Functional proportional hazards mixture cure model and its application to modelling the association between cancer mortality and physical activity in NHANES 2003-2006

We develop a functional proportional hazards mixture cure (FPHMC) model with scalar and functional covariates measured at the baseline. The mixture cure model, useful in studying populations with a cure fraction of a particular event of interest is extended to functional data. We employ the EM algorithm and develop a semiparametric penalized spline-based approach to estimate the dynamic functional coefficients of the incidence and the latency part. The proposed method is computationally efficient and simultaneously incorporates smoothness in the estimated functional coefficients via roughness penalty. Simulation studies illustrate a satisfactory performance of the proposed method in accurately estimating the model parameters and the baseline survival function. Finally, the clinical potential of the model is demonstrated in two real data examples that incorporate rich high-dimensional biomedical signals as functional covariates measured at the baseline and constitute novel domains to apply cure survival models in contemporary medical situations. In particular, we analyze i) minute-by-minute physical activity data from the National Health and Nutrition Examination Survey (NHANES) 2003-2006 to study the association between diurnal patterns of physical activity (PA) at baseline and all cancer mortality through 2019 while adjusting for other biological factors; ii) the impact of daily functional measures of disease severity collected in the intensive care unit on post ICU recovery and mortality event. Our findings provide novel epidemiological insights into the association between daily patterns of PA and cancer mortality. Software implementation and illustration of the proposed estimation method is provided in R.