Demystifying Inbreeding Coefficients
John Armstrong
While most breeders recognize that a mating between halfsibs or cousins represents inbreeding, the majority probably have no idea which represents the closer inbreeding. This is not helped by the nonstandard definition of inbreeding in some books (e.g. Onstott's "Breeding Better Dogs").The standard definition of inbreeding is that it is any scheme which results in the sire and the dam having common ancestors. This common heritage is expressed by a parameter called the inbreeding coefficient, first proposed by Sewell Wright in 1922. Designated F by Wright (but more commonly IC or IBC by breeders), it can theoretically range from 0 to 100%, and indicates the probability that the two alleles for any gene are identical by descent. Though the primary consequence of inbreeding is to increase homozygosity, the IC is not a direct measure of homozygosity because the two alleles may be the same for other reasons. Within a breed, some proportion of all the genes will be the homozygous because there was only one allele to start with. In that sense, the IC may be regarded as indicating what proportion of the remainder have been made homozygous by inbreeding.
The inbreeding coefficient is a function of the number and location of the common ancestors in a pedigree. It is not a function, except indirectly, of the inbreeding of the parents. Thus, one can mate two highly inbred individuals who share little common ancestry and produce a litter with a very low IC. (Because the potential number of ancestors doubles every generation, eventually you reach a point where the number of ancestors exceeds the number of individuals alive at that time. You are, therefore, bound to find some common ancestors if you go back far enough.) Conversely, it is possible to mate two closely related dogs, both of which have low ICs, and boost the IC substantially.
Calculating inbreeding coefficients
The accepted method is best illustrated by a simple example. Suppose we mate halfsibs, the common ancestor, Anson, being the father. Don is the son of Anson and Bess; Eva the daughter of Anson and Claire. Fred is one of their progeny.
To simplify, we don't show the ancestors that aren't shared:
Now we consider a gene for which Anson carries two different alleles, a1 and a2. Whichever one is passed to Don has a 50% probability of being passed to Fred. There is also a 50% probability that the same allele will be passed from Anson to Eva, and a 50% probability of it being passed from Eva to Fred, if Eva got it. When dealing with events that are contingent (this *and* that must happen), we multiply the probabilities  in this case 0.5 x 0.5 x 0.5 = 0.125 (12.5%). This final number is the probability that Fred will be homozygous for either a1 or a2 because of the common grandfather. If this were the only common ancestor, the inbreeding coefficient for Fred would be 12.5%.
In general, Wright's method is to determine the path from Fred to the common ancestor, Anson, and back again on the other side of the pedigree (FredDonAnsonEvaFred), count the number of individuals in the path, excluding Fred (there are 3, DonAnsonEva) and then calculate 1/2 to the power n, where n is that number. So, in the present case, we have (1/2)^{3} or (1/2 x 1/2 x 1/2) = 1/8, which is 12.5% as we calculated above.
Now, suppose the common ancestor was one of the grandfathers of the parents (i.e. a greatgrandfather of the litter). This adds an individual on each side of the pedigree, so that we will get a path of the type FredXDonAnsonEvaYFred, and the inbreeding on Anson will be (1/2)^{5} or 1/32 (3.125%).
Complications
If we had only a single common ancestor to deal with, life would be relatively simple. However, there are two complications to deal with. The first is that there will be more than one common ancestor. Let's consider the case of first cousins. In human populations such a pairing is prohibited in some societies but allowed in others. We have already calculated the inbreeding for a single shared grandparent. First cousins have two shared grandparents, and we simply add the inbreeding coefficient for each to get 6.25%.
The second complication is that the common ancestor may be inbred. If so, his or her inbreeding coefficient will have to be calculated. To account for this we have to multiply the inbreeding coefficient calculated for Fred by (1 + F_{A}), where F_{A} is the inbreeding coefficient calculated for Anson. For example, if Anson is the product of a mating of first cousins, the total inbreeding for Fred will be 0.125 x 1.0625 = 0.133 (13.3%) if there are no other shared ancestors in the pedigree.
Unfortunately, in the average pedigree, there are a large number of shared ancestors. Therefore, the total inbreeding for a dog cannot generally be calculated manually and appropriate software must be used (e.g. CompuPed). Calculating inbreeding for only the first few generations is not particularly useful. If there are more than one or two common ancestors in four or five generation pedigree, the inbreeding is probably already higher than desirable. Unfortunately, having none is no guarantee that common ancestors will not occur in abundance further back, and some pedigrees of this type still achieve moderately high inbreeding coefficients. Neither can be number of shared ancestors be used as a reliable guide, as the inbreeding coefficient is very sensitive to when and where they occur in a pedigree.
© John Armstrong, 1998
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