Distributional outcome regression and its application to modelling continuously monitored heart rate and physical activity

We propose a distributional outcome regression (DOR) with scalar and distributional predictors. Distributional observations are represented via quantile functions and the dependence on predictors is modelled via functional regression coefficients. DOR expands existing literature with three key contributions: handling both scalar and distributional predictors, ensuring jointly monotone regression structure without enforcing monotonicity on individual functional regression coefficients, providing a statistical inference for estimated functional coefficients. Bernstein polynomial bases are employed to construct a jointly monotone regression structure without over-restricting individual functional regression coefficients to be monotone. Asymptotic projection-based joint confidence bands and a statistical test of global significance are developed to quantify uncertainty for estimated functional regression coefficients. Simulation studies illustrate a good performance of DOR model in accurately estimating the distributional effects. The method is applied to continuously monitored heart rate and physical activity data of 890 participants of Baltimore Longitudinal Study of Aging. Daily heart rate reserve, quantified via a subject-specific distribution of minute-level heart rate, is modelled additively as a function of age, gender, and BMI with an adjustment for the daily distribution of minute-level physical activity counts. Findings provide novel scientific insights in epidemiology of heart rate reserve.