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The Probability of Pairwise Shared Ancestry and the Expected number of Pairs of k-th cousins in a Population Sample

Natural Science 2022年第11期14卷 472-482页

作者： Philip M. Service Department of Biological Sciences Northern Arizona University Flagstaff USA

An analytical solution is derived for the probability that a random pair of individuals from a panmictic population of size N will share ancestors who lived G generations previously. The analysis is extended to obtain 1) the probability that a sample of size s will contain at least one pair of (G - 1)th cousins;and 2) the expected number of pairs of (G - 1)th cousins in that sample. Solutions are given for both monogamous and promiscuous (non-monogamous) cases. Simulation results for a population size of N = 20,000 closely approximate the analytical expectations. Simulation results also agree very well with previously derived expectations for the proportion of unrelated individuals in a sample. The analysis is broadly consistent with genetic estimates of relatedness among a sample of 406 Danish school children, but suggests that a different genetic study of a heterogenous sample of Europeans overestimates the frequency of cousin pairs by as much as one order of magnitude.

关键词： Pairwise Shared Ancestry Simulation Denmark number of cousins Population Sample Population Structure

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Pairwise Shared Ancestry in Random-Mating Constant-Size Populations

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Natural Science 2022年第5期14卷 193-202页

作者： Philip M. Service Department of Biological Sciences Northern Arizona University Flagstaff USA

In a panmictic population of constant size N, random pairs of individuals will have a most recent shared ancestor who lived slightly more than 0.5 log2N generations previously, on average. The probability that a random pair of individuals will share at least one ancestor who lived 0.5 log2N generations ago, or more recently, is about 50%. Those individuals, if they do share an ancestor from that generation, would be cousins of degree (0.5 log2N) - 1. Shared ancestry from progressively earlier generations increases rapidly until there is universal pairwise shared ancestry. At that point, every individual has one or more ancestors in common with every other individual in the population, although different pairs may share different ancestors. Those ancestors lived approximately 0.7 log2N generations in the past, or more recently. Qualitatively, the ancestries of random pairs have about 50% similarity for ancestors who lived about 0.9 log2N generations before the present. That is, about half of the ancestors from that generation belonging to one member of the pair are present also in the genealogy of the other member. Qualitative pairwise similarity increases to more than 99% for ancestors who lived about 1.4 log2N generations in the past. Similar results apply to a metric of quantitative pairwise genealogical overlap.

关键词： Pairwise Shared Ancestry Genealogical Overlap Quantitative Ancestry Most Recent Common Ancestor number of cousins

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