Local polynomial mixed-effects models for longitudinal data

We consider a nonparametric mixed-effects model y(i)(t(ij)) = eta(t(ij)) + nu(i)(t(ij)) +epsilon(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 fixed-effects (Population) curve eta(t) and random-effects curves nu(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 eta(t) and nu(i)(t). Simulation studies show that our estimator for eta(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 mu(t). When n(i) is bounded and it 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.