Modelling a response as a function of high-frequency count data: The association between physical activity and fat mass

Accelerometers are widely used in health sciences, ecology and other application areas. They quantify the intensity of physical activity as counts per epoch over a given period of time. Currently, health scientists use very lossy summaries of the accelerometer time series, some of which are based on coarse discretisation of activity levels, and make certain implicit assumptions, including linear or constant effects of physical activity. We propose the histogram as a functional summary for achieving a near lossless dimension reduction, comparability between individual time series and easy interpretability. Using the histogram as a functional summary avoids registration of accelerometer counts in time. In our novel method, a scalar response is regressed on additive multi-dimensional functional predictors, including the histogram of the high-frequency counts, and additive non-linear predictors for other continuous covariates. The method improves on the current state-of-the art, as it can deal with high-frequency time series of different lengths and missing values and yields a flexible way to model the physical activity effect with fewer assumptions. It also allows the commonly made modelling assumptions to be tested. We investigate the relationship between the response fat mass and physical activity measured by accelerometer, in data from the Avon Longitudinal Study of Parents and Children. Our method allows testing of whether the effect of physical activity varies over its intensity by gender, by time of day or by day of the week. We show that meaningful interpretation requires careful treatment of identifiability constraints in the light of the sum-to-one property of a histogram. We find that the (not necessarily causal) effect of physical activity on kg fat mass is not linear and not constant over the activity intensity.