A Measure of Consanguinity

A natural numerical measure of consanguinity is developed that applies to individuals with arbitrary multiple kinship connections. For simple relationships the consanguineal distance specializes to the civil degree, less two if the relationship goes through full siblings. This measure is deduced from axioms motivated by an heuristic picture of blood mixtures. The formula suggests a quantum mechanical probability interpretation whose classical counterpart yields a generalization of the Murdock degree of consanguinity. In this paper we introduce a consanguinity coe cient that measures the degree of consanguinity between people with arbitrary multiple kinships relations. This coe cient does not measure genetic closeness in the manner of Mal ecot's coe cient of coancestry [4]. Rather it corresponds to the naive view that consanguinity means similarity of blood, and that the blood of any individual is a mixture of equal parts of the blood of his two parents. We use this model as the motivation for our computations since it seems to re ect a deeply ingrained attitude toward kinship as evidenced by the expressions \blood relative", \royal blood", etc. In the absence of multiple kinship relations, is 2 d where d is the civil degree of consanguinity, that is, the sum of the removals from the nearest common ancestor [1]. Thus for unrelated individuals, is zero, for a child and either of his unrelated parents is one half, and for half-siblings is one quarter. From a mathematical point of view, multiple kinship relations in our society are the rule rather than the exception, because related individuals are typically common descendants of a pair of spouses, and thus are related in two distinct ways. The canonical example is the relation between full siblings which derives from both the father and the mother. In terms of the blood model, full siblings share two di erent kinds of blood in the same proportions. Since the blood of full siblings is thus identical, we are led to maintain with Kendall [3] that is one for full siblings, and that is 2 d+2 for simple relationships that go through full siblings. Thus is one half for an aunt and niece, just like for a mother and daughter, the two nearest common ancestors being the aunt's parents, and d being 1 + 2 = 3. For persons with multiple kinship relations other than descent from siblings the problem is more complex. The simplest situation is that of multiple ancestry, such as when a maternal great-grandfather is also a paternal great-great-grandfather.