Friedman Test

Procedure

The Friedman test is used to analyze several related (i.e., dependent) samples. Friedman referred to his procedure as the method of ranks in that it is based on replacing the original scores with rank-ordered values. Consider a study in which data are collected within a randomized block design where N blocks are observed over K treatment conditions on a dependent measure that is at least ordinal-level. The first step in the Friedman test is to replace the original scores with ranks, denoted Rjk, within each block; that is, the scores for block j are compared with each other, and a rank of 1 is assigned to the smallest observed score, a rank of 2 is assigned to the second smallest, and so on until the largest value is replaced by a rank of K. In the situation where there are ties within a block (i.e., two or more of the values are identical), the midrank is used. The midrank is the average of the ranks that would have been assigned if there were no ties. Note that this procedure generalizes to a repeated measures design in that the ranks are based on within-participant observations (or, one can think of the participants as defining the blocks). Table 1 presents the ranked data in tabular form.

It is apparent that row means in Table 1 (i.e., mean of ranks for each block) are the same across blocks; however, the column means (i.e., mean of ranks within a treatment condition) will be affected by differences across treatment conditions. Under the null hypothesis that there is no difference due to treatment, the ranks are assigned at random, and thus, an equal frequency of ranks would be expected for each treatment condition. Therefore, if there is no treatment effect, then the column means are expected to be the same for each treatment condition. The null hypothesis may be specified as follows:

To test the null hypothesis that there is no treatment effect, the following test statistic may be

...