Marginal posterior distributions for regression parameters in the Cox model using Dirichlet and gamma process priors

A Bayesian treatment of the proportional hazard (PH) model is revisited using Dirichlet process and gamma process priors for the baseline survival and cumulative hazard functions respectively. Such priors, due to their discrete support, conflict with the absolutely continuous nature of survival responses expressed through the hazard function structure that defines the PH model. We resolve this conflict through the use of a proposed epsilon-grid likelihood approach which we apply to the PH model to consider marginal inference for the regression parameter beta using expansions as epsilon down arrow 0. Using Dirichlet process priors, the epsilon-grid likelihood approach leads to a new explicit marginal posterior density expansion for beta which accommodates the most general setting with arbitrary ties and right censoring. In the previously considered case of gamma process priors, the approach extends the results of Kalbfleisch (1978) to deal with arbitrary arrangements of ties and also provides a rigorous justification for his posterior expressions. As in Kalbfleisch (1978), we show that the leading terms of both expansions approximate Cox's partial likelihood when there are no ties and the process priors are diffuse. The epsilon-grid likelihood approach is similar in concept to the grouped data likelihood approach of Sinha et al. (2003) but differs in the likelihood approximation. Whereas our expansions are Poincare expansions (with relative error O(epsilon) as epsilon down arrow 0), those of Sinha et al. (2003) are stochastic expansions and not proper Poincare expansion (as we show), so that they lack relative error O(epsilon). Using a flat improper prior for beta, the two marginal posterior expressions for beta are shown to be integrable for general arrangements of ties and censoring under weak conditions on the design and failures. (C) 2021 Elsevier B.V. All rights reserved.