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Animal model with dominance effect

Model

Linear mixed models can handle a nonadditive genetic model with dominance or epistatic relationship matrices. In a dominance model, we can consider additive and dominance genetic effects simultaneously. In this example, the author assumes an animal model with one fixed effect, an additive genetic effect, a dominance effect, and a residual effect. The additive variance is \(\sigma_a^2 = 90\), the dominance variance is \(\sigma_d^2 = 80\), and the residual variance is \(\sigma_e^2 = 120\). BLUPF90 does not have a function for calculating \(\mathbf{D}^{-1}\), so we need to supply its elements to the program in a file.

Files

A data file (mr131.dat) should be prepared. The pedigree file (mr131.ped) is also required.

The inverse of the dominance relationship matrix is supplied as a text file (mr131.dom). See the textbook for the actual numerical values (p. 227). The prepared file contains more precise values.

 1  1  1.00000000
 2  2  1.00000000
 ...
11 12 -0.24096386
12 12  1.09236948

The parameter file (mr131.par) defines two random effects: additive and dominance genetic effects. The inverse of the dominance relationship matrix is inserted directly into the equations.

DATAFILE
mr131.dat
NUMBER_OF_TRAITS
1
NUMBER_OF_EFFECTS
3
OBSERVATION(S)
5
WEIGHT(S)

EFFECTS:
4  2 cross
1 12 cross
1 12 cross
RANDOM_RESIDUAL VALUES
120.0
RANDOM_GROUP
2
RANDOM_TYPE
add_animal
FILE
mr131.ped
(CO)VARIANCES
90.0
RANDOM_GROUP
3
RANDOM_TYPE
user_file
FILE
mr131.dom
(CO)VARIANCES
80.0
OPTION solv_method FSPAK

Solutions and remarks

The solutions are identical to the reference values in the textbook (p. 228).

Note that there is a rapid algorithm for calculating the inverse of the dominance relationship matrix while ignoring inbreeding, using a specialized “pedigree” file. The algorithm was suggested by Hoeschele and VanRaden (1991). They focused on the parental subclass effects \(\mathbf{f}\), associated with the dominance effect, and their covariance matrix \(\mathbf{F}\). They developed the algorithm to calculate \(\mathbf{F}^{-1}\), while ignoring inbreeding, using the dominance pedigree file. The inverse \(\mathbf{D}^{-1}\) can be derived from \(\mathbf{F}^{-1}\).

The BLUPF90 family can calculate only \(\mathbf{F}^{-1}\). This matrix itself cannot be used to estimate individual dominance effects, although \(\mathbf{F}^{-1}\) is still useful for estimating variance components. Another drawback of this algorithm is that the dominance pedigree file has a somewhat complicated structure. This “parental dominance” method was explained in the previous edition of the textbook but has been removed from the current edition.

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