A Note on the Asymptotic Properties of Affected-sib-pair Linkage Tests

This article concerns the asymptotic properties of linkage tests for affected‐sib‐pair data under the null hypothesis of no linkage. We consider a popular single‐locus analysis model where the unknown parameters are the disease allele frequency, the three penetrances for the three genotypes at the disease locus, and the recombination fraction between the marker locus and the disease locus. These parameters are completely confounded under the null hypothesis of no linkage. We show that 1) If the total variance of the trait (i.e., the additive variance plus the dominance variance) is “separated” from 0, then the likelihood ratio statistic has an asymptotic 0.5χ²₀+ 0.5χ²₁ distribution; 2) If the prevalence of the trait is “separated” from 0 and the recombination fraction is fixed at 0, then the likelihood ratio statistic has an asymptotic distribution which is a mixture of χ²₀, χ²₁ and χ²₂ . The implications of these results are discussed.