Page 36 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

However, it is desired that there should be no genetic change in trait 2; thus
effectively:
H = w a
1 1
and the index to predict H is:

I = b y + b y
1 1 2 2
To ensure that there is no genetic change in trait 2, cov(I, a ) must be equal to zero.
2
From Eqn 1.20:
cov(I, a ) = b g + b g = 0
2 1 12 2 22
This is included as an extra equation to the normal equations for the b values, and a
dummy unknown, the so-called Lagrange multiplier, is added to the vector of solu-
tions for the index weights (Ronningen and Van Vleck, 1985). The equations for the
index therefore are:

−
1
⎡ 1 b ⎤ ⎡ p 11 p 12 g ⎤ ⎡ g ⎤
11
12
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
2 ⎥ ⎢
⎢ b = p 21 p 22 g 22⎥ ⎢ g 1 12 ⎥ (1.24)
⎢ ⎣ l⎥ ⎦ ⎣ g ⎢ 12 g 22 0⎥ ⎢ 0⎥ ⎦
⎦ ⎣
Example 1.10
Using the same data and parameters as in Example 1.6, construct an index to improve
the aggregate genotype for fast-growing lean cattle using an index consisting of GR
and LP but with no genetic change in LP.
From Eqn 1.23 the index equations are:

⎡
⎡ 6400 − 57.60 62.064⎤ b ⎤ ⎡2752 ⎤
1
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
b
⎥ ⎢
2 ⎥ ⎢
⎢ 57.60 51.80 15.552 b = 62.064 ⎥
⎢
⎢ ⎣ 62.064 15.552 0 ⎥ l⎥ ⎢ 0 ⎥ ⎦
⎦ ⎣
⎦ ⎣
The solutions for b and b from solving the above equations are 0.325 and −1.303.
1 2
Therefore the index is:
) + (−1.303(LP − m ))
ADG LP
I = 0.325(ADG − m
The accuracy of this index (Eqn 1.21) is:
r = ⎣ ( ⎡ 0 325. )(2752 ) +− ( 1 303. (62 064. )) 2752/ ⎤ = 0 544.
⎦
which is lower than the accuracy for the equivalent index in Example 1.6, but with
no restriction on LP, and is also lower than the accuracy of prediction of breeding
value for ADG on the basis of single records. The imposition of a restriction on any
trait in the index will never increase the efficiency of the index but usually reduces it
unless I = 0 for the constrained trait.
i

20 Chapter 1