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 Alzheimers 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.
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