Functional regression models are developed in this project for testing associations between complex traits and genetic variants, which can be rare variants or common variants or the combination of the two. By treating multiple genetic variants of an individual in a human population as a realization of a stochastic process, the genome of an individual in a chromosome region is a continuum of sequence data rather than discrete observations. The genome of an individual is viewed as a stochastic function which contains both linkage and linkage disequilibrium (LD) information of the genetic markers. By using techniques of functional data analysis, functional regression models are built to test the association between complex traits and genetic variants adjusting for covariates. The methods are applied to analyze data from the Trinity Students Study, neural tube defects, and Hirschsprung's disease in Division of Intramural Population Health Research at NICHD. Triad families are routinely used to test association between genetic variants and complex diseases. Triad studies are important and popular since they are robust in terms of being less prone to false positives due to population structure. In practice, one may collect not only complete triads, but also incomplete families such as dyads (affected child with one parent) and singleton monads (affected child without parents). Since there is a lack of convenient algorithms and software to analyze the incomplete data, dyads and monads are usually discarded. This may lead to loss of power and insufficient utilization of genetic information in a study. We develop likelihood-based statistical models and likelihood ratio tests to test for association between complex diseases and genetic markers by using combinations of full triads, parent-child dyads, and affected singleton monads for a unified analysis. A likelihood is calculated directly to facilitate the data analysis without imputation and to avoid computational complexity. This makes it easy to implement the models and to explain the results. By simulation studies, we show that the proposed models and tests are very robust in terms of accurately controlling type I error evaluations, and are powerful by empirical power evaluations. The methods are applied to test for association between transforming growth factor alpha (TGFA) gene and cleft palate in an Irish study. Longitudinal genetic studies provide a valuable resource for exploring key genetic and environmental factors that affect complex traits over time. Genetic analysis of longitudinal data that incorporates temporal variations is important for understanding genetic architecture and biological variations of common complex diseases. Although they are important, there is a paucity of statistical methods to analyze longitudinal human genetic data. In this project, longitudinal methods are developed for temporal association mapping to analyze population longitudinal data. Both parametric and non-parametric models are proposed. The models can be applied to multiple di-allelic genetic markers such as single nucleotide polymorphisms and multi-allelic markers such as micro-satellites. By analytical formulae, we show that the models take both the linkage disequilibrium and temporal trends into account simultaneously. Variance-covariance structure is constructed to model the single measurement variation and multiple measurement correlations of an individual based on the theory of stochastic processes. Novel penalized spline models are used to estimate the time-dependent mean functions and regression coefficients. The methods were applied to analyze Framingham Heart Study data of Genetic Analysis Workshop (GAW) 13 and GAW 16. The temporal trends and genetic effects of the systolic blood pressure are successfully detected by the proposed approaches. Simulation studies were performed to find out that the non-parametric penalized linear model is the best choice in fitting real data. The research sheds light on the important area of longitudinal genetic analysis, and it provides a basis for future methodological investigations and practical applications.
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