Synchronized permutation tests in replicated I×JI×J designs

In this paper we present an extension of the synchronized permutation tests for main factors and interaction introduced by Pesarin [2001. Multivariate Permutation Tests with Applications to Biostatistics. Wiley, Chichester] and Salmaso [2003. Synchronized permutation tests in factorial designs. Comm. Statist. Theory Methods 32, 1419–1437] in the case of a 2k2k complete factorial design with n replicates. This extension considers the more general case when the main factors can assume, respectively, I and J lelvels. An algorithm to obtain Monte Carlo synchronized permutations is introduced and a comparative simulation study with the parametric ANOVA is reported. We also implemented the nonparametric simultaneous confidence intervals based on syncrhonized permutations for main factor effects. A graphical representation for the simultaneous confidence intervals is suggested and illustrated with some examples from the literature.