In large-scale epidemiological studies, it is often of interest to investigate joint relationships between longitudinal and time-to-event outcomes with exposures that are trajectories or functions. Our motivation study is the Objective Physical Activity and Cardiovascular Health (OPACH) Study, which collected accelerometry-measured physical activity in 6,489 older women. One of the scientific aims is to understand sedentary behavior accumulation patterns and its association with physical function (longitudinal) and mortality (time-to-event). We propose a novel approach that first converts raw accelerometry data into daily sitting bout accumulation profiles, which are treated as functional covariates, and then develop a functional joint model for longitudinal and time-to-event outcomes that incorporates a baseline functional covariate. The longitudinal process is modeled using functional data methods and linked to the survival process through functional principal component scores. Both sub-models include interpretable linear scalar-on-function regression coefficients to capture flexible dose-response associations between sitting accumulated across varying bout durations and health outcomes. Estimation is carried out via an efficient expectation-maximization (EM) algorithm with penalized spline approximations. Simulation studies demonstrate accurate parameter estimation and reliable model selection. Application to the OPACH data reveals flexible and interpretable dose-response relationships between sitting bout durations, physical function, and mortality.

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