Offspring risk and sibling risk for multilocus traits

The recurrence risk of a trait in a relative of type R is the probability that an individual who is in relationship of type R to an affected proband has the trait. It is intuitively clear that closer relationships lead to higher recurrence risks. However, no exact analysis of this phenomenon has been presented for multilocus traits. We prove a theorem that shows how recurrence risks are influenced by the degree of closeness of the relationship R. For example, our theorem implies that sibling risk is always higher than offspring risk. The loci influencing the trait are assumed to be autosomal and unlinked, but arbitrary epistasis between the loci is allowed. We give a detailed proof of the theorem by using stochastic matrices. A shorter proof based on the additive and dominance genetic variances is also sketched. Additionally, we also give some empirical results and discuss generalizations of the theorem.