The Inbreeding Coefficient and Probability
from:  Probability and its relationship to parts of Livestock production
by Jori (Garoutte) Butler

The inbreeding coefficient, as used in linebreeding, is a measurement of the probable chance that any genes will double when two animals are mated.  This probability increases with the degree of relationship between the two individuals.  By knowing the probability in such instances, a breeder can safely decide whether to try linebreeding in some cases.

Inbreeding coefficients and their related probabilities (percent chance of genes doubling) can be seen in the following table:

INBREEDING COEFFICIENTS FOR VARIOUS MATINGS

kind of mating                                                                         inbreeding %
*Half first cousin (1 common grandparent)                                        3.12
*first cousin (2 common grandparents)                                              6.25
*Grandparent-Grandoffspring                                                          12.50
*Half brother-sister                                                                         12.50
*Parent-offspring                                                                            25.00
*Full brother-sister (1 generation)                                                     25.00
*Full brother-sister (2 generations)                                                   37.50
*Full brother-sister (3 generations)                                                   50.00
(Boggess)

One can see from the previous table that as relationship increases, so does the inbreeding coefficient and the probability that genes will double in each mating.

Knowing this probability is very important to breeders who incorporate linebreeding into their programs as the doubling of genes can either greatly improve or massively decline the quality of livestock as not only the good genes are doubled.  The bad genes have the same probability of doubling and becoming dominant.

Source:
Boggess, Mark V.  "Application of Genetic Principles" Cattle Producers Library.

(from:  J. Garoutte, Mathematical Applications in Animal Sciences)

*Crossbreeding*
In crossbreeding, heterosis or hybrid vigor, is often a common calculation in determining whether or not to cross two breeds.  Heterosis is the degree to which an offspring deviates from its parent's breeds.  The amount of heterosis is calculated where breed A and breed B and crosses between them are raised as contemporaries.  In this manner, the amount of heterosis is calculated as follows:
AB+AB - A+B
2         2

The percent of heterosis is calculated as follows:
amount of heterosis  * 100
A+H/2

These formulas can be figured with many examples, but a common one is applied to weaning weights.  The percentage of heterosis can also be calculated as the sum of the crossbred average minus the straightbred average multiplied by 100, all divided by the straightbred average.  (Crossbreeding . . . Western Range . . .)

Annual progress for inherited traits can also be calculated.  It is equal to the heritability multiplied by the selection differential, all divided by the generation interval.

The heritability is the proportion of differences between animals which are transmitted to the offspring.  The selection differential is the difference between the average of a group with the selected individual, and the generation interval is the average length of the generation.  (Gregory 9,10)

*Linebreeding*
The math used in linebreeding is not very difficult, but it is quite interesting.  The inbreeding coefficient is a percentage of the probable chance that genes will double when related animals are mated.  It is often used in determining whether or not ot mate two related animals because the higher the inbreeding coefficient is, the more likely it is that genes will double, whether they be good or bad.

The different inbreeding coefficients are related to the different kinds of matings.  In a half-first cousin mating, 1 common grandparent, the inbreeding coefficient is 3.12%.  First cousin matings, 2 common grandparents, have a 6.25% inbreeding coefficient.  Both grandparent-grandoffspring and half-sibling matings have an inbreeding coefficient of 12.5%.  Parent-offspring matings and full-sibling matings, for 1 generation, have an inbreeding coefficient of 25%.  Two generation full-sibling matings have an inbreeding coefficient of 37.5% and full-sibling matings for 3 generations have an inbreeding coefficient of 50%.

By considering the inbreeding coefficient, breeder can determine whether or not it will be beneficial for them to linebreed two animals.  (Boggess 1020-3)

The rate of inbreeding per generation can also be calculated considering that the herd is basically closed to outside influence and all mating is basically random where relationship is concerned.  It is calculated where m is the number of males used each generation and f is the total number of females in the herd each generation by the following forumula:  1/8m+1/8F.  (Gregory 14)

Sources:
Boggess, Mark V.  "Application of Genetic Principles"  Cattle Producer's Library

Crossbreeding Beef Cattle for Western Range Environments.  Nevada Agricultural Experiment Station College of Agriculture University of Nevada-Reno, Nov. 1988

Gregory, Keith E.  Beef Cattle Breeding.  United States Department of Agriculture