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. A coefficient of inbreeding can be calculated for an individual, as a measure for the amount of pedigree collapse within that individual's genealogy. 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.
In general, the higher the level of inbreeding the closer the coefficient of relationship approaches a value of 1, expressed as a percentage, and approaches a value of 0 for individuals with arbitrarily remote common ancestors.
Coefficient of inbreeding
The coefficient of inbreeding ("f") is a measure of the likelihood of genetic effects due to inbreeding to be expected based on a known pedigree (i.e. a fully documented genealogy e.g. due to a fixed system of breeding). The measure expresses the expected percentage of homozygosity arising from a given system of breeding.
For a given gene with equally common dominant and recessive variants A and a, a random-bred stock will be 50% homozygous (25% AA and 25% aa), while a closely inbred population will be 100% homozygous (100% AA or 100% aa). The coefficient of inbreeding f is thus designed to run from 0 for an expected 50% homozygosis to 1 for an expected 100% homozygosis, f=2h-1, where h is the chance of finding homozygosis in this gene.
Note that f is an expectation value for an unspecified, hypothetical, perfectly Mendelian gene. Its definition holds regardless of whether the organism's genome actually contains such a gene. Therefore, the coefficient of inbreeding is a statistical value derived from the individual's pedigree and cannot be verified or "measured" exactly by looking at the individual's genome.
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:
- pAO= 2−n⋅((1+fA)/(1+fO))½
where fA and fO are the coefficients of inbreeding for A and O, respectively.[clarification needed]
The coefficient of relationship rBC is now obtained by summing over all path coefficients:
- rBC = Σ pAB⋅pAC.
By assuming that the pedigree can be traced back to a sufficiently remote population of perfectly random-bred stock (fA=0)[clarification needed] the definition of r may be simplified to
- rBC = Σp 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 less generations ago, their coefficient of relationship would be
- r = 2n⋅2−2n = 2−n = 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.
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.
|0||identical twins; clones||100%|
|2||full siblings||50% (2−2+2−2)|
|2||3/4 siblings or sibling-cousins||37.5% (2−2+2⋅2−4)|
|2||half siblings||25% (2−2)|
|4||double first cousins||25% (2−3+2−3)|
|3||great grandparent-great grandchild||12.5% (2−3)|
|4||first cousins||12.5% (2⋅2−4)|
|6||quadruple second cousins||12.5% (8⋅2−6)|
|6||triple second cousins||9.38% (6⋅2−6)|
|4||half-first cousins||6.25% (2−4)|
|5||first cousins once removed||6.25% (2⋅2−5)|
|6||double second cousins||6.25% (4⋅2−6)|
|6||second cousins||3.13% (2−6+2−6)|
|8||third cousins||0.78% (2⋅2−8)|
|10||fourth cousins||0.20% (2⋅2−10)|
Most legally prohibited degree of kinship regarding incestuous relationships concern relations of r = 25% or higher (3 or 4 degrees of relationship or lower), while most permit unions of individuals with r = 12.5% or lower. An exception are a number of US states where cousin marriage is prohibited.  Also, most legislations make no provision for the rare case of marriage between double first cousins.
It should also be noted that incest laws also include prohibitions of unions between biologically unrelated individuals if there is a close legal relationship such as adoption or affinity. It also covers relationships where there is no likelihood of birth of children.
- Effective population size
- Genetic distance
- Identity by descent
- Malecot's method of coancestry
- strictly speaking, r=1 for clones and identical twins, but since the definition of r is usually intended to estimate the suitability of two individuals for breeding, they are typically taken to be of opposite sex.
- "An inbreeding coefficient to be of most value should measure as directly as possible the effects to be expected on the average from the system of mating in the given pedigree." Wright (1922:331)
- A full family tree of seven generations (128 paths to ancestors of the 7th degree) is unreasonable even for members of high nobility. For example, the family tree of Queen Elizabeth II is fully known for a depth of six generations, but becomes difficult to trace in the seventh generation.
- By replacement in the definition of the notion of "generation" by meiosis". Since identical twins are not separated by meiosis, there are no "generations" between them, hence n=0 and r=1. See genetic-genealogy.co.uk.
- "Kin Selection". Benjamin/Cummings. Retrieved 2007-11-25.
- This degree of relationship is usually indistinguishable from the relationship to a random individual within the same population (tribe, country, ethnic group).
- Wright, Sewall (1921). "Systems of Mating". 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 and 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. Retrieved 17 April 2011.