Likelihood ratio tests in variance components models for identifying genetic risk factors for complex disorders using multiple quantitative traits

This dissertation is about likelihood ratio tests in variance components models for identifying genetic risk factors for complex disorders, and it consists of three chapters---two methodological chapters concerning asymptotic null distributions and asymptotic powers of likelihood ratio tests under nonstandard boundary conditions, and one applied chapter that motivated one of the methodological chapters. The first chapter is about asymptotic null distribution of a likelihood ratio test that jointly tests genetic effects on multiple traits in variance components model. In previous papers, the asymptotic null distribution of this test has been stated to be a mixture of-chi-square distributions with binomial mixing probabilities. All aspects of the previously stated asymptotic null distribution are shown to be incorrect, both by simulations and by theoretical arguments based on the geometry of the parameter space. The asymptotic null distribution has complex features that raise challenges for the assessment of significance, and a new method to generate a true asymptotic null distribution is proposed, and is shown to be much faster than other alternative empirical methods. The second chapter is about a whole genome screen for reading disability in the Russian language for identifying genetic risk factors for the given disorder. Applying standard univariate likelihood ratio tests across the whole genome on five traits in the data, two novel regions on chromosome 19 and 10 are identified. Another type of likelihood ratio test proposed by Marlow et al. (2003) is also performed on these novel regions to examine if the traits share common genetic effects, and the results suggest that the locus on chromosome 19 is a generalist gene, while the locus on chromosome 10 is a specialist gene. The third chapter is about asymptotic power of a likelihood ratio test proposed by Marlow et al. (2003). Marlow's method has been highly influential and widely applied in the literature of neurodevelopmental disorders as well as other disorders including obesity and asthma. In this chapter, we study the validity and the power of this test, providing explanations on the source of power increase using this test over standard methods. We implement this test and a power calculator in software that can assist in designing studies by using pilot data to calculate required sample sizes and choose traits for further studies.