Efficiencies of Interblock Rank Statistics for Repeated Measures Designs

Abstract

For factorial designs, tests that rely on the method of n rankings, for example, the Friedman test, suffer a potential loss of power since they depend only on within-block ranks. This deficiency is overcome by ranking across blocks as is done in the aligned rank tests and the rank transform procedures. This article examines a broad class of powerful interblock rank statistics for testing for both treatment effects and ordered alternatives and for performing multiple comparisons in a two-way repeated measures design. Pitman efficiencies are obtained for tests that closely resemble the rank transform test and for aligned rank tests under weaker hypotheses than assumed by Puri and Sen (1971). In addition, several serious limitations of the rank transform procedure for repeated measures designs are presented. Limitations of the rank transform procedures, in particular, merit such study because they are recommended without reservation by the SAS manual for use with PROC ANOVA and PROC GLM and are very easy to implement. As a result, they are used, often indiscriminately, for both hypothesis testing and estimation by professionals and researchers in such diverse fields as clinical psychology and data processing services. Yet, little is known about their asymptotic (much less small-sample) properties except for a few designs. Whereas in some instances it has been demonstrated that the rank transform is an appropriate test with good power properties, it has been demonstrated in equally many instances that it is not a valid procedure and does not have the expected asymptotic distribution.

Key Words:

• Analysis of variance
• Efficacy
• Linear rank statistics
• Pitman alternatives
• Rank transformation