A general permutation approach for analyzing repeated measures ANOVA and mixed-model designs

Repeated measures ANOVA and mixed-model designs are the main classes of experimental designs used in psychology. The usual analysis relies on some parametric assumptions (typically Gaussianity). In this article, we propose methods to analyze the data when the parametric conditions do not hold. The permutation test, which is a non-parametric test, is suitable for hypothesis testing and can be applied to experimental designs. The application of permutation tests in simpler experimental designs such as factorial ANOVA or ANOVA with only between-subject factors has already been considered. The main purpose of this paper is to focus on more complex designs that include only within-subject factors (repeated measures) or designs that include both within-subject and between-subject factors (mixed-model designs). First, a general approximate permutation test (permutation of the residuals under the reduced model or reduced residuals) is proposed for any repeated measures and mixed-model designs, for any number of repetitions per cell, any number of subjects and factors and for both balanced and unbalanced designs (all-cell-filled). Next, a permutation test that uses residuals that are exchangeable up to the second moment is introduced for balanced cases in the same class of experimental designs. This permutation test is therefore exact for spherical data. Finally, we provide simulations results for the comparison of the level and the power of the proposed methods.