  Wearable data is a rich source of information that can provide deeper
understanding of links between human behaviours and human health. Existing
modelling approaches use wearable data summarized at subject level via scalar
summaries using regression techniques, temporal (time-of-day) curves using
functional data analysis (FDA), and distributions using distributional data
analysis (DDA). We propose to capture temporally local distributional
information in wearable data using subject-specific time-by-distribution (TD)
data objects. Specifically, we propose scalar on time-by-distribution
regression (SOTDR) to model associations between scalar response of interest
such as health outcomes or disease status and TD predictors. We show that TD
data objects can be parsimoniously represented via a collection of time-varying
L-moments that capture distributional changes over the time-of-day. The
proposed method is applied to the accelerometry study of mild Alzheimer's
disease (AD). Mild AD is found to be significantly associated with reduced
maximal level of physical activity, particularly during morning hours. It is
also demonstrated that TD predictors attain much stronger associations with
clinical cognitive scales of attention, verbal memory, and executive function
when compared to predictors summarized via scalar total activity counts,
temporal functional curves, and quantile functions. Taken together, the present
results suggest that the SOTDR analysis provides novel insights into cognitive
function and AD.
