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The coefficient of relationship is a measure of the degree of
consanguinity Consanguinity ("blood relation", from Latin '' consanguinitas'') is the characteristic of having a kinship with another person (being descended from a common ancestor). Many jurisdictions have laws prohibiting people who are related by blood fro ...
(or biological relationship) between two individuals. The term
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
of relationship was defined by
Sewall Wright Sewall Green Wright FRS(For) Honorary FRSE (December 21, 1889March 3, 1988) was an American geneticist known for his influential work on evolutionary theory and also for his work on path analysis. He was a founder of population genetics along ...
in 1922, and was derived from his definition of the coefficient of
inbreeding Inbreeding is the production of offspring from the mating or breeding of individuals or organisms that are closely related genetically. By analogy, the term is used in human reproduction, but more commonly refers to the genetic disorders and ...
of 1921. The measure is most commonly used in
genetics Genetics is the study of genes, genetic variation, and heredity in organisms.Hartl D, Jones E (2005) It is an important branch in biology because heredity is vital to organisms' evolution. Gregor Mendel, a Moravian Augustinian friar workin ...
and
genealogy Genealogy () is the study of families, family history, and the tracing of their lineages. Genealogists use oral interviews, historical records, genetic analysis, and other records to obtain information about a family and to demonstrate kin ...
. A
coefficient of inbreeding The coefficient of inbreeding of an individual is the probability that two alleles at any locus in an individual are identical by descent from the common ancestor(s) of the two parents. The coefficient of inbreeding is: The probability that t ...
can be calculated for an individual, and is typically one-half the coefficient of relationship between the parents. 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.

# 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: :$p_ = 2^ \cdot \sqrt$ where $f_A$ and $f_O$ are the coefficients of inbreeding for A and O, respectively. The coefficient of relationship $r_$is now obtained by summing over all path coefficients: :$r_ = \sum$ By assuming that the pedigree can be traced back to a sufficiently remote population of perfectly random-bred stock (''f''A = 0 for all ''A'' in the sum) the definition of ''r'' may be simplified to :$r_ = \sum_$ 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 give 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 :$r = 2^n \cdot 2^ = 2^$, which for n = 5, is, $2^ = \frac$, or 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 In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire ...
population) falls below one part-per-million.

# Human relationships

The coefficient of relationship is sometimes used to express degrees of kinship in numeric terms in human
genealogy Genealogy () is the study of families, family history, and the tracing of their lineages. Genealogists use oral interviews, historical records, genetic analysis, and other records to obtain information about a family and to demonstrate kin ...
. 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 Adoption is a process whereby a person assumes the parenting of another, usually a child, from that person's biological or legal parent or parents. Legal adoptions permanently transfer all rights and responsibilities, along with filiation, from ...
or affinity. Whether there is any likelihood of conception is generally considered irrelevant.

# Kinship coefficient

The kinship coefficient is a simple measure of relatedness, defined as the
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking ...
that a pair of randomly sampled homologous
alleles An allele (, ; ; modern formation from Greek ἄλλος ''állos'', "other") is a variation of the same sequence of nucleotides at the same place on a long DNA molecule, as described in leading textbooks on genetics and evolution. ::"The chro ...
are identical by descent. More simply, it is the probability that an allele selected randomly from an individual, i, and an allele selected at the same
autosomal An autosome is any chromosome that is not a sex chromosome. The members of an autosome pair in a diploid cell have the same morphology, unlike those in allosomal (sex chromosome) pairs, which may have different structures. The DNA in autosomes ...
locus from another individual, j, are identical and from the same ancestor. The coefficient of relatedness is equal to twice the kinship coefficient.

