Multilevel functional distributional models with application to continuous glucose monitoring in diabetes clinical trials

Continuous glucose monitoring (CGM) is a minimally invasive technology that measures blood glucose every few minutes for weeks or months at a time. CGM data are often collected in the free-living environment and is strongly related to sleep, physical activity and meal intake. As the timing of these activities varies substantially within- and between-individuals, it is difficult to model CGM trajectories as a function of time of day. Therefore, in practice, CGM trajectories are often reduced to one or two scalar summaries of the thousands of measurements collected for a study participant. To alleviate the potential loss of information, the cumulative distribution function (cdf) of the CGM time series was proposed as an alternative. Here we address the problem of conducting inference on cdfs in clinical trials with long follow up and frequent measurements. Our approach provides three major innovations: (1) modeling the entire cdf and preserving its monotonicity; (2) accounting for the cdfs correlation (because they are measured on the same individual), continuity (results are robust to the choice of the probability grid), and differential error (e.g., medians have lower variability than $0.99$ quantiles); and (3) preserving the family-wise error when the observed data are longitudinal samples of cdfs. We focus on modeling data collected by The Juvenile Diabetes Research Foundation Continuous Glucose Monitoring Group in a large clinical trial that collected CGM data every few minutes for 26 weeks. Our basic observation unit is the distribution of CGM observations in a four-week interval. The scientific goals are to: (1) identify and quantify the effects of factors that affect glycaemic control in type 1 diabetes patients (T1D); and (2) identify and characterize the patients who respond to treatment.