Page 93 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

correlation. However, there was an increase of about 20% in reliability for WWG for
each animal under the multivariate analysis compared with the univariate analysis. Again
much of the gain in accuracy from the multivariate analysis is observed in WWG.

5.2.5 Calculating daughter yield deviations in multivariate models

The equations for calculating daughter yield deviations (DYDs) with a multivariate
model are similar to Eqn 3.12 for the univariate model except that the weights are
matrices of order equal to the order of traits. The equations can briefly be derived
(Mrode and Swanson, 2004) as follows.
Given the daughter (prog) of a bull, with no progeny of her own, Eqn 5.8 becomes:
ˆ a = W PA W ( YD) (5.9)
+
prog 1 prog 2 prog
Let PC be expressed as in Eqn 5.7:
− 1
PC =0.5 G ∑ a prog (2 a ˆ prog − a ˆ mate ) (5.10)

Substituting Eqn 5.9 into Eqn 5.10 gives:

PC =0.5G - 1 ∑ a prog ( W 1 prog a ˆ anim + W 1 prog a ˆ mate + W 2 prog 2 YD a ˆ mate ) )
-
Since the daughter has no offspring of her own, W = 0, therefore W = I - W .
3 1prog 2prog
Then:
- 1
PC =0.5 G ∑ I (( - W )a ˆ + W (2YD - a ˆ )) (5.11)
prog 2 prog anim 2prog mate
a
Substituting Eqn 5.11 into Eq 5.7 and moving all terms involving â to the left-
anim
hand side gives:
(ZR Z′ − 1 − 1 +0.5G − 1 ∑ W )a ˆ
par 2 prog a prog anim
+ 2G a
− 1 +(Z R′ − −1 −1 ∑ − a ˆ
)
=2G a par PA ZYD + 0.5G W 2 prog a prog (2YD mate )
Pre-multiplying both sides of the equation by the inverse coefficient matrix gives:
(
(
(
ˆ a = M PA)+ M YD)+ M DYD) (5.12)
anim 1 2 3
where:
DYD = ∑ W 2 prog a prog (2 YD - ˆ u mate ) ∑ W 2 prog a prog (5.13)

−1
−1
−1
−1
and M + M + M = I, with M = (DIAG) 2G a , M = (DIAG) (Z ¢ R Z) and M =
1 2 3 1 par 2 3
¢
-
-
1
1
−1
−1
(DIAG) 0.5G SW where (DIAG )=(Z R Z - 1 +0.5G å W ).
2prog prog +2G a par 2prog a prog
a
The matrix W in the equation for DYD is not symmetrical and is of the order of traits
2prog
and the full matrix has to be stored. This could make the computation of DYD cum-
bersome, especially with a large multivariate analysis or when a random regression
model is implemented (see Chapter 9). For instance, in the Canadian test day
model, which involves analysing milk, fat and protein yields and somatic cell count
(SCC) in the first three lactations, it is a matrix of order 36 (Jamrozik et al., 1997).
Multivariate Animal Models 77