# Inbreeding and Ancestor Loss Coefficients

Von Thomas Bez am 02.08.2018

Weblog Barnimer Land </weblog>

This is an explanation of terms that were used in breeders section of the Pedigree Database. Recommended reading is the article "Coefficients of inbreeding and relationship" by Sewall Wright that can be found in the internet.

# Ancestor-loss Coefficient

The ancestor loss coefficient is a measurement for the number of different ancestors in both the paternal and the maternal line. It is calculated by dividing the number of different ancestors by the total number of ancestors over X generations. (According to this calculation method the coefficient would have to be called strictly speaking "ancestor variety coefficient", because with its maximum value of 100% the ancestor loss is zero.)

The ALC is only meaningful together with the indication of the number of regarded generations. With increasing number of generations the ALC declines usually very fast. This is caused by the fact that the distance of the ancestors counted is not taken into account.

The advantage of the ALC is that it (with limited precision) permits to estimate the variety of the gene pool of an animal.

# Inbreeding Coefficient (COI)

The in-breeding coefficient does not refer to the entire gene pool of an animal. The IC measures the probability that the two existing alleles of any gene (except coincidental mutations) are identical, i.e. they are inherited from the same ancestor. Since equal alleles can be inherited only once over the father and once over the mother, only those ancestors which are represented in the paternal as well as in the maternal line contribute to the IC value.

The computation of the IC also incorporates the distance of the common ancestors from the test subject. This has the consequence that the IC (contrary to the ALC) converges and changes less with increasing number of regarded generations.

This is the formula for the computation of the IC:

1...m are the ancestors who are common to maternal and paternal line. n(a,s) and n(a,d) are the distances (in generations) of the common ancestors from the test subject. Theoretically, also the IC of the respective common ancestors should have to be included into the calculation. Since they have only marginal effect on the result, we exclude this component from the calculation, i.e. we assume F(a)=0.

In the extreme case of mating brother and sister the IC would rise to the maximum possible value of 0.25 (25%). A special effect of the IC is that it can be brought to zero. If we crossed into an established European breeding line a Briard from the rear angle of the world which is really a Briard but not related to our line for X generations, the IC would become zero.

The method for computing the COI is based on the study "Coefficients of Inbreeding and Relationship" by Sewall Wright <https://aipl.arsusda.gov/publish/other/wright1922.pdf> in The American Naturalist 1922. As the calculation of the complete formula, which also includes the individual COI of every ancestor, would be time-consuming and therefore slow, we use a simplified calculation method in the database. The method used by us is usually called "isonomic method".

For Briards we consider COI up to 2% over 7 generations as normal and common. In the range of such low inbreeding degrees, both formulas (the exact and the isonomic) practically deliver the same values. In livestock management, high levels of inbreeding are often desired to achieve a particular result or performance value. (We do not want to rate this here.) The Wright study shows a good example with 46% over four generations. In such an extreme case, both formulas actually produce different results, but we wonder why one should ask for a COI in this range of values ??at all.

The COI converges with increasing numbers of generations being considered. More precisely: The marginal change of the COI, if another generation is included in the calculation, decreases with increasing number of generations. In contrast, the ALC diverges and then always goes towards zero, which it must do, since all Briards are related and the ALC does not consider the distance of the individuals in the generation sequence.

Example: Herzogin aus dem Barnimer Land = Fausto vom Wälkesberg x Clarissa aus dem Barnimer Land

Generations COI ALC
4 0,00% 100,00%
5 0,00% 98,39%
6 0,78% 85,71%
7 0,92% 68,90%
8 0,95% 52,94%
9 0,98% 38,06%
10 0,99% 24,98%

About the same ancestor-loss coefficient Maria Theresa had, and she even became Archduchess.