Canadian Journal of Statistics

The universal validity of the possible triangle constraint for affected sib pairs

Journal Article

Abstract

The authors consider affected‐sib‐pair analysis, in which genetic marker data are collected from families with at least two sibs affected by a disease under investigation. At any locus not linked to the disease gene, a sib pair shares 0, 1 and 2 alleles identical by descent (IBD) with probabilities of 1/4, 1/2 and 1/4, respectively. With linkage, the IBD value increases stochastically. Louis, Payami & Thomson (1987) and Holmans (1993) were the first ones who discovered that the IBD distribution satisfies the “possible triangle constraint” in some situations. Consequently, more powerful statistical procedures can be designed in detecting linkage. It is of statistical and genetical importance to investigate whether the possible triangle constraint remains true under general genetic models. In this paper, the authors introduce a new technique to prove the possible triangle constraint. Their proof is particularly simple for the single disease locus case. The general case is proved by linking IBD distributions between marker loci through a transition probability matrix.

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