  The advent of wearable and sensor technologies now leads to functional
predictors which are intrinsically infinite dimensional. While the existing
approaches for functional data and survival outcomes lean on the
well-established Cox model, the proportional hazard (PH) assumption might not
always be suitable in real-world applications. Motivated by physiological
signals encountered in digital medicine, we develop a more general and flexible
functional time-transformation model for estimating the conditional survival
function with both functional and scalar covariates. A partially functional
regression model is used to directly model the survival time on the covariates
through an unknown monotone transformation and a known error distribution. We
use Bernstein polynomials to model the monotone transformation function and the
smooth functional coefficients. A sieve method of maximum likelihood is
employed for estimation. Numerical simulations illustrate a satisfactory
performance of the proposed method in estimation and inference. We demonstrate
the application of the proposed model through two case studies involving
wearable data i) Understanding the association between diurnal physical
activity pattern and all-cause mortality based on accelerometer data from the
National Health and Nutrition Examination Survey (NHANES) 2011-2014 and ii)
Modelling Time-to-Hypoglycemia events in a cohort of diabetic patients based on
distributional representation of continuous glucose monitoring (CGM) data. The
results provide important epidemiological insights into the direct association
between survival times and the physiological signals and also exhibit superior
predictive performance compared to traditional summary based biomarkers in the
CGM study.
