The nonparametric procedure known as Friedman's test is often used when we have one observation in each cell of a two-way table. Some time ago, Benard and van Elteren generalized this "method of m rankings" to tables containing arbitrary numbers of observations per cell, though their work apparently remains little known by experimenters. An example shows how the calculation is performed, a number of other nonparametric tests are shown to be special cases of Benard and van Elteren's, and there is a brief discussion of related topics. In an appendix, a FORTRAN IV program for applying this test is given. A novel feature of this program is that the significance level of xrz may be estimated directly by repeatedly assigning random ranks within each row of the table. (It is shown that this is important because the assumption that xr2 is distributed as x2 tends to be conservative for tables with few rows.) Because the following are similar or equivalent to this, the program is also suitable when they might be used: Friedman's test; Kruskal-Wallis test; Mann-Whitney U (Wilcoxon) test: sign test for matched pairs; Spearman's rank correlation; Wilcoxon's stratified test; Meddis's test: and Jonckheere's generalization of the Kruskal-Wallis test to the case of ordered alternatives.
When we have a two-way table of observed measurements, one per cell, and we wish to determine whether the column variable affects the level of our dependent measurement, we can use the Analysis of Variance. Alternatively, we can use a nonparametric test due to Friedman (7). This has the advantage of being applicable when the data are only ranked within each row, rather than being measured absolutely, and is much quicker and easier to carry out. Furthermore it does not require the assumptions of Normality and homoscedasticity of errors.
Friedman's test is performed by ranking the observations within each row, and then adding up the ranks columnwise. The squares of these column totals are then summed. If the total is sufficiently large we can conclude that our observations are dependent on the column variable. Friedman's test statistic, usually denoted by xr2, is given by the formula x,' = [ 1 ZR/kN(N + l)] -3k(N + I),