Bayesian estimation of rates using a hierarchical log-linear model.

Abstract

The standard model for the analysis of rates is the log-linear model where counts are assumed to follow the Poisson distribution and exposures are given. Under this model the rate at which events occur is represented by a regression function that describes the relation between a set of known predictor variables and unknown parameters. While parameter estimates computed by the method of maximum likelihood usually perform well, interval estimates based on the Poisson assumption do not. In the presence of overdispersion, the model is misspecified, generally resulting in subnominal coverage probability for the parameter estimates. This thesis is concerned with the Bayesian estimation of rates when prior belief is expressed in a log-linear model. A Poisson likelihood is specified for the random counts and modeling proceeds in accordance with a Bayesian two-stage prior distribution. At the first stage, the unobserved rates are assigned conjugate Gamma densities with means that satisfy a log-linear model. At the second stage, there are p $+$ 1 unknown hyperparameters: the p-dimensional vector of regression coefficients, and a scale parameter for accommodating overdispersion. The second stage hyperprior is given a vague specification. Two solutions are proposed for obtaining the marginal posterior densities which require evaluating high-dimensional integrals. One solution involves integrating out the regression coefficients using the Laplace method of integrals. The scale parameter is integrated out numerically. The second solution involves implementing several versions of the Gibbs sampling algorithm using the exact conditionals, and conditionals based on analytic approximations. Illustrative examples show that the analytic approximations are quite accurate. The final topic covered in this thesis is model evaluation. The issues of covariate selection and outlier detection are addressed by considering the predictive distribution, and more specifically, the cross-validation distribution (CVD). The CVD is proper even under improper priors and leads to calculation of the pseudo-Bayes factor for model selection. Diagnostics for detecting outliers are also developed by assessing a set of N CVD's against observed data values. The cross-validatory approach has points in common with frequentist strategies aimed at detecting influential observations based on case deletion.