# Coefficient of relationship

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In population genetics, Sewall Wright's coefficient of relationship or coefficient of relatedness or relatedness or r is a measure for the level of consanguinity between two given individuals.

The coefficient of inbreeding is calculated for a single individual, and is a measure for the amount of pedigree collapse within that individual's genealogy.

Roughly speaking, the coefficient of relationship approaches a value of 1 for individuals from a completely inbred population,[1] and approaches a value of 0 for individuals with arbitrarily remote common ancestors.

## Definition

The coefficient of correlation as defined by Wright (1922) is derived from the definition of the coefficient of inbreeding f as defined in Wright (1921).

### Inbreeding coefficient

The purpose of such a coefficient is to express the likelihood of 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).[2] The coefficient introduced by Wright (1921) expresses the expected percentage of homozygosity arising from a given system of breeding.

For a given gene with 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.

### Coefficient of relationship

The coefficient of relationship rBC 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 a 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.Template:Huh

The coefficient of correlation rBC is now obtained by summing over all path coefficients:

rBC = Σ pABpAC.

By assuming that the pedigree can be traced back to a sufficiently remote population of perfectly random-bred stock (fA=0)Template:Huh 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.

## Human genealogy

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

In human genealogy, 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.[3]

rrelationshipdegree of relationship
100%identical twins; clones 0[4]
50% (2-1)parent-offspring[5] 1
50% (2-2+2-2)full siblings 2
25% (2-2)grandparent-grandchild 2
25% (2-2)half siblings 2
25% (2⋅2-3)aunt/uncle-nephew/niece 3
25% (2-3+2-3)double first cousins 4
12.5% (2-3)great grandparent-great grandchild 3
12.5% (2⋅2-4)first cousins 4
12.5% (8⋅2-6)quadruple second cousins  6
9.38% (6⋅2-6)triple second cousins  6
6.25% (2-4)half-first cousins 4
6.25% (2⋅2-5)first cousins once removed 5
6.25% (4⋅2-6)double second cousins  6
3.13% (2-6+2-6)second cousins 6
0.78% (2⋅2-8)third cousins  8
0.20% (2⋅2-10)fourth cousins 10[6]

From the above table, it can be seen that most legislation regarding incestuous unions concern relations of r=25% or higher, while most permit unions of individuals with r=12.5% or lower. An exception are certain 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 unrelated individuals if there is a close legal relationship such as adoption.

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## References

1. 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.
2. "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)
3. 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.
4. 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.
5. Kin Selection. Benjamin/Cummings. URL accessed on 2007-11-25.
6. this degree of relationship is usually indistinguishable from the relationship to a random individual within the same population (tribe, country, ethnic group).
• [[Sewall Wright|]] (1921). . Genetics 6: 111–178.
• [[Sewall Wright|]] (1922). Coefficients of inbreeding and relationship. American Naturalist 56: 330–338.
• 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.
Topics in population genetics(edit)
Key concepts: Hardy-Weinberg law | linkage disequilibrium | Fisher's fundamental theorem | neutral theory
Selection: natural | sexual | artificial | ecological
Genetic drift: small population size | population bottleneck | founder effect | coalescence
Founders: Ronald Fisher | J.B.S. Haldane | Sewall Wright
Related topics: evolution | microevolution | evolutionary game theory | fitness landscape
List of evolutionary biology topics