Bayesian Functional Mixed Models for Survival Responses with Application to Prostate Cancer

In this chapter, we propose a flexible approach to model functional measurements for survival outcomes. Often the class of models for functional observations are assumed to be linear, which may be too restrictive in some cases. We propose an alternative model, in which the simple linear mixed model has been modified by a more flexible semiparametric spline-based functional mixed model, wherein the usage of splines simplifies parameterizations and the joint modeling framework allows synergistic benefit between the regression of functional predictors and the modeling of survival data. We explicitly model the number and location of change points such that our formulation allows for an unknown set of basis functions characterizing the population-averaged and patient-specific trajectories. In addition, we propose a novel auxiliary variable scheme for a fully Bayesian estimation of our model, which not only allows dimension reduction of the parameter space but also allows efficient sampling from the conditional distributions. We illustrate our approach with a recent prostate cancer clinical trial study.