Fully nonparametric hypotheses for factorial designs I: Multivariate repeated measures designs

We introduce nonparametric versions for many of the hypotheses tested in analysis of variance and repeated measures models, such as the hypotheses of no main effects, no interaction effects, and no factor effects. These natural extensions of the nonparametric hypothesis of equality of the k distributions in the k sample problem have appealing practical interpretations. We concentrate on multivariate repeated measures designs and obtain simple rank statistics for testing these hypotheses. These statistics are the rank transform (RT) versions of the classical statistics for testing hypotheses in repeated measures designs. We emphasize that even though recent research has demonstrated the inappropriateness of the RT method for many parametric hypotheses, the RT procedure is always valid for testing our nonparametric hypotheses. We show that the rank statistics converge in distribution to central chi-squared distributions under their respective nonparametric null hypotheses. The noncentrality parameters under nonparametric contiguous alternatives are obtained. In addition, we present an interpretation of simultaneous confidence intervals and multiple comparison procedures based on rank statistics. Finally, we illustrate the proposed rank tests with a real data set from the statistical literature.