A TWO-STEP SMOOTHING METHOD FOR VARYING-COEFFICIENT MODELS WITH REPEATED MEASUREMENTS

COLIN O. WU^{1}* , KAI FUN YU^{2} AND CHIN-TSANG CHIANG^{3}

1Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, MD 21218, U.S.A. 2Division of Epidemiology, Statistics and Prevention Research, National Institute of Child Health and Human Development, Bethesda, MD 20852, U.S.A. 3Department of Statistics, Tunghai University, Taichung, Taiwan

(Received May 6, 1998; revised February 15, 1999)

Abstract. Datasets involving repeated measurements over time are common in medical trials and epidemiological cohort studies. The outcomes and covariates are usually observed from randomly selected subjects, each at a set of possibly unequally spaced time design points. One useful approach for evaluating the effects of covariates is to consider linear models at a specific time, but the coefficients are smooth curves over time. We show that kernel estimators of the coefficients that are based on ordinary local least squares may be subject to large biases when the covariates are time-dependent. As a modification, we propose a two-step kernel method that first centers the covariates and then estimates the curves based on some local least squares criteria and the centered covariates. The practical superiority of the two-step kernel method over the ordinary least squares kernel method is shown through a fetal growth study and simulations. Theoretical properties of both the two-step and ordinary least squares kernel estimators are developed through their large sample mean squared risks.

Key words and phrases: Bandwidth, kernel, longitudinal data, mean squared error, ultrasound measurement, varying coefficient models.