Page 101 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

However, var(a), where a¢ = [a , a ,...,a ], the vector of breeding values, is:
1 2 n
var(a) = A*G
where * refers to the direct product, A is the relationship matrix and G the covariance
matrix for additive genetic effects. Schaeffer et al. (1978) discussed this model in
detail but from the standpoint of variance component estimation.
Assuming there are two traits, the model for the analysis is as given in Eqn 5.1
but with R and G defined as above. The MME for the BLUP of a and estimable func-
tions of b are:

′
ˆ ⎡ 1 b ⎤ r ⎡ 11 XX 1 0 r 11 X ′ 1 Z 1 0⎤ ⎡ ⎢ X 1 r ′ 11 ⎤ ⎥
1 y
1
⎢ ⎥ ⎢ 22 ′ 22 X ′ ⎥ ⎢ ⎢ X ′ 22 2 y ⎥
⎢ ˆ 2 b ⎥ ⎢ 0 r XX 2 0 r 2 Z 2 2⎥ = ⎢ 2 r ⎥
2
1 12⎥
⎢
⎢ 1 ˆ a ⎥ r 11 ZX 0 r 11 ZZ + − 1 11 A g ⎢ Z 1 r ′ 11 y 1⎥
′
−
′
1 A g
⎢ ⎥ ⎢ 1 1 1 ⎥ ⎢ ⎥
′
2 ˆ a ⎣ ⎦ ⎢ ⎣ 0 r 22 ZX 2 A g r 22 Z Z + A g ⎦ ⎥ ⎣ ⎢ Z r 22 y 2⎥ ⎦
′
′
−
−1 22
1
1 21
2
2
2
2
An illustration
Example 5.4
Consider the following data on the progeny of three sires born in the same herd;
assuming that selection is for dual-purpose sires, such that the male and female calves
are raised on different feeding regimes, with males recorded for yearling weight and
females for fat yield:
Calf Sex Sire Dam HYS Yearling weight (kg) Fat yield (kg)
4 Female 1 Unknown – – –
9 Male 1 4 1 375.0 −
10 Male 2 5 2 250.0 −
11 Male 1 6 2 300.0 −
12 Male 3 Unknown 1 450.0 −
13 Female 1 7 1 − 200.0
14 Female 3 8 2 − 160.0
15 Female 2 Unknown 3 − 150.0
16 Female 2 13 2 − 250.0
17 Female 3 15 3 − 175.0
HYS, herd–year–season.
The aim is to estimate HYS effects for both traits and predict breeding values for
yearling weight and fat yield for all animals, carrying out a multivariate analysis.
Note that animal 4 is just an ancestor and has no yield record for either trait. Assume
that the additive genetic covariance matrix (G) is:

⎡ 43 18⎤
0
,
G = ⎢ ⎥ and R = diag(77 7 )
⎣ 18 30 ⎦
Multivariate Animal Models 85