Smoothing spline estimation for varying coefficient models with repeatedly measured dependent variables

Longitudinal samples, i.e., datasets with repeated measurements over time, are common in biomedical and epidemiological studies such as clinical trials and cohort observational studies. An exploratory tool for the analyses of such data is the varying coefficient model Y(t) = X-T(t)beta (t) + is an element of (t), where Y(t) and X(t) = (X-(0) (t),...,X-(k)(t))(T) are the response and covariates at time t, beta (t) = (beta (0)(t),...., beta (k) (t))(T) are smooth coefficient curves of t and E(t) is a mean zero stochastic process. A special case that is of particular interest in many situations is data with time-dependent response and time-independent covariates. We propose in this article a componentwise smoothing spline method for estimating beta (0)(t),..., beta (k)(t) nonparametrically based on the previous varying coefficient model and a longitudinal sample of (t, Y(t),X) with time-independent covariates X = (X-(0),...,X-(k))(T) from n independent subjects. A "leave-one-subject-out" cross-validation is suggested to choose the smoothing parameters. Asymptotic properties of our spline estimators are developed through the explicit expressions of their asymptotic normality and risk representations, which provide useful insights for inferences. Applications and finite sample properties of our procedures are demonstrated through a longitudinal sample of opioid detoxification and a simulation study.