Estimation of Covariance Structures in Functional Mixed Models with Application to Heritability Estimation
Patwary, Mohammad Shaha Alam
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When the response on a subject can be naturally viewed as a smooth curve or function, it is said to be a functional response. The response may be observed at a set of discrete times, possibly with noise. Functional data arise when observations of a functional response are available from a sample of subjects. Thus, the functional data essentially consist of a sample of curves. One example of such data is the usual longitudinal data where a variable of interest is measured over time on a sample of subjects. Functional data arise in a variety of disciplines, including economics, environmental science, public health, medicine, and genetics. Analysis of such data is currently an active area of statistical research. Functional data are often analyzed by modeling them as a functional mixed model. This model commonly assumes that the within-subject errors are homoscedastic and uncorrelated. But this assumption is often violated in practice, which may sometimes lead to potentially misleading inferences.This is especially an issue if the object of inference is a function of both random effect and error autocovariance functions. One such quantity is heritability function, defined as the proportion of variance explained by the random effect. In genetics, the random effect can be interpreted as the additive genetic component of a quantitative trait. This way, heritability is the ratio of additive genetic variance to the total phenotypic variance of the trait. It measures the extent to which individuals’ phenotypes are determined by the genes transmitted from the parents. This makes heritability a fundamental quantity of interest in genetics. This dissertation makes three contributions toward the issue of estimating both random effect and error covariance structures in a functional mixed model. First, it develops a methodology for modeling functional data from independent subjects that incorporates parametric models for error covariance structure. The methodology is evaluated using a simulation study. Its application is illustrated by analyzing a growth curve data. Next, this methodology is extended for family data where the subjects may be grouped into families and subjects from the same family are dependent. This methodology is also evaluated using simulation. Finally, it introduces the novel notion of a singular mixed model, whose further development in future may allow modeling the error covariance structure nonparametrically, enhancing the flexibility of functional mixed models.