On Synchronized Permutation Tests in Two-Way ANOVA

Abstract

In I × J balanced factorial designs units are not exchangeable between blocks since their expected values depend on received treatments. It does not seem possible, therefore, to obtain exact and separate tests to respectively assess main factor and interaction effects. Parametric two-way ANOVA F tests are exact tests only under assumption of normal homoschedastic errors, but they are also positively correlated. Instead, it is possible to obtain exact, separate and uncorrelated permutation tests at least for main factors by introducing a restricted kind of permutations, named synchronized permutations. Since these tests are conditional on observed data, they are distribution-free and may be shown to be almost as powerful as their parametric counterpart under normal errors. We obtain the expression of the correlation between the main factor ANOVA tests as a function of the number of replicates in each block, the number of main factor levels and their noncentrality parameters.