Smoothing Spline Estimation for Varying Coefficient Models With Repeatedly Measured Dependent Variables

• Published in: Journal of the American Statistical Association Vol. 96; no. 454; pp. 605 - 619
• Main Authors: Chiang, Chin-Tsang, Rice, John A, Wu, Colin O
• Format: Journal Article
• Language: English
• Published: Alexandria, VA Taylor & Francis 01.06.2001; American Statistical Association; Taylor & Francis Ltd
• Subjects: Applications; Asymptotic normality; Asymptotic properties; Clinical research; Clinical trials; Componentwise operations; Confidence bands; Data analysis; Data smoothing; Estimation methods; Estimators; Exact sciences and technology; Logical proofs; Longitudinal data; Mathematical procedures; Mathematics; Mean squared errors; Medical sciences; Modelling; Nonparametric inference; Observational studies; Opioid analgesics; Opioids; Parametric inference; Probability and statistics; Sciences and techniques of general use; Simulation; Smoothing parameters; Statistical analysis; Statistical data; Statistical models; Statistical variance; Statistics; Theory and Methods; Variables; Biometrics; Parameter estimation; Confidence limit; Non parametric estimation; Tolerance limit; Cross validation; Covariate; Stochastic process; Penalty method; Spline; Spline approximation; Mean square error; Probability density; Uniform distribution; Repeated measurement; Bootstrap; Clinical trial; Smoothing methods; Asymptotic behavior; Statistical estimation; Mean value; Green function; Time response; Medicine; Statistical method; Simulation; Response time; Smoothing; Asymptotic normality
• ISSN: 0162-1459; 1537-274X
• DOI: 10.1198/016214501753168280

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)β(t) + ϵ(t), where Y(t) and X(t) = (X (0) (t),...,X (k) (t)) T are the response and covariates at time t, β(t) = (β 0 (t),...,β k (t)) T are smooth coefficient curves of t and ϵ(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 β 0 (t),...,β 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.