On Distribution-Free Rank Tests for Two-Way Layouts

Abstract

A class of distribution-free statistics using scores based on ranks for testing main effects in two-way layouts, with no interactions and more than one observation per cell, is discussed. Asymptotic distributions are developed under the null and alternative hypotheses for the case in which cell size is fixed and the number of blocks is allowed to become large, and as cell size becomes large with the number of blocks fixed. Efficiencies are considered under both asymptotic situations. We show that under the case in which the number of blocks becomes large, it is possible to choose a set of scores that is optimal in the sense of achieving the highest possible asymptotic relative efficiency for a given distribution.

Key Words:

• Friedman test
• Aligned ranks
• Normal scores
• Asymptotic relative efficiencies