On the number of siblings and p-th cousins in a large population sample

Abstract

The number of individuals in a random sample with close relatives in the sample is a quantity of interest when designing Genome Wide Association Studies and other cohort based genetic, and non-genetic, studies. In this paper, we develop expressions for the distribution and expectation of the number of p-th cousins in a sample from a population of size N under two diploid Wright-Fisher models. We also develop simple asymptotic expressions for large values of N. For example, the expected proportion of individuals with at least one p-th cousin in a sample of K individuals, for a diploid dioecious Wright-Fisher model, is approximately [Formula: see text]. Our results show that a substantial fraction of individuals in the sample will have at least a second cousin if the sampling fraction (K / N) is on the order of [Formula: see text]. This confirms that, for large cohort samples, relatedness among individuals cannot easily be ignored.

Keywords: Cousins; Dioecious Wright–Fisher model; Pedigree; Siblings; Stirling numbers of the second kind.