Hierarchical functional data with mixed continuous and binary measurements

Motivated by objective measurements of physical activity, we take a functional data approach to longitudinal data with simultaneous measurement of a continuous and a binary outcomes. The regression structures are specified as smooth curves measured at various time-points with random effects that have a hierarchical correlation structure. The random effect curves for each variable are summarized using a few important principal components, and the association of the two longitudinal variables is modeled through the association of the principal component scores. We use penalized splines to model the mean curves and the principal component curves, and cast the proposed model into a mixed effects model framework for model fitting, prediction and inference. Via a quasilikelihood type approximation for the binary component, we develop an algorithm to fit the model. Data-based transformation of the continuous variable and selection of the number of principal components are incorporated into the algorithm. The method is applied to the motivating physical activity data and is evaluated empirically by a simulation study. Extensions for different types of outcomes are also discussed.