Likelihood Ratio Test for Multi-Sample Mixture Model and Its Application to Genetic Imprinting

Genomic imprinting is a known aspect of the etiology of many diseases. The imprinting phenomenon depicts differential expression levels of the allele depending on its parental origin. When the parental origin is unknown, the expression level has a finite normal mixture distribution. In such applications, a random sample of expression levels consists of three subsamples according to the number of minor alleles an individual possesses, of which one is the mixture and the other two are homogeneous. This understanding leads to a likelihood ratio test (LRT) for the presence of imprinting. Because of the nonregularity of the finite mixture model, the classical asymptotic conclusions on likelihood-based inference are not applicable. We show that the maximum likelihood estimator of the mixing distribution remains consistent. More interestingly, thanks to the homogeneous subsamples, the LRT statistic has an elegant and rather distinct 0.5χ² ₁ + 0.5χ² ₂ null limiting distribution. Simulation studies confirm that the limiting distribution provides precise approximations of the finite sample distributions under various parameter settings. The LRT is applied to expression data. Our analyses provide evidence for imprinting for a number of isoform expressions.