Coefficient of inbreeding

An individual is said to be inbred if there is a loop in its pedigree chart. A loop is defined as a path that runs from an individual up to the common ancestor through one parent and back down to the other parent, without going through any individual twice. The number of loops is always the number of common ancestors the parents have. If an individual is inbred, the coefficient of inbreeding is calculated by summing all the probabilities that an individual receives the same allele from its father's side and mother's side. As every individual has a 50% chance of passing on an allele to the next generation, the formula depends on 0.5 raised to the power of however many generations separate the individual from the common ancestor of its parents, on both the father's side and mother's side. This number of generations can be calculated by counting how many individuals lie in the loop defined earlier. Thus, the coefficient of inbreeding f of an individual X can be calculated with the following formula:^[7][1]

$${\displaystyle f_{X}=\sum _{N}0.5^{n-1}\cdot (1+f_{A})}$$ where ${\displaystyle n}$ is the number of individuals in the aforementioned loop,
${\displaystyle N}$ is the number of common ancestors (loops),
and ${\displaystyle f_{A}}$ is the coefficient of inbreeding of the common ancestor of X's parents.

To give an example, consider the following pedigree.

In this pedigree chart, G is the progeny of C and F, and C is the biological uncle of F. To find the coefficient of inbreeding of G, first locate a loop that leads from G to the common ancestor through one parent and back down to the other parent without going through the same individual twice. There are only two such loops in this chart, as there are only 2 common ancestors of C and F. The loops are G – C – A – D – F and G – C – B – D – F, both of which have 5 members.

Because the common ancestors of the parents (A and B) are not inbred themselves, ${\displaystyle f_{A}=0}$. Therefore the coefficient of inbreeding of individual G is ${\displaystyle f_{G}=(0.5^{4}+0.5^{4})\cdot (1+0)=12.5\%}$.

If the parents of an individual are not inbred themselves, the coefficient of inbreeding of the individual is one-half the coefficient of relationship between the parents. This can be verified in the previous example, as 12.5% is one-half of 25%, the coefficient of relationship between an uncle and a niece.

One simple special case is starting by mating two unrelated parents ("the founders", but then in each generation mate two siblings from the previous generation. This case may be analysed directly from the loop summation formula. The offspring of the mating of the founders are not inbred (although they have a positive coefficient of relationship); therefore, they may be called "generation 0". (However, note, that this technically makes the founders to "generation -1".) Thus, the inbreeding coefficient ${\displaystyle f_{g}}$ equals 0 for ${\displaystyle g=0}$ (and for ${\displaystyle g=-1}$). For any positive integer g, any individual in generation g, and any generation h earlier than the parents of that individual, there are exactly ${\displaystyle 2^{g-h-1}}$ loops through that individual connecting to an 'earliest common common ancestor' of generation h; and each such loop contains in total ${\displaystyle 2(g-h)-1}$ ancestors. Thus, the inbreeding coefficient ${\displaystyle f_{g}}$ for this individual may be calculated recursively:

${\displaystyle f_{g}=\sum _{h=-1}^{g-2}2^{g-h-1}2^{-(2(g-h)-1)}(1+f_{h})=\sum _{h}2^{-(g-h)}(1+f_{h})=\sum _{i=2}^{n+1}0.5^{i}(1+f_{g-i})\,.}$

(The simplifications employ some exponentiation rules, and substituting i for ${\displaystyle g-h}$.)

In other words, we get ${\displaystyle f_{1}=0.5^{2}=0.25}$, ${\displaystyle f_{2}=0.5^{2}+0.5^{3}=0.375}$, ${\displaystyle f_{3}=0.5^{2}\cdot 0.25+0.5^{3}+0.5^{4}=0.5}$, and so on. Expressing the coefficients as rounded percents yields the following table:

^A After 20 generations, the individuals are considered to be part of an inbred strain.^[8] Experiments in mice have shown some heterozygosity can still be measured until the 40th generation.^[8]

• Coefficient of relationship
• F-statistics
• Hardy–Weinberg principle
• QST (genetics)