Large sample properties and confidence bands for component-wise varying-coefficient regression with longitudinal dependent variable
Longitudinal studies with repeatedly measured dependent variable (out-come) and time-invariant covariates are common in biomedical and epidemi-ological studies. A useful statistical tool to evaluate the effects of covariates on the outcome variable over time is the varying-coefficient regression, which considers a linear relationship between the covariates and the outcome at a specific time point but assumes the linear coefficients to be smooth curves over time. In order to provide adequate smoothing for each coefficient curve, Wu and Chiang ( 1999 ) proposed a class of component-wise kernel estimators and determined the large sample convergence rates and some of the constant terms of the mean squared errors of their estimators. In this paper we calcu¬late the explicit large sample mean squared errors, including the convergence rates and ail the constant terms, and the asymptotic distributions of the kernel estimators of Wu and Chiang ( 1999 ). These asymptotic distributions are used to construct point-wise confidence intervals and Bonferroni-type confidence bands for the coefficient curves. Through a Monte Carlo simulation, wre show that our confidence regions have adequate coverage probabilities. Applying our procedures to a NIH fetal growth study, we show that our procedures are useful to determine the effects of maternal height, cigarette smoking and al¬cohol consumption on the growth of fetal abdominal circumference over time during pregnancy.