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We develop a functional proportional hazards mixture cure (FPHMC) model withscalar and functional covariates measured at the baseline. The mixture curemodel, useful in studying populations with a cure fraction of a particularevent of interest is extended to functional data. We employ the EM algorithmand develop a semiparametric penalized spline-based approach to estimate thedynamic functional coefficients of the incidence and the latency part. Theproposed method is computationally efficient and simultaneously incorporatessmoothness in the estimated functional coefficients via roughness penalty.Simulation studies illustrate a satisfactory performance of the proposed methodin accurately estimating the model parameters and the baseline survivalfunction. Finally, the clinical potential of the model is demonstrated in tworeal data examples that incorporate rich high-dimensional biomedical signals asfunctional covariates measured at the baseline and constitute novel domains toapply cure survival models in contemporary medical situations. In particular,we analyze i) minute-by-minute physical activity data from the National Healthand Nutrition Examination Survey (NHANES) 2003-2006 to study the associationbetween diurnal patterns of physical activity (PA) at baseline and all cancermortality through 2019 while adjusting for other biological factors; ii) theimpact of daily functional measures of disease severity collected in theintensive care unit on post ICU recovery and mortality event. Our findingsprovide novel epidemiological insights into the association between dailypatterns of PA and cancer mortality. Software implementation and illustrationof the proposed estimation method is provided in R.

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