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 speci ed using a exible 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 coe cients to vary in a smooth way with time. Penalized B-splines models will be
proposed for each of these coe cients. We show how the regression parameters and the penalty weights
can be estimated e ciently using Bayesian inference tools based on the Metropolis-adjusted Langevin
algorithm.