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Any positive definite, symmetric matrix can be written as the product of a
matrix times its transpose. Henderson noticed that
When animals are inbred, then the elements of
many different values. Quaas (1976) noted that the diagonals of
say Bii were
One of the more efficient algorithms for calculating inbreeding
coefficients is that of Meuwissen and Luo (1992).
Animals should be in chronological
order, as for the Tabular Method.
To illustrate consider the example given
in the Tabular Method section.
The corresponding elements of
for animals A to F would be
The efficiency of this algorithm depends on the number of generations in each pedigree. If each pedigree is 10 generations deep, then each of the vectors above could have over 1000 elements for a single animal. To obtain greater efficiency, animals with the same parents could be processed together, and each would receive the same inbreeding coefficient, so that it only needs to be calculated once. For situations with only 3 or 4 generation pedigrees, this algorithm would be very fast and the amount of computer memory required would be low compared to other algorithms (Golden et al. (1991), Tier(1990)). Additive Matrix
Consider the pedigrees in the table below:
Animals with unknown parents may or may not be selected individuals, but their parents (which are unknown) are assumed to belong to a em base generation of animals, i.e. a large, random mating population of unrelated individuals. Animal 3 has one parent known and one parent unknown. Animal 3 and its sire do not belong to the base generation, but its unknown dam is assumed to belong to the base generation. If these assumptions are not valid, then the concept of phantom parent groups needs to be utilized (covered later in these notes). Using the tabular method, the matrix for the above seven animals is given below.
Now partition into and giving:
The breeding value of any animal with both parents known can be
written as a linear function of the breeding values of animals
with unknown parents in the list of pedigrees, plus Mendelian
sampling effects. Take animal 6 as an example,
Take another example using animal 3:
The following conclusion can be made and it is important to keep in mind the provisions under which it is true. The additive genetic numerator relationship matrix properly accounts for the variances of breeding values under non random mating provided that both parents for all animals are known, except for parents of animals originating from the base population.
The Inverse of Relationship Matrix
The inverse of the additive genetic numerator relationship matrix is needed for the prediction of breeding values in some methods that will be studied. Henderson (1975) made an important (landmark) discovery that allows the inverse of this matrix to be calculated very quickly from a list of animals and their parents. This discovery allowed animal models to be applied to large populations of animals for genetic evaluation.
Recall that the relationship matrix could be written as the
product of triangular matrices and a diagonal matrix as
If none of the animals are inbred, then bii can have only three possible values, i.e. .5, .75, or 1.0.
The inverse of the relationship matrix can be constructed very readily by a set of easy rules. Recall the previous example of seven animals with the following values for bii.
Each animal in the pedigree is processed one at a time, but in any order can be taken. Let's start with animal 6 as an example. The sire is animal 1 and the dam is animal 4. In this case, . Following the rules and starting with an inverse matrix that is empty, after handling animal 6 the inverse matrix should appear as follows:
After processing all of the animals, then the inverse of the relationship matrix for these seven animals should be as follows:
The reader should verify that the product of the above matrix with the original relationship matrix, , gives an identity matrix.
In situations where unknown parents could have resulted from non random matings or prior selection of some type, then it is not appropriate to assume that they belong to the base population. However, when their identity is unknown assumptions are still needed. Westell (1984) and Robinson (1986) assigned phantom parents in place of real parents, or today one could think of them as virtual parents. Each phantom parent is assumed to have only one progeny and all phantom parents are assumed to be unrelated to all other real or phantom animals.
The next assumption was that phantom parents of animals that were born in a particular time period probably underwent the same degree of selection intensity, but perhaps differently for phantom sires versus phantom dams. Thus, the phantom parents were assigned to phantom parent genetic groups depending on whether they were sires or dams and on the year of birth of their (real and only) progeny. In application, genetic groups may also be formed depending on breed composition and/or regions within a country. The basis for further groups depends on the belief in the existence of different selection intensities involved in arriving at those particular phantom parents.
Phantom parent genetic groups are best handled by considering them
as additional animals in the pedigree. Then the inverse of the
relationship matrix can be constructed using the same rules as
before. These results are due to Quaas (1984). To illustrate,
use the same seven animals as before. Assign the unknown sires of
animals 1 and 2 to genetic group 1 (G1) and the unknown dams to
genetic group 2 (G2). Assign the unknown dam of animal 3 to
genetic group 3 (G3). The resulting matrix will be of order
10 by 10 :
There is another potential problem with phantom parent genetic
groups, and that is in the variance of breeding values of all
animals. Take animal 4 from the example, and represent its
breeding value as
Phantom parent genetic groups are used in many genetic evaluation systems today. The phantom parents that are assigned to a genetic group are assumed to be the outcome of non random mating and similar selection differentials on their parents. This assumption, while limiting, is not as severe as assuming that all phantom parents belong to one base population. The effects on variances of breeding values needs to be explored further.
Other Genetic Matrices
Let represent the matrix of dominance genetic relationships among animals. may be constructed from the gametic relationship matrix given earlier. This may or may not be a simple problem. Usually, researchers have only looked at dominance relationships within herds of dairy cattle, for example, and have ignored dominance relationships between herds. Other researchers have changed the model to include a sire-dam interaction and a sire by maternal grandsire interaction (J. Dairy Sci. 74:557), but the procedure only works for noninbred populations.
The problem is that while it may be possible to construct a general with some effort, there has not been any discovery of a simple method to obtain as easily as obtaining . Jamrozik and Schaeffer looked at possibly using the full gametic relationship matrix to account for both additive and dominance relationships simultaneously, but further work is needed.
Additive by Additive Genetic
VanRaden and Hoeschele (1991) published a method to invert the
epistatic matrix for additive by additive genetic effects
(J. Dairy Sci. 74:570). The additive by additive genetic relationship
matrix is given by
Other Epistatic Effects
Except for possibly small experimental situations, there have not been any attempts to go beyond dominance genetic and additive by additive genetic effects. Part of the reason is that estimates of dominance genetic variances have been small in many cases (but not all), and the variances of higher order epistatic effects are expected to be even smaller, making the complicated process of estimating them less appealing. Also, the assumptions needed (in the VanRaden and Hoeschele papers, for example) require a noninbred population, and with real field data there has often been many years of selection and therefore, inbreeding cannot be ignored. Finally, with selection and inbreeding there is joint disequilibrium which creates nonzero covariances between different types of genetic effects where previously these covariances have been assumed to be zero (i.e. joint equilibrium was assumed). Research into this area of study gets very complicated very quickly.
This LaTeX document is available as postscript or asAdobe PDF.Larry Schaeffer