum of all the ranks across the k conditions. The deviation of the mean of the ranks in the ith condition is $\left[\left(R_i/n\right)-\left(R/kn\right)\right]$. Friedman defined a statistic that he denoted by the symbol ${\chi }_r^2$ (chi-r-square) as the sum of the squared standardized deviations across the k conditions.

However, this can be simplified computationally because the data in each row are simply the integers from 1 to k. Within each row, the sum of all the ranks is k(k+1)/2, the mean of all the ranks is (k+1)/2, and the variance of all the ranks is (k2−1)/12. This enabled Friedman to express his test statistic in terms of the following formula:

$${\chi }_r^2=\left\{12/[nk(k+1)]\right\}\left[{\displaystyle \sum \left(R_i^2\right)}\right]-3n\left(k+1\right),$$

where summation is carried out across the k conditions. For k = 2, Friedman noted that his test was formally equivalent to the sign test, whose properties had already been well-documented. For k = 3 and values of n between 2 and 9, and for k = 4 and values of n between 2 and 4, he presented tables showing the exact probability of obtaining a value of ${\chi }_r^2$ or a higher value under the null hypothesis that the observations in the different conditions were drawn from the same population. For larger values of k and n, Friedman proposed that the means of the ranks would be normally distributed under the null hypothesis; ${\chi }_r^2$ would therefore be distributed as chi-square (χ2) with (k − 1) degrees of freedom, and it could be evaluated using existing tables of χ2.

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