Bayesian Analysis

Prior Effective Sample Size in Conditionally Independent Hierarchical Models

Abstract

Prior effective sample size (ESS) of a Bayesian parametric model was defined by Morita, et al. (2008, Biometrics, 64, 595-602). Starting with an $\varepsilon$-information prior defined to have the same means and correlations as the prior but to be vague in a suitable sense, the ESS is the required sample size to obtain a hypothetical posterior very close to the prior. In this paper, we present two alternative definitions for the prior ESS that are suitable for a conditionally independent hierarchical model. The two definitions focus on either the first level prior or second level prior. The proposed methods are applied to important examples to verify that each of the two types of prior ESS matches the intuitively obvious answer where it exists. We illustrate the method with applications to several motivating examples, including a single-arm clinical trial to evaluate treatment response probabilities across different disease subtypes, a dose-finding trial based on toxicity in this setting, and a multicenter randomized trial of treatments for affective disorders.

Article information

Source
Bayesian Anal., Volume 7, Number 3 (2012), 591-614.

Dates
First available in Project Euclid: 28 August 2012

https://projecteuclid.org/euclid.ba/1346158777

Digital Object Identifier
doi:10.1214/12-BA720

Mathematical Reviews number (MathSciNet)
MR2981629

Zentralblatt MATH identifier
1330.62147

Citation

Morita, Satoshi; Thall, Peter F.; Müller, Peter. Prior Effective Sample Size in Conditionally Independent Hierarchical Models. Bayesian Anal. 7 (2012), no. 3, 591--614. doi:10.1214/12-BA720. https://projecteuclid.org/euclid.ba/1346158777

References

• Berlin, J.A. and Colditz, G.A. 1999. The role of meta-analysis in the regulatory process for foods, drugs, and devices. Journal of the American Medical Association 281: 830-834.
• Berry, D. A., and Stangl, D. K. 1996. Bayesian Biostatistics. New York: Marcel Dekker.
• Congdon, P. 2005. Bayesian Statistical Modelling (2nd Edition). Chichester: John Wiley and Sons.
• Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. 2004. Bayesian Data Analysis (2nd Edition). New York: Chapman and Hall/CRC.
• Gelman A. 2006. Prior distributions for variance parameters in hierarchical models. Bayesian Analysis 1: 515-533.
• Gray, R. J. 1994. A Bayesian analysis of institutional effects in a multicenter cancer clinical trial. Biometrics 50: 244-253.
• Kass, R. E. and Steffey, D. 1989. Approximate Bayesian Inference in Conditionally Independent Hierarchical Models. Journal of the American Statistical Association 84: 717-726.
• Morita, S., Thall, P. F., Müller, P. 2008. Determining the effective sample size of a parametric prior. Biometrics 64: 595-602.
• O’Quigley J, Pepe M, Fisher L. 1990. Continual reassessment method: a practical design for phase I clinical trials in cancer. Biometrics 46: 33-48.
• Spiegelhalter, D.J., Freedman, L.S. and Parmar, M.K.B. 1994. Bayesian approaches to randomized trials. Journal of the Royal Statistical Society, Series A 157: 357-416.
• Stangl, D. K. 1996. Hierarchical Analysis of Continuous-Time Survival Models. In: Berry, D. A., and Stangl, D. K. eds. Bayesian Biostatistics: 429-450.
• Thall, P. F., Wathen, K., 1, Bekele, B. N., Champlin, R. E., Baker, L. H., Benjamin, R. S. 2003. Hierarchical Bayesian approaches to phase II trials in diseases with multiple subtypes. Statistics in Medicine 22: 763-780.