Bayesian proportional hazards model with time-varying regression coefficients: a penalized Poisson regression approach

One can fruitfully approach survival problems without covariates in an actuarial way. In narrow time bins, the number of people at risk is counted together with the number of events. The relationship between time and probability of an event can then be estimated with a parametric or semi-parametric model. The number of events observed in each bin is described using a Poisson distribution with the log mean specified using a flexible penalized B-splines model with a large number of equidistant knots. Regression on pertinent covariates can easily be performed using the same log-linear model, leading to the classical proportional hazard model. We propose to extend that model by allowing the regression coefficients to vary in a smooth way with time. Penalized B-splines models will be proposed for each of these coefficients. We show how the regression parameters and the penalty weights can be estimated efficiently using Bayesian inference tools based on the Metropolis-adjusted Langevin algorithm.