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

Quantifying heritability is the first step in understanding the contribution of geneticvariation to the risk architecture of complex human diseases and traits. Heritabilitycan be estimated for univariate phenotypes from non-family data using linear mixedeffects models. There is, however, no fully developed methodology for defining orestimating heritability from longitudinal studies. By examining longitudinal studies,researchers have the opportunity to better understand the genetic influence on the tem-poral development of diseases, which can be vital for populations with rapidly changingphenotypes such as children or the elderly. To define and estimate heritability for lon-gitudinally measured phenotypes, we present a framework based on functional dataanalysis, FDA. While our procedures have important genetic consequences, they alsorepresent a substantial development for FDA. In particular, we present a very generalmethodology for constructing optimal, unbiased estimates of variance components infunctional linear models. Such a problem is challenging as likelihoods and densities donot readily generalize to infinite dimensional settings. Our procedure can be viewedas a functional generalization of the minimum norm quadratic unbiased estimationprocedure, MINQUE, presented by C. R. Rao, and is equivalent to residual maximumlikelihood, REML, in univariate settings. We apply our methodology to the ChildhoodAsthma Management Program, CAMP, a four year longitudinal study examining thelong term effects of daily asthma medications on children.