Flexible estimation of covariance function by penalized spline with application to longitudinal family data
Article first published online: 13 APR 2011
Copyright © 2011 John Wiley & Sons, Ltd.
Statistics in Medicine
Volume 30, Issue 15, pages 1883–1897, 10 July 2011
How to Cite
Wang, Y. (2011), Flexible estimation of covariance function by penalized spline with application to longitudinal family data. Statist. Med., 30: 1883–1897. doi: 10.1002/sim.4236
- Issue published online: 10 JUN 2011
- Article first published online: 13 APR 2011
- Manuscript Accepted: 24 JAN 2011
- Manuscript Received: 18 FEB 2010
- multi-level functional data;
- Cholesky decomposition;
- age-specific heritability;
- Framingham Heart Study
Longitudinal data are routinely collected in biomedical research studies. A natural model describing longitudinal data decomposes an individual's outcome as the sum of a population mean function and random subject-specific deviations. When parametric assumptions are too restrictive, methods modeling the population mean function and the random subject-specific functions nonparametrically are in demand. In some applications, it is desirable to estimate a covariance function of random subject-specific deviations. In this work, flexible yet computationally efficient methods are developed for a general class of semiparametric mixed effects models, where the functional forms of the population mean and the subject-specific curves are unspecified. We estimate nonparametric components of the model by penalized spline (P-spline, Biometrics 2001; 57:253–259), and reparameterize the random curve covariance function by a modified Cholesky decomposition (Biometrics 2002; 58:121–128) which allows for unconstrained estimation of a positive-semidefinite matrix. To provide smooth estimates, we penalize roughness of fitted curves and derive closed-form solutions in the maximization step of an EM algorithm. In addition, we present models and methods for longitudinal family data where subjects in a family are correlated and we decompose the covariance function into a subject-level source and observation-level source. We apply these methods to the multi-level Framingham Heart Study data to estimate age-specific heritability of systolic blood pressure nonparametrically. Copyright © 2011 John Wiley & Sons, Ltd.