Local Polynomial Mixed-Effects Models for Longitudinal Data

• Published in: Journal of the American Statistical Association Vol. 97; no. 459; pp. 883 - 897
• Main Authors: Wu, Hulin, Zhang, Jin-Ting
• Format: Journal Article
• Language: English
• Published: Alexandria, VA Taylor & Francis 01.09.2002; American Statistical Association; Taylor & Francis Ltd
• Subjects: Bandwidth selection; Bandwidths; Correlations; Data models; Data smoothing; Distribution theory; Estimation methods; Estimators; Exact sciences and technology; Leave-one-point-out cross-validation; Leave-one-subject-out cross-validation; Linear inference, regression; Linear mixed-effects model; Local polynomial smoothing; Longitudinal data; Mathematical methods; Mathematics; Multivariate analysis; Nonparametric mixed-effects model; Nonparametric models; Parametric inference; Polynomials; Population estimates; Probability and statistics; Sciences and techniques of general use; Statistical analysis; Statistical methods; Statistical variance; Statistics; Theory and Methods; Rank statistic; Statistical distribution; Error estimation; Bias; Selection; Bandwith selection; Non parametric estimation; Cross validation; Polynomial method; Multivariate analysis; Statistical simulation; Asymptotic variance; Linear model; Parametric method; Asymptotic bias; Fixed effect model; Consistent estimator; Bandwidth; Random effect; Approximation theory; Data structure; Covariance analysis; Linear regression; Statistical association; Statistical estimation; Fixed effect; Variance analysis; Mean estimation; Kernel method; Statistical regression; Selection problem; Correlation analysis; Smoothing; Asymptotic normality; Mixed model; Influence; Estimating equation; Models; Asymptotic approximation; Mean error; Polynomial regression; Biased estimation
• ISSN: 0162-1459; 1537-274X
• DOI: 10.1198/016214502388618672

We consider a nonparametric mixed-effects model y i (t ij )=η(t ij )+v i (t ij )+ϵ i (t ij ),j=1,2,...,n i ;i=1,2,...,n for longitudinal data. We propose combining local polynomial kernel regression and linear mixed-effects (LME) model techniques to estimate both fixedeffects (population) curve η(t) and random-effects curves v i (t). The resulting estimator, called the local polynomial LME (LLME) estimator, takes the local correlation structure of the longitudinal data into account naturally. We also propose new bandwidth selection strategies for estimating η(t) and v i (t). Simulation studies show that our estimator for η(t) is superior to the existing estimators in the sense of mean squared errors. The asymptotic bias, variance, mean squared errors, and asymptotic normality are established for the LLME estimators of η(t). When n i is bounded and n tends to infinity, our LLME estimator converges in a standard nonparametric rate, and the asymptotic bias and variance are essentially the same as those of the kernel generalized estimating equation estimator proposed by Lin and Carroll. But when both n i and n tend to infinity, the LLME estimator is consistent with a slower rate of n 1/2 compared to the standard nonparametric rate, due to the existence of within-subject correlations of longitudinal data. We illustrate our methods with an application to a longitudinal dataset.