Using Wavelet-Based Functional Mixed Models to Characterize Population Heterogeneity in Accelerometer Profiles: A Case Study.

We present a case study illustrating the challenges of analyzing accelerometer data taken from a sample of children participating in an intervention study designed to increase physical activity. An accelerometer is a small device worn on the hip that records the minute-by-minute activity levels of the child throughout the day for each day it is worn. The resulting data are irregular functions characterized by many peaks representing short bursts of intense activity. We model these data using the wavelet-based functional mixed model. This approach incorporates multiple fixed effect and random effect functions of arbitrary form, the estimates of which are adaptively regularized using wavelet shrinkage. The method yields posterior samples for all functional quantities of the model, which can be used to perform various types of Bayesian inference and prediction. In our case study, a high proportion of the daily activity profiles are incomplete, i.e. have some portion of the profile missing, so cannot be directly modeled using the previously described method. We present a new method for stochastically imputing the missing data that allows us to incorporate these incomplete profiles in our analysis. Our approach borrows strength from both the observed measurements within the incomplete profiles and from other profiles, from the same child as well as other children with similar covariate levels, while appropriately propagating the uncertainty of the imputation throughout all subsequent inference. We apply this method to our case study, revealing some interesting insights into children's activity patterns. We point out some strengths and limitations of using this approach to analyze accelerometer data.