Page 107 - Linear Models for the Prediction of Animal Breeding Values

In the next section, some of the bull solutions are partitioned to contributions
from various sources to gain a better understanding of MACE.

Equations for partitioning bull evaluations from MACE

The equations for sire proofs from Eqn 5.15 are:
(5.16)
−1
−1
−1
−1
−1
ˆ
(Z′R Z + A −1 Ä G )a = (A Q Ä G )gˆ + Z′R (y - Xcˆ)
where:
ˆ a = Qg ˆ + ˆ s
Thus Eqn 5.16 can be expressed as:
(Z′R Z + A −1 Ä G )a = (A Q Ä G )gˆ + Z′R −1 Z(CD) (5.17)
−1
−1
ˆ
−1
−1
where:
−1
−1
−1
CD = (Z′R Z) (Z′R (y - Xcˆ))
CD (country deviation) is simply a vector of weighted average of corrected DRP in
all countries where the bull has a daughter, the weighting factor being the reciprocal
−1
of EDC multiplied by the residual variance in each country. Since R is diagonal, CD
is equal to the vector (y − Xc).
ˆ
For a particular bull with a direct progeny (e.g. son), Eqn 5.17 can be written as:
−1 −1 )a −1 (a ˆ + 0.5(a ˆ + g)) + Z′R Z(CD)
−1
ˆ
ˆ
(Z′R Z + G a bull bull = G a par sire mgs
−1 (a ˆ - 0.25a ˆ ) (5.18)
+ G ∑a prog prog mate
1
8
2
8
where a = , , or if both sire and MGS (maternal grandsire), only MGS, only sire
par 11 15 3 2
= if bull’s mate (MGS of the progeny)
8
or no parents are known, respectively; and a prog 11
2
3 par prog
is known or if unknown. The above values for a and a are based on the assump-
−1
tion that A has been calculated without accounting for inbreeding. Note that in Eqn 5.18:
bull par prog
a = 2a + 0.5a
Equation 5.18 can be expressed as:
−1 −1 −1 −1
ˆ
bull bull par
(Z′R Z + G a )a = 2G a (PA) + (Z′R Z)CD
−1
prog prog mate
+ 0.5G ∑a (2a ˆ - 0.5a ˆ )
where:
PA = 0.5ˆ a sire + 0.25(ˆ a mgs + ) ˆ g
−1 −1 −1
Pre-multiplying both sides of the equation by (Z¢R Z + G a bull ) gives:
ˆ a = W PA + W YD + W PC (5.19)
bull 1 2 3
where:
)¤
ˆ
ˆ
PC = ∑ a prog (2a prog - 0.5a mate ∑ a prog and W + W + W = I
3
2
1
Multivariate Animal Models 91