Inference for the Extended Functional Cox Model: A UK Biobank Case Study

Multiple studies have shown that scalar summaries of objectively measured physical activity (PA) using accelerometers are the strongest predictors of mortality, outperforming all traditional risk factors, including age, sex, body mass index (BMI), and smoking. Here we show that diurnal patterns of PA and their day-to-day variability provide additional information about mortality. To do that, we introduce a class of extended functional Cox models and corresponding inferential tools designed to quantify the association between multiple functional and scalar predictors with time-to-event outcomes in large-scale (large $n$) high-dimensional (large $p$) datasets. Methods are applied to the UK Biobank study, which collected PA at every minute of the day for up to seven days, as well as time to mortality ($93{,}370$ participants with good quality accelerometry data and $931$ events). Simulation studies show that methods perform well in realistic scenarios and scale up to studies an order of magnitude larger than the UK Biobank accelerometry study. Establishing the feasibility and scalability of these methods for such complex and large data sets is a major milestone in applied Functional Data Analysis (FDA).