Page 108 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

−1
−1 −1 ) and 2G a ,
−1
2
1
3
The matrices W , W and W are the product of (Z¢R Z + G a bull par
−1 − 1
Z¢R Z, and 0.5G ∑a prog , respectively. Using Eqn 5.19, the contributions from
different sources of information from different countries to the MACE of a bull can
be computed.
If the progeny in Eqn 5.19 is not a direct progeny of the bull but a maternal
4
4
equals if mate (sire) is known or if unknown.
grandson of the bull then a prog 11 15
Then Eqn 5.19 becomes:
−1 −1 )a ˆ −1 (a ˆ + 0.5(a ˆ + g)) + Z′R Z(CD)
−1
ˆ
(Z′R Z + G a bull bull = G a par sire mgs
−1 (a - 0.5a )
ˆ
ˆ
+ G ∑a prog prog mate
and 0.5â = 0.5â , the sire of the progeny.
par
and a bull now equals 2a + 0.25a prog mate s
The above can be expressed as:
−1 −1 −1 −1
bull bull a par
(Z′R Z + G a )a ˆ = 2G (PA) + (Z′R Z)CD
−1
ˆ
prog prog mate
+ 0.25G ∑a (4a - 2a ˆ )
−1 −1 −1
bull
Pre-multiplying both sides by (Z′R Z + G a ) gives the same equation as
Eqn 5.19 but with:
PC = ∑ a prog a ( 4ˆ prog − 2ˆ a ) ∑ a prog (5.20)
mate
and W now equals ( ′ −1 −1 ) (0.25 G −1 ∑ )
−1
ZR Z
3 + G a bull a prog
The use of Eqn 5.19 to partition proofs from MACE is illustrated for two
bulls, one with no progeny and another with a maternal grandson. First, consider
bull 3 in Example 5.5 that has DRPs only in country 1 and has no progeny. Therefore,
CD for bull 3 in country i is:
3i
CD = y − m = 19.2651 − 7.268 = 11.997 and CD = 0
31 31 1 32
Parent average for bull 3 (PA ) in country i is:
3i
PA = 0.5(aˆ ) + 0.25(aˆ + gˆ ) = 0.5(4.310) + 0.25(2.176 + (−0.067)) = 2.68225
31 71 21 G51
and:
PA = 0.5(a ) + 0.25(a + g ˆ ) = 0.5(3.071) + 0.25(0.403 + (−0.010)) = 1.63375
ˆ
ˆ
32 72 22 G52
ˆ
ˆ
where a is the breeding value of animal j in country i and g is the solution for
ji Gji
group j and in the ith country.
1
The residual variance for bull 3 in country 1 (r ) = ( )206.5 = 10.325 and its
31 20
16
= .
inverse equals 0.09685. Both sire and MGS of bull 3 are known, therefore a bull 11
Then:
′ (
.
−
−1
1
ZR Z + G a bull ) = ⎛0 09685 0. ⎞ ⎟ + ⎛ ⎜ 0 4620 −0 62436. ⎞ ⎟
⎜
⎝0 ⎠ 0 ⎝ −0 62436 0. ..99693⎠
⎛ . 0 55884 − . 0 62436 ⎞
= ⎜ ⎝ − . 0 62436 . 0 99693⎠ ⎟
92 Chapter 5