Bayesian two-stage modeling of longitudinal and time-to-event data with an integrated fractional Brownian motion covariance structure.

It is difficult to characterize complex variations of biological processes, often longitudinally measured using biomarkers that yield noisy data. While joint modeling with a longitudinal submodel for the biomarker measurements and a survival submodel for assessing the hazard of events can alleviate measurement error issues, the continuous longitudinal submodel often uses random intercepts and slopes to estimate both between- and within-patient heterogeneity in biomarker trajectories. To overcome longitudinal submodel challenges, we replace random slopes with scaled integrated fractional Brownian motion (IFBM). As a more generalized version of integrated Brownian motion, IFBM reasonably depicts noisily measured biological processes. From this longitudinal IFBM model, we derive novel target functions to monitor the risk of rapid disease progression as real-time predictive probabilities. Predicted biomarker values from the IFBM submodel are used as inputs in a Cox submodel to estimate event hazard. This two-stage approach to fit the submodels is performed via Bayesian posterior computation and inference. We use the proposed approach to predict dynamic lung disease progression and mortality in women with a rare disease called lymphangioleiomyomatosis who were followed in a national patient registry. We compare our approach to those using integrated Ornstein-Uhlenbeck or conventional random intercepts-and-slopes terms for the longitudinal submodel. In the comparative analysis, the IFBM model consistently demonstrated superior predictive performance.