Page 224 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

where n = 2a /wt, n = 1/wt, n = 0.5a /wt, with wt equal to the sum of the numerator
1 1 2 3 1
of n , n and 3(n ).
1 2 3
The solution for the dominance effect of animal i from the MME is:
⎡ ⎛ ⎞ ⎤
i)
ˆ ⎜∑ ˆ j ⎟ ( y − b − ˆ a ⎥ ( + ii )
ˆ
+
i
ij
⎢ ⎣ ⎝ j ⎠ ⎥ ⎦
d =− ⎢ a 2 c d i k n c a 2
where c is the inverse element of D between animal i and j, and n is the number of
ij
records. For instance, the dominance effect of animal 6 is:
ˆ
d = (0 + (20 − 16.980 − 1.259))/(1 + 1.5) = 0.705
6
The dominance effect for an individual represents interactions of pairs of genes
from both parents and Mendelian sampling; it therefore gives an indication of how
well the genes from two parents combine. This could be used in the selection of
mates.
12.3.2 Solving for total genetic merit directly

Example 12.2
Using the same data and genetic parameters as in Example 12.1, solving directly for
total genetic merit (aˆ + d) applying Eqn 12.4 is illustrated.

SETTING UP THE MME

The design matrices X and Z are exactly the same as in Eqn 12.3. However, in Eqn 12.4,
2
2
G = As + Ds . The matrix D has been given earlier and A can be calculated as outlined
a d
−1
2
in Section 2.2. Then G s is added to Z′Z to obtain the MME (Eqn 12.4). Solving the
e
MME by direct inversion of the coefficient matrix gives the following solutions:
Effects Solutions
Sex
Female 16.980
Male 20.030
Animal + dominance Animal + dominance
1 −0.160 7 0.792
2 −0.160 8 −1.991
3 0.059 9 −0.349
4 0.819 10 −2.683
5 −0.184 11 2.489
6 1.963 12 −0.078

The vector of solutions for additive genetic effects can then be calculated as aˆ =
2
2
−1
−1
s AG g and as d = s DG g for dominance effects, as mentioned earlier. It should
a d
be noted that the sum of a and d for animal i in Example 12.1 equals the solution
ˆ
i i
for animal i above, indicating that the two sets of results are equivalent. The advan-
tage of using Eqn 12.4 is the reduction in the number of equations to be solved.
208 Chapter 12