## Institute of Mathematical Statistics Lecture Notes - Monograph Series

- Optimality
- 2009, 326-332

### Bayesian Decision Theory for Multiple Comparisons

Charles Lewis and Dorothy T. Thayer

#### Abstract

Applying a decision theoretic approach to multiple comparisons very similar to that described by Lehmann [*Ann. Math. Statist.* **21** (1950) 1–26; *Ann. Math. Statist.* **28** (1975a) 1–25; *Ann. Math. Statist.* **28** (1975b) 547–572], 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 Benjamini and Hochberg [*J. Roy. Statist. Soc. Ser. B* **57** (1995) 289–300] 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.

#### Chapter information

**Source***Optimality: The Third Erich L. Lehmann Symposium* (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2009)

**Dates**

First available in Project Euclid: 3 August 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.lnms/1249305337

**Digital Object Identifier**

doi:10.1214/09-LNMS5719

**Mathematical Reviews number (MathSciNet)**

MR2681679

**Zentralblatt MATH identifier**

1271.62020

**Subjects**

Primary: 62J15: Paired and multiple comparisons 62C10: Bayesian problems; characterization of Bayes procedures 62F15: Bayesian inference

**Keywords**

Bayesian decision theory loss function multiple comparisons false discovery rate

**Rights**

Copyright © 2009, Institute of Mathematical Statistics

#### Citation

Rojo, Javier. Bayesian Decision Theory for Multiple Comparisons. Optimality, 326--332, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2009. doi:10.1214/09-LNMS5719. https://projecteuclid.org/euclid.lnms/1249305337

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