Friedman Test

The Friedman test is a rank-based, nonparametric test for several related samples. This test is named in honor of its developer, the Nobel laureate and American economist Milton Friedman, who first proposed the test in 1937 in the Journal of the American Statistical Association. A researcher may sometimes feel confused when reading about the Friedman test because the “related samples” may arise from a variety of research settings. A very common way to think of Friedman's test is that it is a test for treatment differences for a randomized complete block (RCB) design. The RCB design uses blocks of participants who are matched closely on some relevant characteristic. Once the blocks are formed, participants within each block are assigned randomly to the treatment conditions. In the behavioral and health sciences, a common procedure is to treat a participant as a “block,” wherein the participant serves in all the treatment conditions of an independent variable—also commonly referred to as a repeated measures design or a within-subjects design.

Although it is seen relatively rarely in the research literature, there is another research situation in which the Friedman test can be applied. One can use it in the context in which one has measured two or more comparable (also referred to as “commensurable”) dependent variables from the same sample, usually at the same time. In this context, the data are treated much like a repeated measures design wherein the commensurable measures are levels of the repeated measures factor.

There is an additional source of confusion when one thinks about the Friedman test for repeated measures designs because for the RCB design, the parametric method for testing the hypothesis of no differences between treatments is the two-way ANOVA, with treatment and block factors. The Friedman test, which depends on the ranks of the dependent variable within each block, may therefore be considered a two-way ANOVA on ranks.

It is known in theoretical statistics that the Friedman test is a generalization of the sign test and has similar modest statistical power for most distributions that are likely to be encountered in [Page 381] behavioral and health research. For normal distributions, the asymptotic relative efficiency (ARE) of the Friedman test with respect to the F test, its counterpart among parametric statistical tests, is 0.955k / (k + 1), where k is the number of treatment groups. When k = 4, the ARE of the Friedman test relative to the F test is 0.764. There is evidence from computer simulation studies that the ARE results, which are for very large sample sizes, are close to the relative efficiency to be expected for small and moderate sample sizes.

It is useful to note that in our desire to select the statistical test with the greatest statistical power, we often select the test with the greatest ARE. Therefore, for a normal distribution, the parametric test is more statistically powerful, leading us to recommend the ANOVA F test over the Friedman test. However, in the case of nonnormal distributions of dependent variables, the recommendation favors the Friedman test. Given the frequency at which nonnormal distributions are encountered in research, it is remarkable that nonparametric tests, such as the Friedman test, or other more-powerful tests, such as (a) the Zimmerman-Zumbo repeated measures ANOVA on ranks, wherein the scores in all treatment groups are combined in a single group and ranked, or (b) the Quade test, which uses information about the range of scores in the blocks relative to each other, are not used more often.

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