STATISTICS IN MEDICINE, VOL. 16, 1121—1133 (1997) BAYESIAN INFERENCE IN TWO-PHASE PREVALENCE STUDIES ALAATTIN ERKANLI*, REFIK SOYER AND DALENE STANGL  Developmental Epidemiology Program, Department of Psychiatry and Behavioral Sciences, Duke University Medical Center, Box 3454, Durham, NC 27710, U.S.A.  Management Science Department, School of Business and Public Management, George Washington University, Washington, DC 20052, U.S.A.  Institute of Statistics and Decision Sciences, Duke University, Durham, NC 27708, U.S.A. SUMMARY This paper discusses Bayesian methods for the assessment of the prevalence of a disorder based on data from a two-phase design with a short screening instrument administered at the first phase followed by an in-depth diagnostic instrument given at the second phase. In calculating the posterior distributions of the quantities of interest, for example, the prevalence, sensitivity and specificity, and predictive distributions, we used the Gibbs sampler. We illustrate our approach by assessing the prevalence of depression in adolescents with use of data attained from a two-phase design.  1997 by John Wiley & Sons, Ltd. Stat. Med., Vol. 16, 1121—1133 (1997). (No. of Figures: 5 No. of Tables: 2 No. of Refs: 17) 1. INTRODUCTION In this paper, we develop Bayesian methods for the estimation of the prevalence of a rare disorder in a population. There is a recent interest in the application of Bayesian methodology for screening tests and the prevalence estimation. Notably, Gastwirth, Gatsonis and Iyengar and Johnson and Gastwirth have used Bayesian methods to assess the accuracy of screening tests for AIDS. These authors used large sample approximations, such as the Laplace approximation, to the posterior distributions of the parameters of interest. The Laplace approximation is complic- ated and can break down if the MLE or the posterior mode is close to boundary values that can happen when the likelihood of the prevalence is asymmetric and highly peaked around the boundary values. Another potential problem with the Laplace-type methods is their sensitivity to reparameterization. Here, we present Bayesian analyses for two-phase prevalence tests using Markov chain Monte Carlo methods, such as Gibbs sampling. The Gibbs sampler is straightforward to implement and produces accurate approximations to the posterior and predictive distributions of interest. We consider Bayesian models for the estimation of prevalence and efficiency of the screening instrument in identifying the diagnosis. We can assess efficiency by investigating the posterior distributions of the predictive value positive, predictive value negative, or the sensitivity and specificity of the screening test. We also calculate the predictive distribution of the number of cases with a positive diagnosis and other quantities of interest for a future test. * Correspondence to: A. Erkanli CCC 0277—6715/97/101121—13$17.50 Received August 1995  1997 by John Wiley & Sons, Ltd. Revised June 1996