We study a Bayesian analysis of generalized linear mixed models using non-informative priors. Mixed models (also known as hierarchical or random-coefficient models) extend the usual generalized linear regression by incorporation of a rich and flexible pattern of correlations among outcomes. The classic non-informative prior for the Gaussian family (Box and Tiao, pg 58, 1992) is currently used as conventional objective priors for these models. In previously published work, the Principal Investigator has shown that this family of priors does not lead to proper posterior distributions for mixed models for binomial responses, thus making them undesirable; further, their use in some situations can lead to seriously misleading results in practice. This research extends the Principal Investigators' previous work by exploring alternative approaches for a non-informative Bayesian analysis of hierarchical models. Four objective specifications will be investigated: Jeffreys's prior, the uniform shrinkage prior, flat priors and the reference conjugate prior, and theoretical and practical recommendations made for their use. The methods developed in this proposal will be used to analyze real data from a double-blind clinical trial to study the efficacy of Rifabutin in the treatment of Crohn's disease. More generally, there will be a horizontal shift of these methods to various disciplines where correlated data routinely arise. Planning activities include: deriving the reference priors and investigating propriety of the resulting posterior distributions; examining methods for sampling from the posterior distributions.
