# Coefficient of Inbreeding

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Inbreeding occurs when relatives unite to produce children. The degree of inbreeding can be measured by the coefficient of inbreeding[1], which is defined as:

(1) $F = \sum_{i=1}^{N} \frac{1}{2}^{n_{f,i}+n_{m,i}+1} (1+F_{i})$

where

• $F = [0$ $1]$ is the coefficient of inbreeding of the individual;
• $N$ is the number of common ancestors of the individual's parents;
• $i = 1, 2, ..., N$ indexes the common ancestors;
• $n_{f,i}$ is the number of generations from the father to common ancestor $i$;
• $n_{m,i}$ is the number of generations from the mother to common ancestor $i$; and
• $F_{i}$ is the coefficient of inbreeding of common ancestor $i$.

Equation (1) is hard to compute and has been criticised by more recent writers. The coefficient of inbreeding of the children of unrelated but inbred parents is set to zero. Therefore, at Familypedia we use the following coefficient:

(2) $C_{child} = 0.5*C_{father} + 0.5*C_{mother} + F_{father x mother}$

where

• $C_i = [0$ $1]$ is the Familypedia coefficient of inbreeding for $i=child, father, mother$; and
• $F = [0$ $1]$ is the Wright coefficient of inbreeding of the parents as defined by Equation (1) for $F_{i}=0$.

Equation (2) is easy to compute as it only uses information about the parents and their relationship. The Wright coefficients can be seen from Table 1.

Table 1. Wright coefficient of inbreeding for selected cases

Relationship between parentsCoefficient of inbreeding
Parent/child0.500000000
Grandparent/grandchild0.250000000
Siblings0.250000000
Great-grandparent/great-grandchild0.125000000
Half-siblings0.125000000
Aunt/nephew, Uncle/niece0.125000000
Double first cousins0.125000000
First cousins0.062500000
First cousins once removed0.031250000
First cousins twice removed0.015625000
Second cousins0.015625000
Second cousins once removed0.007812500
Second cousins twice removed0.003906250
Third cousins0.003906250
Third cousins once removed0.001953125
Fourth cousins0.000976563