A dynamic model for survival data with longitudinal covariates

Analyses involving both longitudinal and time-to-event data are quite common in medical research. The primary goal of such studies may be to simultaneously study the effect of treatment on both the longitudinal covariate and survival, but secondary objectives, such as understanding the within-patients patterns of change of the time-dependent marker, or the relationship between the marker's profiles and the occurrence of the event of interest, are often considered. Currently available methods of analyzing survival and longitudinal data usually introduce many undesirable and sometimes unreasonable assumptions. We introduce two flexible Bayesian hierarchical modeling approaches for analyzing these two types of data by use of dynamic models and survival analysis methods. In both approaches the longitudinal covariate is modeled via dynamic hierarchical models which allow the shapes of the longitudinal trajectories to be determined by the data rather than by the assumed parametric model. The trajectories are patient specific, and the link between them is provided by the hierarchical structure of the model, allowing borrowing of strength across patients. In the survival part of the model, the first method, referred to as 2-Stage model, uses the estimates of the longitudinal trajectories obtained in the first stage of the analysis, as a time-dependent covariate in the Cox PH model, to find the estimates of the corresponding survival model parameters. The second approach, called the joint model, assumes the piecewise exponential distribution for the event times of the patients and uses a discretized version of the Cox PH model. Some of the parameters of this survival model are also allowed to change over time, and again, dynamic models provide the description of the stochastic evolution of these parameters. A combination of various MCMC techniques is used to obtain a sample from the joint posterior distributions of all the model parameters. This distribution combines the likelihood of the longitudinal and survival data and the prior knowledge about the parameters. Simulation studies provide the measure of the quality of the method and both models are compared to one of the currently existing approaches