Theory and Applications of Bayesian Lag Joint Model (BLJM) for Right-censored Time-to-event Data

Fluctuation often implies unknown risks among all walks of life, especially in stock market, global temperature and medicine. I propose to utilize a new measurement, defined by arc length in mathematical literature, to capture fluctuations or variations of longitudinal risk factors. Specifically, its advantages can be fused into our proposed Bayesian Lag Joint Model (BLJM) to address the research problem of interest in survival analysis. Most statistical methods built upon the Cox proportional hazards model assume the hazard rate at a specific time is impacted by either the baseline or the instantaneous value (time-dependent Cox model) of covariate. But this assumption is often questionable when covariates demonstrate cumulative effects. That is, both the past and the current status of the covariate impact the hazard. Meanwhile, the lag models in survival analysis by Gasparrini et al. (2010, 2014, 2017) and Bender et al. (2018, 2019) utilized a general functional form to capture cumulative effects and added this function to the hazard modeling. However, their specific requirements, such as the fine partition of follow-up times, can become problematic. A parallel extensive literature on joint modeling of longitudinal and survival data provides nice model structures for comprehensive applications. Their advantages rely on the unique association structure (current value or the area under the curve) between the event and the longitudinal trajectory. The established structures, nevertheless, fail to capture fluctuations in one way or another. In BLJM, the three parallel components (Survival Joint Model, Distributed Lag Nonlinear Model and Arc Length) are fused into one united framework. The main purpose of arc length is to capture and reflect the cumulative effect of variations of covariates. The larger value of arc length often suggests the greater risk. Our proposed model can capture more useful information from these longitudinal covariates to improve the estimation of hazard rates. The cumulative effect of variation assessed by arc length can be calculated as an indicator to identify high-risk populations. In summary, BLJM consists of two submodels within the context of joint models: one for survival data and the other for longitudinal data. The two submodels are linked by cumulative effects through arc length. Its major goal is to achieve the high-accuracy in dynamic prediction of hazards. When variations of a longitudinal factor pose a significant impact on the hazard, BLJM outperforms joint models with AUC. It can be used to identify possible biomarkers for future cancer treatments. It can be performed in various applications and disciplines. We also illustrate its usage in both simulation and clinical studies.