Application of asymptotic LRTS results to mixture problems in genetic research

This dissertation considers two mixture problems. The first has known mixing proportions, fixed but arbitrary number of components, and component density function from the one parameter exponential family. The asymptotic null distribution of the likelihood ratio test statistic (LRTS) of the null hypothesis of a single component distribution versus the alternative of two or more components in a mixture is proven to be ½ c20 + ½ c21 . The result is shown to hold when the component density function has a nuisance parameter satisfying a linearly independence condition on the partial derivatives. These results are applied to an F-2 breeding experiment in which the mixing proportions are known to be 1/4, 1/2, and 1/4. The scientific problem underlying the second problem is that there is a continuous variable used in assigning a subject in a genetic study to a genotype, which can be modeled by a mixture distribution with the same known number of components in controls and cases. The component parameters are assumed to be equal but unknown in the two groups. The null hypothesis is that the case mixing proportions are equal to the control mixing proportions. The LRTS for this hypothesis is presented, and the non-centrality parameter of the power function is derived. We compare the power of the LRTS to the power of the chi-square test of independence using genotype classifications from two classification rules, a half-way rule and a Bayesian rule. The LRTS is more powerful asymptotically than the test of independence using either classification rule, with increasing superiority as the frequency of the least common component becomes small.;Keywords. F-2 breeding experiment, genetic association study, case control study, statistical power, non-central parameter, classification rule.