Weighted biomarker variability in joint analysis of longitudinal and time-to-event data

Motivated by the clinical evidence that the biomarker variability may have prognostic value for a related disease, we extend the standard joint model for longitudinal and time-to-event outcomes to incorporate the weighted cumulative effects of both biomarker level and variability on the survival hazard. A mixed-effects model is specified for biomarker observations wherein the subject-specific trajectories are modelled by spline functions with random coefficients. Borrowing ideas from smoothing splines, we propose a new variability measure which characterizes the roughness of the subject-specific biomarker trajectory by the integrated amount of its second derivatives over time. The inclusion of weight functions in cumulative quantities permits the importance of biomarker history to vary with time. To reduce computational complexity, we confine the weight functions to a particular parametric family with scale parameters to be estimated. Asymptotic properties of maximum likelihood estimators are established with a discussion on the identification issue of the scale parameters. We use EM algorithm in estimation with initial values obtained from a two-stage method. Simulation studies have been conducted under different settings. Finally, we apply our model to investigate the weighted cumulative effects of systolic blood pressure level and variability on cardiovascular events in the Medical Research Council trial.