Triangle constraints for sib-pair identity by descent probabilities under a general multilocus model for disease susceptibility
In this paper, we study sib-pair IBD probabilities under a general multilocus model for disease susceptibility which doesn't assume random mating, linkage equilibrium or Hardy-Weinberg equilibrium. We derive the triangle constraints satisfied by affected, discordant and unaffected sib-pair IBD probabilities, as well as constraints distinguishing between sharing of maternal and paternal DNA, under general monotonicity assumptions concerning the penetrance probabilities. The triangle constraints are valid for age and sex-dependent penetrances, and in the presence of parental imprinting. We study the parameterization of sib-pair IBD probabilities for common models, and present examples to demonstrate the impact of non-random mating and the necessity of our assumptions for the triangle constraints. We prove that the affected sib-pair possible triangle is covered by the IBD probabilities of two types of models, one with fixed mode of inheritance and general mating type frequencies, the other with varying mode of inheritance and random mating. Finally, we consider IBD probabilities at marker loci linked to disease susceptibility loci and derive the triangle constraints satisfied by these probabilities.