Coefficient of relationship

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The coefficient of relationship is a measure of the degree of consanguinity (or biological relationship) between two individuals. The term coefficient of relationship was defined by Sewall Wright in 1922, and was derived from his definition of the coefficient of inbreeding of 1921. The measure is most commonly used in genetics and genealogy. A coefficient of inbreeding can be calculated for an individual, and is typically one-half the coefficient of relationship between the parents. Note that the remainder of this article is confused about this.

In general, the higher the level of inbreeding the closer the coefficient of relationship between the parents approaches a value of 1, expressed as a percentage, and approaches a value of 0 for individuals with arbitrarily remote common ancestors.

Video Coefficient of relationship

Coefficient of relationship

The coefficient of relationship ("r") between two individuals B and C is obtained by a summation of coefficients calculated for every line by which they are connected to their common ancestors. Each such line connects the two individuals via a common ancestor, passing through no individual which is not a common ancestor more than once. A path coefficient between an ancestor A and an offspring O separated by n generations is given as:

p_AO= 2^-n?((1+f_A)/(1+f_O))^½

where f_A and f_O are the coefficients of inbreeding for A and O, respectively.

The coefficient of relationship r_BC is now obtained by summing over all path coefficients:

r_BC = ? p_AB?p_AC.

By assuming that the pedigree can be traced back to a sufficiently remote population of perfectly random-bred stock (f_A=0) the definition of r may be simplified to

r_BC = ?_p 2^-L(p),

where p enumerates all paths connecting B and C with unique common ancestors (i.e. all paths terminate at a common ancestor and may not pass through a common ancestor to a common ancestor's ancestor), and L(p) is the length of the path p.

To given an (artificial) example: Assuming that two individuals share the same 32 ancestors of n=5 generations ago, but do not have any common ancestors at four or fewer generations ago, their coefficient of relationship would be

r = 2ⁿ?2^-2n = 2^-n which is, because n = 5,

r = 2⁵o2^-10 = 2^-5 = 1/32 which is approximately 0.0313 or 3%.

Individuals for which the same situation applies for their 1024 ancestors of ten generations ago would have a coefficient of r = 2^-10 = 0.1%. If follows that the value of r can be given to an accuracy of a few percent if the family tree of both individuals is known for a depth of five generations, and to an accuracy of a tenth of a percent if the known depth is at least ten generations. The contribution to r from common ancestors of 20 generations ago (corresponding to roughly 500 years in human genealogy, or the contribution from common descent from a medieval population) falls below one part-per-million.

Maps Coefficient of relationship

Human relationships

The coefficient of relationship is sometimes used to express degrees of kinship in numeric terms in human genealogy.

In human relationships, the value of the coefficient of relationship is usually calculated based on the knowledge of a full family tree extending to a comparatively small number of generations, perhaps of the order of three or four. As explained above, the value for the coefficient of relationship so calculated is thus a lower bound, with an actual value that may be up to a few percent higher. The value is accurate to within 1% if the full family tree of both individuals is known to a depth of seven generations.

Most incest laws concern the relationships where r = 25% or higher, although many ignore the rare case of double first cousins. Some jurisdictions also prohibit sexual relations or marriage between cousins of various degree, or individuals related only through adoption or affinity. Whether there is any likelihood of conception is generally considered irrelevant.

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See also

• Effective population size
• F-statistics
• Genetic distance
• Identity by descent
• Malecot's method of coancestry

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References

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Bibliography

• Wright, Sewall (1921). "Systems of Mating" (PDF). Genetics. 6: 111-178.  five papers:
  • I) The biometric relations between offspring and parent
  • II) The effects of inbreeding on the genetic composition of a population
  • III) Assortative mating based on somatic resemblance
  • IV) The effects of selection
  • V) General considerations
• Wright, Sewall (1922). "Coefficients of inbreeding and relationship". American Naturalist. 56: 330-338. doi:10.1086/279872. 
• Malécot, G. (1948) Les mathématiques de l'hérédité, Masson et Cie, Paris. Lange, K. (1997) Mathematical and statistical methods for genetic analysis, Springer-Verlag, New-York.
• Oliehoek, Pieter; Jack J. Windig; Johan A. M. van Arendonk; Piter Bijma (May 2006). "Estimating Relatedness Between Individuals in General Populations With a Focus on Their Use in Conservation Programs". Genetics. 173: 483-496. doi:10.1534/genetics.105.049940. PMC 1461426 . PMID 16510792. 

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