Permutational multiple testing adjustments with multivariate multiple group data

We consider the multiple comparison problem where multiple outcomes are each compared among several different collections of groups in a multiple group setting. In this case there are several different types of hypotheses, with each specifying equality of the distributions of a single outcome over a different collection of groups. Each type of hypothesis requires a different permutational approach. We show that under a certain multivariate condition it is possible to use closure over all hypotheses, although intersection hypotheses are tested using Boole's inequality in conjunction with permutation distributions in some cases. Shortcut tests are then found so that the resulting testing procedure is easily performed. The error rate and power of the new method is compared to existing competitors through simulation of correlated data. An example is analyzed, consisting of multiple adverse events in a clinical trial.