Permutation Tests for Multivariate Stratified Data: Synchronized or Unsynchronized Permutations?

In the present work, we adopt a method based on permutation tests aimed at facing stratified experiments. The method consists in computing permutation tests separately for each strata and then combining the results.We know that by perform ing simultaneously permutation tests (synchronized) in different strata, we maintain the underlying dependence structure and we can properly adopt the nonparametric combination of dependent tests procedure. But when strata have different sample sizes, performing the same permutations is not allowed. On the other hand, if units in different strata can be assumed independent we can think to perform permutation tests independently (unsynchronized) for each strata, and then combining the result ing p-values. In this work, we show that when strata are independent we can adopt equivalently both synchronized and unsynchronized permutations.