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Two-Stage Functional Mixed Models for Evaluating the Effect of Longitudinal Covariate Profiles on a Scalar Outcome

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The Daily Hormone Study, a substudy of the Study of Women's Health Across the Nation (SWAN) consisting of more than 600 pre- and perimenopausal women, includes a scalar measure of total hip bone mineral density (BMD) together with repeated measures of creatinine-adjusted follicle stimulating hormone (FSH) assayed from daily urine samples collected over one menstrual cycle. It is of scientific interest to investigate the effect of the FSH time profile during a menstrual cycle on total hip BMD, adjusting for age and body mass index. The statistical analysis is challenged by several features of the data: (1) the covariate FSH is measured longitudinally and its effect on the scalar outcome BMD may be complex; (2) due to varying menstrual cycle lengths, subjects have unbalanced longitudinal measures of FSH; and (3) the longitudinal measures of FSH are subject to considerable among- and within-subject variations and measurement errors. We propose a measurement error partial functional linear model, where repeated measures of FSH are modeled using a functional mixed effects model and the effect of the FSH time profile on BMD is modeled using a partial functional linear model by treating the unobserved true subject-specific FSH time profile as a functional covariate. We develop a two-stage nonparametric regression calibration method using period smoothing splines. Using the connection between smoothing splines and mixed models, we show that a key feature of our approach is that estimation at both stages can be conveniently cast into a unified mixed model framework. A simple testing procedure for constant functional covariate effect is also proposed. The proposed methods are evaluated using simulation studies and applied to the SWAN data.
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Keywords: Longitudinal covariates; Mixed effects models; Nonparametric regression; Periodic smoothing spline; Restricted maximum likelihood; Variance components

Document Type: Research Article

Affiliations: 1: Department of Biostatistics, Harvard School of Public Health, Boston, Massachusetts 02115, U.S.A. 2: Department of Epidemiology, University of Michigan, Ann Arbor, Michigan 48109, U.S.A.

Publication date: 2007-06-01

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