Bayesian quantile regression joint models: Inference and dynamic predictions

In the traditional joint models (JM) of a longitudinal and time-to-event data, a linear mixed model (LMM) assuming normal random error is frequently used to model the longitudinal continuous outcome. However, in many circumstances, the normality assumption cannot be satisfied and LMM is not appropriate to use. In addition, as a mean regression based methods, LMM only models the conditional mean of the longitudinal outcome, thus its application is limited when clinical interest lies in making inference or predictions on median, lower, or upper ends of the outcome variable. In contrast, quantile regression (QR) models provide a more flexible, distribution-free way to study covariate effects at different conditional quantiles of the outcome and it is robust against deviations from normality as well as outlying observations. In addition, the JM framework provides a convenient way to make subject-specific predictions of event probability. However, current predictive algorithms are all based on the traditional JM that uses LMM. In the first paper, we proposed a new version of JM that adopts a linear quantile mixed model (LQMM) for the longitudinal process and we named it quantile regression joint models (QRJM). We developed a Gibbs sampling algorithm based on the location-scale representation of the asymmetric Laplace distribution, assessed its performance through extensive simulation studies, and demonstrated how the QRJM approach can be used for making subject-specific dynamic predictions of the risk of Huntington\u27s disease onset. As another type of time-to-event outcome, recurrent events are commonly encountered in longitudinal biomedical studies. In contrast to survival data, multiple event times are observed in a single subject during the study follow-up. In the second paper, we extended the proposed QRJM in paper 1 to joint analysis of longitudinal and recurrent event data and developed a fully Bayesian algorithm for model inference. In the third paper, we developed a subject-specific dynamic prediction algorithm for recurrent event probability based on the QRJM proposed in paper 2. We conducted extensive simulation studies to validate the proposed algorithm in inference (paper 2) and to quantify its predictive performance (paper 3). In the data applications of paper 2 and 3, we illustrated the flexibility of the QRJM and its advantages over the traditional JM by jointly modeling the risk of coronary heart disease (CHD) recurrences and longitudinal systolic blood pressure (SBP) measurements (paper 2) and by making predictions of the risk of CHD recurrences (paper 3). QRJM was able to provide more insight into the disease progression and the association between the two disease processes in terms of various quantile-based estimations and dynamic predictions