Thecoefficient of inbreeding (COI) is a number measuring howinbred an individual is. Specifically, it is theprobability that twoalleles at anylocus in an individual are identical by descent from acommon ancestor of the two parents.^[1][2][3][4] A higher COI will make the traits of the offspring more predictable, but also increases the risk of health issues. In dog breeding, it is recommended to keep the COI less than 5%; however, in some breeds this may not be possible withoutoutcrossing.^[5] The average COI for dog breeds is 25%.^[6]

An individual is said to be inbred if there is a loop in itspedigree chart. A loop is defined as a path that runs from an individual up to thecommon 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 inbreedingf 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 biologicaluncle of F. To find the coefficient of inbreeding of G, first locate a loop that leads from G to thecommon 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${\displaystyle f_G=({0.5}^4+{0.5}^4)\cdot (1+0)=12.5\mathrm{\%}}$.

If the parents of an individual are not inbred themselves, the coefficient of inbreeding of the individual is one-half thecoefficient 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 positivecoefficient 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 integerg, any individual in generationg, and any generationh 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 generationh; 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 someexponentiation rules, and substitutingi 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 aninbred 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)

• Genealogy
• Kinship and descent
• Breeding
• Incest
• Population genetics

• Articles with short description
• Short description matches Wikidata