Testing each hypothesis marginally at alpha while still controlling FWER: how and when

This paper proposes a multiple testing procedure that allows one to reject each individual hypothesis at a prespecified level α, while still controlling the familywise error rate at α in the strong sense. Typically, rejecting a hypothesis when its marginal p-value is ⩽α in a multiple hypothesis testing setting will lead to an inflation of familywise error rate. However, this inflation can be avoided if a particular consistency criterion is prespecified and incorporated in the testing algorithm. The criterion is equivalent to requiring that all p-values be smaller than or equal to a particular threshold in the one-sided hypothesis testing setting. Extensions to the two-sided hypothesis testing setting and extensions to situations where the criterion can be chosen per user's preference are also presented. Copyright © 2012 John Wiley & Sons, Ltd.