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On weighted Hochberg procedures
Department of Industrial Engineering, and Management Sciences, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, U.S.A. ajit{at}iems.northwestern.edu
Department of Statistics, Northwestern University, 2008 Sheridan Road, Evanston, Illinois 60208, U.S.A. lingyunliu{at}northwestern.edu
Received for publication 1 March 2007. Revision received 1 December 2007.
We consider different ways of constructing weighted Hochberg-type step-up multiple test procedures including closed procedures based on weighted Simes tests and their conservative step-up short-cuts, and step-up counterparts of two weighted Holm procedures. It is shown that the step-up counterparts have some serious pitfalls such as lack of familywise error rate control and lack of monotonicity in rejection decisions in terms of p-values. Therefore an exact closed procedure appears to be the best alternative, its only drawback being lack of simple stepwise structure. A conservative step-up short-cut to the closed procedure may be used instead, but with accompanying loss of power. Simulations are used to study the familywise error rate and power properties of the competing procedures for independent and correlated p-values. Although many of the results of this paper are negative, they are useful in highlighting the need for caution when procedures with similar pitfalls may be used.
Key Words: Bonferroni test • Closed procedure • Familywise error rate • Holm procedure • Multiple comparisons • p-Value • Simes test • Step-down procedure • Step-up procedure
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