US EPA

3242930 

Journal Article 

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

Chiang, CT; Rice, JA; Wu, CO 

2001 

Yes 

Journal of the American Statistical Association
ISSN: 0162-1459
EISSN: 1537-274X 

96 

454 

605-619 

WOS:000168986400026 

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. 

asymptotic normality; clinical trials; confidence bands; longitudinal data; mean squared errors; smoothing parameters