Affected‐sib‐pair test for linkage based on constraints for identical‐by‐descent distributions corresponding to disease models with imprinting

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Abstract
Holmans' possible triangle test for affected sib pairs has proven to be a powerful tool for linkage analysis. This test is a likelihood‐ratio test for which maximization is restricted to the set of possible sharing probabilities. Here, we extend the possible triangle test to take into account genomic imprinting, which is also known as parent‐of‐origin effect. While the classical test without imprinting looks at whether affected sib pairs share 0, 1, or 2 alleles identical‐by‐descent, the likelihood‐ratio test allowing for imprinting further distinguishes whether the sharing of exactly one allele is through the father or mother. Thus, if the disease gene is indeed subject to imprinting, the extended test presented here can take into account that affecteds will have inherited the mutant allele preferentially from one particular parent. We calculate the sharing probabilities at a marker locus linked to a disease susceptibility locus. Using our formulation, the constraints on these probabilities given by Dudoit and Speed ([1999] Statistics in Genetics; New York: Springer) can easily be verified. Next, we derive the asymptotic distribution of the restricted likelihood‐ratio test statistic under the null hypothesis of no linkage, and give LOD‐score criteria for various test sizes. We show, for various disease models, that the test allowing for imprinting has significantly higher power to detect linkage if imprinting is indeed present, at the cost of only a small reduction in power in case of no imprinting. Altogether, unlike many methods currently available, our novel model‐free sib‐pair test adequately models the epigenetic parent‐of‐origin effect, and will hopefully prove to be a useful tool for the genetic mapping of complex traits.