Bayesian survival analysis in proportional hazard models with logistic relative risk

The traditional Cox proportional hazards regression model uses an exponential relative risk function. We argue that under various plausible scenarios, the relative risk part of the model should be bounded, suggesting also that the traditional model often might overdramatize the hazard rate assessment for individuals with unusual covariates. This motivates our working with proportional hazards models where the relative risk function takes a logistic form. We provide frequentist methods, based on the partial likelihood, and then go on to semiparametric Bayesian constructions. These involve a Beta process for the cumulative baseline hazard function and any prior with a density, for example that dictated by a Jeffreys-type argument, for the regression coefficients. The posterior is derived using machinery for Levy processes, and a simulation recipe is devised for sampling from the posterior distribution of any quantity. Our methods are illustrated on real data. A Bernsteın–von Mises theorem is reached for our class of semiparametric priors, guaranteeing asymptotic normality of the posterior processes.