A Bayesian semiparametric survival model with longitudinal markers.

We consider inference for data from a clinical trial of treatments for metastatic prostate cancer. Patients joined the trial with diverse prior treatment histories. The resulting heterogeneous patient population gives rise to challenging statistical inference problems when trying to predict time to progression on different treatment arms. Inference is further complicated by the need to include a longitudinal marker as a covariate. To address these challenges, we develop a semiparametric model for joint inference of longitudinal data and an event time. The proposed approach includes the possibility of cure for some patients. The event time distribution is based on a nonparametric Pólya tree prior. For the longitudinal data we assume a mixed effects model. Incorporating a regression on covariates in a nonparametric event time model in general, and for a Pólya tree model in particular, is a challenging problem. We exploit the fact that the covariate itself is a random variable. We achieve an implementation of the desired regression by factoring the joint model for the event time and the longitudinal outcome into a marginal model for the event time and a regression of the longitudinal outcomes on the event time, i.e., we implicitly model the desired regression by modeling the reverse conditional distribution.