An Approximate Bivariate Bayesian Method for Analyzing Small Frequencies

Abstract

An approximate Bayesian method is developed to study joint mortality rates of two cancer sites over several geographic areas. A conditionally independent two-stage hierarchical model is used to model sources of variability in these data. At first stage, conditional on area-specific parameters, within-area variation in joint frequencies of death from two sites is modeled by a bivariate Poisson distribution with means proportional to populations at risk. At second stage, conditional on parameters for prior, between-area variation in joint logits of true mortality rates is modeled by a bivariate normal distribution, the prior distribution. In estimating area mortality rates of one site, it is shown that simultaneous use of data on both sites is superior to that based on a single site only. The relation between two sites is studied using correlation between logits of true mortality rates. A Monte Carlo simulation is conducted to investigate performance of this method as a function of prior correlation. The method is applied to two data sets.