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)=XT(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) = (β0(t),…,βk(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 β0(t),…,βk(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.

[1]  Ing Rj Ser.  Approximation Theorems of Mathematical Statistics , 1980 .

[2]  Jianqing Fan.  Design-adaptive Nonparametric Regression , 1992 .

[3]  B. Silverman,et al.  Spline Smoothing: The Equivalent Variable Kernel Method , 1984 .

[4]  D. Redmond,et al.  CLONIDINE BLOCKS ACUTE OPIATE-WITHDRAWAL SYMPTOMS , 1978, The Lancet.

[5]  Karen Messer,et al.  A Comparison of a Spline Estimate to its Equivalent Kernel Estimate , 1991 .

[6]  B. Yandell.  Spline smoothing and nonparametric regression , 1989 .

[7]  Chin-Tsang Chiang,et al.  Asymptotic Confidence Regions for Kernel Smoothing of a Varying-Coefficient Model With Longitudinal Data , 1998 .

[8]  P. Diggle,et al.  Semiparametric models for longitudinal data with application to CD4 cell numbers in HIV seroconverters. , 1994, Biometrics.

[9]  J. Hart,et al.  Kernel Regression Estimation Using Repeated Measurements Data , 1986 .

[10]  M. Wand,et al.  Multivariate Locally Weighted Least Squares Regression , 1994 .

[11]  Naomi Altman,et al.  Kernel Smoothing of Data with Correlated Errors , 1990 .

[12]  L. Skovgaard.  NONLINEAR MODELS FOR REPEATED MEASUREMENT DATA. , 1996 .

[13]  Peter Craven,et al.  Smoothing noisy data with spline functions , 1978 .

[14]  K. Preston,et al.  Naltrexone shortened opioid detoxification with buprenorphine. , 1999, Drug and alcohol dependence.

[15]  M. C. Jones,et al.  Spline Smoothing and Nonparametric Regression. , 1989 .

[16]  Peter J. Diggle,et al.  RATES OF CONVERGENCE IN SEMI‐PARAMETRIC MODELLING OF LONGITUDINAL DATA , 1994 .

[17]  L. Goldstein,et al.  A New Class of Kernels for Nonparametric Curve Estimation , 1993 .

[18]  Lee-Jen Wei,et al.  Inferences for a semiparametric model with panel data , 2000 .

[19]  G. Wahba.  Spline models for observational data , 1990 .

[20]  Jianqing Fan,et al.  Two‐step estimation of functional linear models with applications to longitudinal data , 1999 .

[21]  Douglas W. Nychka,et al.  Splines as Local Smoothers , 1995 .

[22]  D. Cox.  Asymptotics for $M$-Type Smoothing Splines , 1983 .

[23]  Joan G. Staniswalis,et al.  Nonparametric Regression Analysis of Longitudinal Data , 1998 .

[24]  E. Vonesh,et al.  Linear and Nonlinear Models for the Analysis of Repeated Measurements , 1996 .

[25]  R. Serfling.  Approximation Theorems of Mathematical Statistics , 1980 .

[26]  Chin-Tsang Chiang,et al.  KERNEL SMOOTHING ON VARYING COEFFICIENT MODELS WITH LONGITUDINAL DEPENDENT VARIABLE , 2000 .

[27]  B. Silverman,et al.  Estimating the mean and covariance structure nonparametrically when the data are curves , 1991 .

[28]  R. Carroll,et al.  Nonparametric Function Estimation for Clustered Data When the Predictor is Measured without/with Error , 2000 .

[29]  P. Diggle.  Analysis of Longitudinal Data , 1995 .

[30]  Ordinary Differential Equations, a first course. By F. Brauer and J. A. Nohel. Pp. xvi, 457. 1967. (W. A. Benjamin Inc., New York.) , 1972 .

[31]  Jianqing Fan,et al.  Local polynomial modelling and its applications , 1994 .

[32]  P. Diggle.  An approach to the analysis of repeated measurements. , 1988, Biometrics.

[33]  Li Ping Yang,et al.  Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data , 1998 .

[34]  J. Ware,et al.  Random-effects models for longitudinal data. , 1982, Biometrics.

[35]  R. Tibshirani,et al.  Varying‐Coefficient Models , 1993 .

[36]  S. Zeger,et al.  Longitudinal data analysis using generalized linear models , 1986 .