Closed testing using surrogate hypotheses with restricted alternatives

The closed testing principle provides strong control of the type I error probabilities of tests of a set of hypotheses that are closed under intersection such that a given hypothesis H can only be tested and rejected at level α if all intersection hypotheses containing that hypothesis are also tested and rejected at level α. For the higher order hypotheses, multivariate tests (> 1df) are generally employed. However, such tests are directed to an omnibus alternative hypothesis of a difference in any direction for any component that may be less meaningful than a test directed against a restricted alternative hypothesis of interest. Herein we describe applications of this principle using an α-level test of a surrogate hypothesis [Formula: see text] such that the type I error probability is preserved if [Formula: see text] such that rejection of [Formula: see text] implies rejection of H. Applications include the analysis of multiple event times in a Wei-Lachin test against a one-directional alternative, a test of the treatment group difference in the means of K repeated measures using a 1 df test of the difference in the longitudinal LSMEANS, and analyses within subgroups when a test of treatment by subgroup interaction is significant. In such cases the successive higher order surrogate tests can be aimed at detecting parameter values that fall within a more desirable restricted subspace of the global alternative hypothesis parameter space. Closed testing using α-level tests of surrogate hypotheses will protect the type I error probability and detect specific alternatives of interest, as opposed to the global alternative hypothesis of any difference in any direction.