Estimating Variance Components in Functional Linear Models With Applications to Genetic Heritability

Quantifying heritability is the first step in understanding the contribution of genetic variation to the risk architecture of complex human diseases and traits. Heritability can be estimated for univariate phenotypes from nonfamily data using linear mixed effects models. There is, however, no fully developed methodology for defining or estimating heritability from longitudinal studies. By examining longitudinal studies, researchers have the opportunity to better understand the genetic influence on the temporal development of diseases, which can be vital for populations with rapidly changing phenotypes such as children or the elderly. To define and estimate heritability for longitudinally measured phenotypes, we present a framework based on functional data analysis, FDA. While our procedures have important genetic consequences, they also represent a substantial development for FDA. In particular, we present a very general methodology for constructing optimal, unbiased estimates of variance components in functional linear models. Such a problem is challenging as likelihoods and densities do not readily generalize to infinite-dimensional settings. Our procedure can be viewed as a functional generalization of the minimum norm quadratic unbiased estimation procedure, MINQUE, presented by C. R. Rao, and is equivalent to residual maximum likelihood, REML, in univariate settings. We apply our methodology to the Childhood Asthma Management Program, CAMP, a 4-year longitudinal study examining the long term effects of daily asthma medications on children.