Summary We consider likelihood ratio tests (LRT) and their modifications for homogeneity in admixture models. The admixture model is a two‐component mixture model, where one component is indexed by an unknown parameter while the parameter value for
the other component is known. This model is 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 χ2 distribution. In this article, 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 discuss the connection
and comparison between LRT and alternative approaches such as modifications of LRT and score tests, including the modified LRT (Fu, Chen, and Kalbfleisch, 2006, Statistica Sinica16, 805–823). The LRT is an omnibus test that is powerful
to detect general alternative hypotheses. In contrast, alternative approaches may be slightly more powerful to detect certain type of alternatives, but much less powerful for others. Our results are illustrated by simulation studies and an application to a genetic linkage study of schizophrenia.
Division of Public Health Sciences, Fred Hutchinson Cancer Research Center, 1100 Fairview Avenue North, M2-B500, Seattle, Washington 98109, U.S.A. 2:
Department of Life Sciences/Institute of Genome Sciences, National Yang-Ming University, Taiwan, R.O.C.