Pairwise comparisons using ranks in block designs

Friedman’s rank test and the associated aligned rank test are the standard rank alternatives to the classical linear models F statistic for the randomized complete block design (RCBD). However, in current practice there are no good rank alternatives to the Tukey–Kramer all pairwise comparisons procedure for the normal linear model. For example, the standard rank method found in Section 7.3 of Hollander et al. [John Wiley & Sons, Inc., Hoboken, New Jersey; 2014] is based on the permutation distribution under the complete null of no differences and thus cannot strongly control the family-wise error rate (FWER). It also has low power. However, we show that the closed method introduced by Marcus et al. [Biometrika 63 (3);1976:655–60] applied to aligned ranks has strong FWER control and good power compared to the Tukey–Kramer method for long-tailed error distributions.