Chapter 7 One-Way and Higher Designs for Independent Groups 359 7.9 A Rank-Based Method for a Two-Way Design This section describes a rank-based method for a two-way design derived by Akritas, Arnold, and Brunner (1997). The basic idea stems from Akritas and Arnold (1994) and is based on the following point of view. For any value x, let 1 ¯ F j. (x) = K K F jk (x) k=1 be the average of the distributions among the K levels of Factor B corresponding to the jth level of Factor A. The hypothesis of no main effects for Factor A is ¯ ¯ ¯ H 0 : F 1. (x) = F 2. (x) = · · · = F J. (x). for any x. Letting 1 ¯ F .k (x) = J J F jk (x) j=1 be the average of the distributions for the kth level of Factor B, the hypothesis of no main effects for Factor B is