Resumen
Testing for stochastic ordering is of considerable importance when increasing does of a treatment are being compared, but in applications involving multivariate responses has received much less attention. We propose a permutation test for testing against multivariate stochastic ordering. This test is distribution-free and no assumption is made about the dependence relations among variables. A comparative simulation study shows that the proposed solution exhibits a good overall performance when compared with existing tests that can be used for the same problem.

Resumen
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 2k 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.