Additive Functional Cox Model

We propose the additive functional Cox model to flexibly quantify the association between functional covariates and time to event data. The model extends the linear functional proportional hazards model by allowing the association between the functional covariate and log hazard to vary nonlinearly in both the functional domain and the value of the functional covariate. Additionally, we introduce critical transformations of the functional covariate which address the weak model identifiability in areas of information sparsity and discuss their impact on interpretation and inference. We also introduce a novel estimation procedure that accounts for identifiability constraints directly during model fitting. Methods are applied to the National Health and Nutrition Examination Survey 2003–2006 accelerometry data and quantify new and interpretable circadian patterns of physical activity that are associated with all-cause mortality. We also introduce a simple and novel simulation framework for generating survival data with functional predictors which resemble the observed data. The accompanying inferential R software is fast, open source, and publicly available. Our data application and simulations are fully reproducible through the accompanying vignette. Supplementary materials for this article are available online.