Relating the Friedman test adjusted for ties, the Cochran–Mantel–Haenszel mean score test and the ANOVA F test

The Friedman test is used to nonparametrically test the null hypothesis of equality of the treatment distributions in the randomized block design. The simple form of the test statistic is for data that is untied within blocks. When ties occur and mid-ranks are used, an adjustment to the simple form of the test statistic is needed. Here such adjustments are given, and it is shown that the Friedman tests, both with untied rank data and with tied data using mid-ranks, are Cochran–Mantel–Haenszel mean score tests. Additionally, for the randomized block design, the Cochran–Mantel–Haenszel mean score statistic is shown to be a simple function of the ANOVA F statistic. Using this relationship for the Friedman tests is shown to give more accurate p-values close to the nominal significance level. Moreover, since the ANOVA F test null hypothesis specifies equality of mean treatment ranks, so does the Friedman test. Therefore the null hypothesis of the Friedman test is sharper than equality of the treatment distributions.