Coefficient of relationship

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, 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). 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.

The coefficient of correlation  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) 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.

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.