Logical consistency in simultaneous statistical test procedures


Many authors have argued that, when performing simultaneous statistical test procedures, one should seek for solutions that lead to decisions that are consistent and, consequently, easier to communicate to practitioners of statistical methods. In this way, the set of hypotheses that are rejected and the set of hypotheses that are not rejected by a testing procedure should be consistent from a logic standpoint. For instance, if hypothesis A implies hypothesis B, a procedure that rejects B should also reject A, a property not always met by multiple test procedures. We contribute to this discussion by exploring how far one can go in constructing coherent procedures while still preserving statistical optimality. This is done by studying four types of logical consistency relations. We show that although the only procedures that satisfy more than (any) two of these properties are simple tests based on point estimation, it is possible to construct various interesting methods that fulfil one or two of them while preserving different statistical optimality criteria. This is illustrated with several Bayesian and frequentist examples. We also characterize some of these properties under a decision-theoretic framework.

Logic Journal of the IGPL