Coefficient of relationship

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.

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

{\textstyle r}

between two people 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 people via a common ancestor, passing through no person who is not a common ancestor more than once. A path coefficient between an ancestor A and an offspring O separated by

n

{\displaystyle n}

generations is given as:

p

A

O

=

2

−

n

⋅

1

+

f

A

1

+

f

O

{\displaystyle p_{AO}=2^{-n}\cdot {\sqrt {\frac {1+f_{A}}{1+f_{O}}}}}

where

f

A

{\displaystyle f_{A}}

and

f

O

{\displaystyle f_{O}}

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

The coefficient of relationship

r

B

C

{\displaystyle r_{BC}}

is now obtained by summing over all path coefficients:

r

B

C

=

∑

p

A

B

⋅

p

A

C

{\displaystyle r_{BC}=\sum p_{AB}\cdot p_{AC}}

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

r

B

C

=

∑

p

2

−

L

(

p

)

{\displaystyle r_{BC}=\sum _{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.