Learning in hierarchical Bayesian models for longitudinal and survival outcomes
Bradley P. Carlin, Division of Biostatistics, University of Minnesota School of Public Health 
*Laura A. Hatfield, Harvard Medical School 
James S. Hodges, Division of Biostatistics, University of Minnesota School of Public Health 

Keywords: repeated measurements, time-to-event data, latent variables, Bayesian learning

In studying the evolution of a disease and effects of treatment on it, investigators may consider repeated measures of disease severity and the time to occurrence of a significant clinical event. The development of joint models for such longitudinal and survival data has benefited from the use of a latent variable framework. This approach posits an individual-specific latent process that evolves in time and contributes to both the longitudinal and survival outcomes. This allows substantial flexibility to incorporate association among outcomes, including association across repeated measurements, among multiple longitudinal outcomes, and between longitudinal and survival outcomes.

The literature has extended the joint modeling framework in many ways, for example semi- and non-parametric approaches to time trends and latent variable distributions, competing events or cure fractions in the survival model, and so on. However, less attention has been paid to the properties of such models. In particular, we are interested in the contributions of each type of data (longitudinal and survival) to Bayesian learning about individual- and population-level parameters. We consider the problem of attributing informational content to the data inputs of joint models, developing analytical and numerical approaches and demonstrating their use.

As a motivating application, we consider a clinical trial for treatment of mesothelioma, a rapidly fatal lung disease. The protocol included patient-reported outcome collection throughout the treatment phase and followed patients until progression or death to determine progression-free survival times. We develop a joint model that can produce clinically relevant treatment effect estimates on several aspects of disease simultaneously and study individual-level variation in disease processes. Then we quantify the contributions of the data sources to inference on parameters of scientific interest and compare the real data results to a few small simulation studies.