Likelihood Ratio Testing for Admixture Models with Application to Genetic Linkage Analysis

Abstract

We consider likelihood ratio tests (LRT) and their modifications for homogeneity in admixture models. The admixture model is a special case of two component mixture model, where one component is indexed by an unknown parameter while the parameter value for the other component is known. It has been widely used in genetic linkage analysis under heterogeneity, in which the kernel distribution is binomial. For such models, it is long recognized that testing for homogeneity is nonstandard and the LRT statistic does not converge to a conventional 2 distribution. In this paper, we investigate the asymptotic behavior of the LRT for general admixture models and show that its limiting distribution is equivalent to the supremum of a squared Gaussian process. We also provide insights on the connection and comparison between LRT and alternative approaches in the literature, mostly modifications of LRT and score tests, including the modified or penalized LRT (Fu et al., 2006). The LRT is an omnibus test that is powerful against general alternative hypothesis. In contrast, alternative approaches may be slightly more powerful against certain type of alternatives, but much less powerful for other types. Our results are illustrated by simulation studies and an application to a genetic linkage study of schizophrenia