Joint modelling of longitudinal and survival data in the presence of competing risks with applications to prostate cancer data

Abstract

This research is motivated from the data from a large Selenium and Vitamin E Cancer Prevention Trial (SELECT). The prostate specific antigens (PSAs) were collected longitudinally, and the survival endpoint was the time to low-grade cancer or the time to high-grade cancer (competing risks). In this article, the goal is to model the longitudinal PSA data and the time-to-prostate cancer (PC) due to low- or high-grade. We consider the low-grade and high-grade as two competing causes of developing PC. A joint model for simultaneously analysing longitudinal and time-to-event data in the presence of multiple causes of failure (or competing risk) is proposed within the Bayesian framework. The proposed model allows for handling the missing causes of failure in the SELECT data and implementing an efficient Markov chain Monte Carlo sampling algorithm to sample from the posterior distribution via a novel reparameterization technique. Bayesian criteria, $\Delta$ DIC ${}_{\mathit{Surv}}$ , and $\Delta$ WAIC ${}_{\mathit{Surv}}$ , are introduced to quantify the gain in fit in the survival sub-model due to the inclusion of longitudinal data. A simulation study is conducted to examine the empirical performance of the posterior estimates as well as $\Delta$ DIC ${}_{\mathit{Surv}}$ and $\Delta$ WAIC ${}_{\mathit{Surv}}$ and a detailed analysis of the SELECT data is also carried out to further demonstrate the proposed methodology.