Exchangeable random variables

If Equation (1) be satisfied, then all sets S are equally likely under the null hypothesis. To obtain a permutation test that is both unbiased and most powerful, one need only select the set S so that its probability under the alternative >0 is a maximum. This is accomplished in most cases by rejecting the hypothesis for all values of the test statistic that lie in the upper tail of the permutation distribution. As research during the intervening half century has amply demonstrated, permutation tests should be used to do all of the following: