Page 90 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

Multivariate analysis traits Univariate analysis traits
Effects WWG PWG WWG PWG

Sex
Male 4.361 6.800 4.358 6.798
Female 3.397 5.880 3.404 5.879
Animals
1 0.151 0.280 0.098 0.277
2 −0.015 −0.008 −0.019 −0.005
3 −0.078 −0.170 −0.041 −0.171
4 −0.010 −0.013 −0.009 −0.013
5 −0.270 −0.478 −0.186 −0.471
6 0.276 0.517 0.177 0.514
7 −0.316 −0.479 −0.249 −0.464
8 0.244 0.392 0.183 0.384

The differences between the solutions for males and females for WWG and PWG in
the multivariate analysis are more or less the same as those obtained in the univariate
analyses of both traits. The solutions for fixed effects in the multivariate analysis
from the MME can be calculated as:

ˆ ⎡ ⎤ ⎡ 11 12 − 1 ⎡ ⎡ 12 ⎤ ⎤
b 1j j n r j n r ⎤ y − a ˆ − g a ˆ 2j
1 1j
⎢ ⎥ = ⎢ ⎥ ⎢ R − 1 ⎢ 1j ⎥ ⎥ (5.4)
⎥
22
21
⎣ b ⎢ ˆ 2j ⎦ ⎣ ⎢ j n r 21 j n r ⎦ ⎥ ⎢ ⎣ ⎣ y ⎢ 2j − g a 1j ˆ − a ˆ ⎥ ⎥
⎦ ⎦
2j
where y and aˆ are the sums of observations and EBVs, respectively, for calves for
ij ij
ˆ
trait i in sex subclass j, b is the solution for trait i in sex subclass j and n is the num-
ij j
ber of observations for sex subclass j. Using the above equation, the solutions for sex
effects for males for WWG and PWG are:
ˆ ⎡ ⎤ ⎡ 11 3r ⎤ − 1 ⎡ 11 12 ⎡ 13 0 . − 0 082) − g 12 ( −0 10)⎤⎤ ⎡4 .361 ⎤
⎤
.
12
( .
b 11 3r r ⎡ r ⎤
=
⎢ ⎥ ⎢ 22 ⎥ ⎢ ⎢ 22 ⎥ ⎢ ⎥⎥ = ⎢ ⎥
− −0 10)⎥⎥ ⎣
b ⎣ ˆ 21⎦ ⎣ 3r 21 3r ⎦ ⎢ ⎣ r ⎣ 21 r ⎦ 20 3 . − g 21 ( −0 082) ( . ) ⎦⎦ . 6 800 ⎦
⎣ ⎢
.
5.2.3 Partitioning animal evaluations from multivariate analysis
An equation similar to Eqn 3.8 for the partitioning of evaluations from multivariate
model was presented by Mrode and Swanson (2004) in the context of a random regres-
sion model (see Chapter 9). Since the yield records of animals contribute to the breeding
values through the vector of yield deviations (YD), equations for calculating YD are
initially presented. From Eqn 5.1, the equations for the breeding values of animals are:
ˆ
-1
-1
-1
(Z′R Z + A -1 G )â = Z′R (y - Xb)
Therefore:
-1
(Z′R Z + A -1 G )â = (Z′R Z) YD (5.5)
-1
-1
with:
ˆ
-1
YD = (Z′R Z) (Z′R (y - Xb)) (5.6)
-1
-1
74 Chapter 5