Longitudinal functional additive model with continuous proportional outcomes for physical activity data.

Motivated by physical activity data obtained from the BodyMedia FIT device (www.bodymedia.com), we take a functional data approach for longitudinal studies with continuous proportional outcomes. The functional structure depends on three factors. In our three-factor model, the regression structures are specified as curves measured at various factor-points with random effects that have a correlation structure. The random curve for the continuous factor is summarized using a few important principal components. The difficulties in handling the continuous proportion variables are solved by using a quasilikelihood type approximation. We develop an efficient algorithm to fit the model, which involves the selection of the number of principal components. The method is evaluated empirically by a simulation study. This approach is applied to the BodyMedia data with 935 males and 84 consecutive days of observation, for a total of 78, 540 observations. We show that sleep efficiency increases with increasing physical activity, while its variance decreases at the same time.