Statistics Colloquium : Dr. John Lachin

Abstract : The closed testing principle (Marcus et al., Biometrika, 1976) provides a general framework for conducting multiple tests in a manner that controls the family-wise type I error probability without the need for an adjustment for multiple tests such as the Holm improved Bonferroni procedure. Closed testing declares a test of a given hypothesis to be statistically significant at level α when it and the tests of the intersection hypotheses containing it are significant at level α. The process generally starts with a K-level omnibus test, such as a T2-test. However, we show that any α-level test of the joint null hypothesis, such as a 1 df test against a one-directional alternative, can be employed in lieu of the omnibus test.

One application is a subgroup analysis where we wish to test the difference between treatment groups separately within subgroups, such as among males and females where a 1 df test of homogeneity is used as the “gatekeeping” test of the interaction hypothesis rather than a 2 df omnibus test.

Another important application is to the tests of the components of a composite hypothesis such as the components of the cardiovascular MACE (major adverse cardiovascular event) outcome consisting of either a fatal cardiovascular event or non-fatal myocardial infarction or non-fatal stroke. In this case the composite time to first event or the Wei-Lachinmultivariate one-directional test, each on 1 df, can be employed to test the higher order intersection hypotheses rather than joint omnibus tests.

Time permitting, closed testing will also be applied to the analysis of each treatment group versus the others combined in a K>2 group trial in lieu of all K(K-1)/2 pair-wise tests.