A discussion on multi-way ANOVA using a permutation approach

Abstract

We deal with permutation testing for balanced and unbalanced repeated measures designs and we consider a replicated homoscedastic (balanced or unbalanced) factorial design with fixed effects (Milliken et al., 1984) as the basic experimental plan. The design responses are measured in L time occasions. The usual linear model for responses is: 
y = {y _{ij(l) r} = .........................................................................

..................  so that, the null hypothesis H₀ is true if all partial sub hypotheses are true. In order to test separately the overall null hypotheses, we jointly consider  the L measures, then the permutation solution is based on the nonparametric combination methodology (Pesarin, 2001). It is worth noting that the new permutation approach, presented here, is exact and, being conditional to the sufficient statistic represented by the data matrix it does not require normality of error terms in the linear model for responses.