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Bayesian decision theory for multiple comparisons. (English) Zbl 1271.62020

Rojo, Javier (ed.), Optimality. The third Erich L. Lehmann symposium. Selected papers based on the presentations at the symposium, Rice University, Houston, TX, USA, May 16–19, 2007. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 978-0-940600-77-5/pbk). Institute of Mathematical Statistics Lecture Notes - Monograph Series 57, 326-332 (2009).
Summary: Applying a decision theoretic approach to multiple comparisons very similar to that described by E. L. Lehmann [Ann. Math. Stat. 21, 1–26 (1950; Zbl 0036.09501); ibid. 28, 1–25 (1957; Zbl 0078.33402); ibid. 28, 547–572 (1957; Zbl 0080.35704)], we introduce a loss function based on the concept of the false discovery rate (FDR). We derive a Bayes rule for this loss function and show that it is very closely related to a Bayesian version of the original multiple comparisons procedure proposed by Y. Benjamini and Y. Hochberg [J. R. Stat. Soc., Ser. B 57, No. 1, 289–300 (1995; Zbl 0809.62014)] to control the sampling theory FDR. We provide the results of a Monte Carlo simulation that illustrates the very similar sampling behavior of our Bayes rule and Benjamini and Hochberg’s procedure when applied to making all pair-wise comparisons in a one-way fixed effects analysis of variance setup with 10 and with 20 means.
For the entire collection see [Zbl 1203.62003].

MSC:

62C10 Bayesian problems; characterization of Bayes procedures
62J15 Paired and multiple comparisons; multiple testing
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