A Marginal Approach to Reduced-Rank Penalized Spline Smoothing With Application to Multilevel Functional Data

Multilevel functional data are collected in many biomedical studies. For example, in a study of the effect of Nimodipine on patients with subarachnoid hemorrhage (SAH), patients underwent multiple 4-hr treatment cycles. Within each treatment cycle, subjects' vital signs were reported every 10 min. These data have a natural multilevel structure with treatment cycles nested within subjects and measurements nested within cycles. Most literature on nonparametric analysis of such multilevel functional data focuses on conditional approaches using functional mixed effects models. However, parameters obtained from the conditional models do not have direct interpretations as population average effects. When population effects are of interest, we may employ marginal regression models. In this work, we propose marginal approaches to fit multilevel functional data through penalized spline generalized estimating equation (penalized spline GEE). The procedure is effective for modeling multilevel correlated generalized outcomes as well as continuous outcomes without suffering from numerical difficulties. We provide a variance estimator robust to misspecification of correlation structure. We investigate the large sample properties of the penalized spline GEE estimator with multilevel continuous data and show that the asymptotics falls into two categories. In the small knots scenario, the estimated mean function is asymptotically efficient when the true correlation function is used and the asymptotic bias does not depend on the working correlation matrix. In the large knots scenario, both the asymptotic bias and variance depend on the working correlation. We propose a new method to select the smoothing parameter for penalized spline GEE based on an estimate of the asymptotic mean squared error (MSE). We conduct extensive simulation studies to examine property of the proposed estimator under different correlation structures and sensitivity of the variance estimation to the choice of smoothing parameter. Finally, we apply the methods to the SAH study to evaluate a recent debate on discontinuing the use of Nimodipine in the clinical community. Supplementary materials for this article are available online.