Abstract
IntroductionThe 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.MethodsHerein we describe applications of this principle using an α-level test of a surrogate hypothesis such that the type I error probability is preserved if such that rejection of 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.ConclusionClosed 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.
Highlights
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 tested and rejected at level α
We describe applications of this principle using an α-level test of a surrogate hypothesis H~ such that the type I error probability is preserved if H ) H~ such that rejection of H~ implies rejection of H
We present specific applications, starting with the analysis of multiple event-time outcomes (e.g. Major Adverse Cardiovascular Event (MACE) in a cardiovascular trial) following a one-directional Wei-Lachin multivariate test of a combination of outcomes, with a computational example
Summary
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 tested and rejected at level α. For the higher order hypotheses, multivariate tests (> 1df) are generally employed. 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
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