## Calculation

The kinship coefficient between two individuals, i and j, is represented as Φij. The kinship coefficient between a non-inbred individual and itself, Φii, is equal to 1/2. This is due to the fact that humans are
diploid Ploidy () is the number of complete sets of chromosomes in a cell, and hence the number of possible alleles for autosomal and pseudoautosomal genes. Sets of chromosomes refer to the number of maternal and paternal chromosome copies, respecti ...
, meaning the only way for the randomly chosen alleles to be identical by descent is if the same allele is chosen twice (probability 1/2). Similarly, the relationship between a parent and a child is found by the chance that the randomly picked allele in the child is from the parent (probability 1/2) and the probability of the allele that is picked from the parent being the same one passed to the child (probability 1/2). Since these two events are independent of each other, they are multiplied Φij = 1/2 X 1/2 = 1/4.

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Accidental incest Accidental incest is sexual activity or marriage between persons who were unaware of a family relationship between them which would be considered incestuous. The laws of many jurisdictions void incestuous marriages, even if entered into without ...
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Effective population size The effective population size (''N'e'') is a number that, in some simplified scenarios, corresponds to the number of breeding individuals in the population. More generally, ''N'e'' is the number of individuals that an idealised population w ...
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F-statistics In population genetics, ''F''-statistics (also known as fixation indices) describe the statistically expected level of heterozygosity in a population; more specifically the expected degree of (usually) a reduction in heterozygosity when compared ...
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Genetic distance Genetic distance is a measure of the genetic divergence between species or between populations within a species, whether the distance measures time from common ancestor or degree of differentiation. Populations with many similar alleles have sm ...
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Genetic diversity Genetic diversity is the total number of genetic characteristics in the genetic makeup of a species, it ranges widely from the number of species to differences within species and can be attributed to the span of survival for a species. It is dis ...
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Genetic sexual attraction Genetic sexual attraction is a concept in which a strong sexual attraction may develop between close blood relatives who first meet as adults. There is no evidence for genetic sexual attraction being an actual phenomenon, and the hypothesis is re ...
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Inbreeding Inbreeding is the production of offspring from the mating or breeding of individuals or organisms that are closely related genetically. By analogy, the term is used in human reproduction, but more commonly refers to the genetic disorders and ...
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Inbreeding avoidance Inbreeding avoidance, or the inbreeding avoidance hypothesis, is a concept in evolutionary biology that refers to the prevention of the deleterious effects of inbreeding. Animals only rarely exhibit inbreeding avoidance. The inbreeding avoidance ...
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Inbreeding depression Inbreeding depression is the reduced biological fitness which has the potential to result from inbreeding (the breeding of related individuals). Biological fitness refers to an organism's ability to survive and perpetuate its genetic material. ...
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Incest Incest ( ) is human sexual activity between family members or close relatives. This typically includes sexual activity between people in consanguinity (blood relations), and sometimes those related by affinity ( marriage or stepfamily), ad ...
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Incest taboo An incest taboo is any cultural rule or norm that prohibits sexual relations between certain members of the same family, mainly between individuals related by blood. All human cultures have norms that exclude certain close relatives from ...
* Legality of incest *
Malecot's method of coancestry Malecot's coancestry coefficient, f, refers to an indirect measure of genetic similarity of two individuals which was initially devised by the French mathematician Gustave Malécot. f is defined as the probability that any two alleles, sampled ...
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Pedigree collapse In genealogy, pedigree collapse describes how reproduction between two individuals who share an ancestor causes the number of distinct ancestors in the family tree of their offspring to be smaller than it could otherwise be. Robert C. Gunderson c ...
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Phylogenetics In biology, phylogenetics (; from Greek φυλή/ φῦλον [] "tribe, clan, race", and wikt:γενετικός, γενετικός [] "origin, source, birth") is the study of the evolutionary history and relationships among or within groups ...
* Prohibited degree of kinship *
Proximity of blood Proximity of blood, or proximity by degree of kinship, is one of the ways to determine hereditary succession based on genealogy. In effect, the application of this rule is a refusal to recognize the right of representation, a component of primogen ...

# Bibliography

* 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 * * 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. * {{DEFAULTSORT:Coefficient Of Relationship Genealogy Kinship and descent Breeding Dog breeding Incest Population genetics