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Biometrika Advance Access originally published online on November 19, 2007
Biometrika 2007 94(4):965-975; doi:10.1093/biomet/asm067
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© 2007 Biometrika Trust

Articles

Hochberg's Step-Up Method: Cutting Corners Off Holm's Step-Down Method

Yifan Huang

H. Lee Moffitt Cancer Center and Research Institute, University of South Florida, Tampa, Florida 33612, U.S.A. yifan.huang{at}moffitt.org

Jason C. Hsu

Department of Statistics, Ohio State University, Columbus, Ohio 43210, U.S.A. jch{at}stat.ohio-state.edu

Received for publication 1 April 2006. Revision received 1 March 2007.
  Abstract

Holm's method and Hochberg's method for multiple testing can be viewed as step-down and step-up versions of the Bonferroni test. We show that both are special cases of partition testing. The difference is that, while Holm's method tests each partition hypothesis using the largest order statistic, setting a critical value based on the Bonferroni inequality, Hochberg's method tests each partition hypothesis using all the order statistics, setting a series of critical values based on Simes' inequality. Geometrically, Hochberg's step-up method ‘cuts corners’ off the acceptance regions of Holm's step-down method by making assumptions on the joint distribution of the test statistics. As can be expected, partition testing making use of the joint distribution of the test statistics is more powerful than partition testing using probabilistic inequalities. Thus, if the joint distribution of the test statistics is available, through modelling for example, we recommend partition step-down testing, setting exact critical values based on the joint distribution.

Key Words: Hochberg's method • Holm's method • Multiple testing • Partition testing • Step-down test • Step-up test


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