Functional varying coefficient models for longitudinal data

tion (DMS-08-06199). We are grateful to two referees and an Associate Editor for their constructive comments and careful reading. Summary The proposed functional varying coefficient model provides a versatile and flexible analysis tool for relating longitudinal responses to longitudinal predictors. Specifically , this approach provides a novel representation of varying coefficient functions through suitable auto-and cross-covariances of the underlying stochastic processes, which is particularly advantageous for sparse and irregular designs, as often encountered in longitudinal studies. Existing methodology for varying coefficient models is not adapted to such data. The proposed approach extends the customary varying coefficient models to a more general setting, in which not only current but also recent past values of the predictor time course may have an impact on the current value of the response time course. The influence of past predictor values is modeled by a smooth history index function, while the effects on the response are described by smooth varying coefficient functions. The resulting estimators for varying coefficient and history index functions are shown to be asymptotically consistent for sparse designs. In addition, prediction of unobserved response trajectories from sparse measurements on a predictor trajectory is obtained, along with asymptotic pointwise confidence bands. The proposed methods perform well in simulations, especially when compared with commonly used local polynomial smoothing methods for varying coefficient models, and are illustrated with longitudinal primary biliary liver cirrhosis data. The data application and detailed assumptions and proofs can be found in online Supplemental Material.