Page 228 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

Since the inheritance of dominance effects is from subclass effects, dominance
effects can be predicted by the inclusion of the inverse of the relationship matrix (D )
*
among dominance effects and subclass effects in the MME. From Eqns 12.5 and 12.13,
the dominance (d) and subclass effect (f) may be predicted as:

d ⎡ ⎤ ⎡ 0 S⎤ ⎡ d⎤ ⎡ ⎤ b
+
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
=
f
e
⎣ ⎦ ⎣ 0Q ⎦ ⎣ ⎦ ⎣ ⎦
f
with:
b
d ⎡ ⎤ ⎡ ⎤ ⎡ 0.75I 0⎤
=
var ⎢ ⎥ = D s 2 d and var ⎢ ⎥ ⎢ 2 d
f
⎣ ⎦ * ⎣ ⎦ ⎣ 00.25R ⎦ ⎥ s
e
where S is the incidence matrix relating d to f, and b equals d minus Sf. Therefore:
−
1
d ⎡ ⎤ ⎡ I − S⎤ ⎡ ⎤ b
=
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
f
⎣ ⎦ ⎣ 0 I − Q ⎦ ⎣ ⎦
e
and the inverse of D can be computed as:
*
⎡
4
⎡ I 0 ⎤ () I 0 ⎤ I ⎡ − S ⎤
3
D −1 = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (12.15)
*
⎣ − S′ I − Q′ ⎦ ⎢ 0 4 R ⎥ ⎣ 0 I − Q ⎦
−1
⎦
⎣
−1
4
From the above, the inverse of D is similar to F with coefficients of on the
* 3
diagonals of dominance effects, − of off-diagonals linking dominance to subclass
4
3
effects, and the coefficients contributed by the subclass effects are multiplied by 4. The
−1
matrix D can then be included in the MME, resulting in the prediction of both
*
dominance and subclass effects. The only disadvantage is that the inclusion of sub-
class effects in the MME will increase the order of equations, but the method can
easily be applied to large data sets.
12.4.3 Calculating the inverse of the relationship matrix among
dominance and subclass effects for example data
Example 12.3
−1
−1
Using the pedigree information in Example 12.1, the calculations of F and D are
*
illustrated.
SETTING UP F −1
−1
Application of rules 1 to 2 in Section 12.4.1 for calculating F generated Table 12.1.
Creating a list of filled subclasses in the first pass (pass 1) through the pedigree in
reverse order generated subclasses A to E (sorted by sire) in Table 12.1. Passes 2 and
3 through this list identified all ancestor subclasses (subclasses F to N). Counts to
212 Chapter 12