Two-stage functional mixed models for evaluating the effect of longitudinal covariate profiles on a scalar outcome.
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.
Zhang
Daowen
D
Department of Statistics, North Carolina State University, Raleigh, North Carolina 27695, USA. zhang@stat.ncsu.edu
Lin
Xihong
X
Sowers
MaryFran
M
D000704:Analysis of Variance; D001699:Biometry; D015519:Bone Density / Q000502:physiology; D005260:Female; D005640:Follicle Stimulating Hormone / Q000652:urine; D006801:Humans; D016013:Likelihood Functions; D016014:Linear Models; D008137:Longitudinal Studies; D008593:Menopause; D008597:Menstrual Cycle; D015233:Models, Statistical; D012044:Regression Analysis; D018709:Statistics, Nonparametric
D005640:Follicle Stimulating Hormone