Exact testing with random permutations

The way in which random permutations have been used in various permutation-based methods leads to anti-conservativeness, especially in multiple testing contexts. Problems arise in particular for Westfall and Young's maxT method, a more recent method by Meinshausen and a global test that we introduce. We illustrate this using simulations. We solve the problem of anti-conservativeness by proving that an exact test is obtained, if the identity map is added to the randomly drawn permutations.