Synchronized Permutation Tests in Two-way ANOVA

In experimental planning, factorial experiments are of particular interest, as they allow us to separately examine the effects of two or more factors in all their possible combinations. In the usual linear model for the analysis of variance, if the error components are not normally distributed, parametric analysis may not be appropriate, even if Rasch and Guiard (2004), recalling previous results by Ito (1969), show that parametric tests are generally robust also in the presence of some nonnormal distributions. When we wish to apply nonparametric tests to the two-way ANOVA layout, problems arise with the exchangeability of the response, which is not satisfied since observations with different treatments might have different expected values. To cope with this problem, either a restricted kind of randomization or the use of residuals is required in order to obtain separate tests for main factors and interaction. In this chapter, based on the concept of synchronized permutations, we introduce an exact permutation solution Pesarin, 2001; Salmaso, 2003; Basso et al., 2007 for testing for fixed effects in replicated two-way factorial designs with continuous responses. This permutation solution, since it is conditional on a set of sufficient statistics, is a distribution-free nonparametric test. It is worth noting that asymptotically distribution-free (but not exact) tests could also be developed using the approach by Draper (1988) or the recent development of a generalization of the Kruskal-Wallis approach to two- and three-way layouts given in Brunner and Puri (2001).