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Biometrika Advance Access originally published online on April 21, 2009
Biometrika 2009 96(2):399-410; doi:10.1093/biomet/asp006
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© 2009 Biometrika Trust

Article

Optimal testing of multiple hypotheses with common effect direction

Richard M. Bittman

Bittman Biostat, Inc., Glencoe, Illinois 60022, U.S.A. rmb{at}bittmanbiostat.com

Joseph P. Romano

Department of Statistics, Stanford University, Stanford, California 94305, U.S.A. romano{at}stanford.edu

Carlos Vallarino

Analytical Science, Takeda Global Research and Development, Deerfield, Illinois 60015, U.S.A. cvallarino{at}tpna.com

Michael Wolf

Institute for Empirical Research in Economics, University of Zurich, CH-8006 Zurich, Switzerland mwolf{at}iew.uzh.ch

Received for publication 1 February 2007. Revision received 1 August 2008.
  Abstract

We present a theoretical basis for testing related endpoints. Typically, it is known how to construct tests of the individual hypotheses, but not how to combine them into a multiple test procedure that controls the familywise error rate. Using the closure method, we emphasize the role of consonant procedures, from an interpretive as well as a theoretical viewpoint. Surprisingly, even if each intersection test has an optimality property, the overall procedure obtained by applying closure to these tests may be inadmissible. We introduce a new procedure, which is consonant and has a maximin property under the normal model. The results are then applied to PROactive, a clinical trial designed to investigate the effectiveness of a glucose-lowering drug on macrovascular outcomes among patients with type 2 diabetes.

Key Words: Closure method • Consonance • Familywise error rate • Multiple endpoints • Multiple testing • O'Brien's method


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