The rank transform method in some two-factor designs

The use of the rank transform for testing problems in some two-factor designs is considered. The rank transform procedure consists of replacing the observations by their ranks in the combined sample and performing one of the standard analysis of variance (ANOVA) procedures on these ranks. The asymptotic version of the rank transform is introduced, and its usefulness as a means for understanding the nature of the rank transform is examined. It is demonstrated that the asymptotic version of the transformation helps identify the testing problems where the rank transform method works and helps suggest which ANOVA procedure should be used on the ranks. This approach is applied on the balanced and unbalanced nested model and the balanced and unbalanced two-way layout with and without interaction. The extension of the rank transform method to some of these models is new. The proposed procedures do not share the known simplicity of existing rank transform statistics, but they do allow heteroscedasticity in the errors. The asymptotic power of the procedures is derived, and comparisons with aligned rank and F tests are made.