Page 32 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

Using similar arguments:
2
2
p = var(y˜ ) = s + 1/n(s )
22 2 B W
2
2
where s is the between-cow variance and 1/n(s ) is the mean of the within-cow
B W
variance. From Section 1.4:
2 1 2
s = s a
B 4
and for cow i in the group of five cows:
2
2
s = var(y˜ − s )
W 2i B
˜
where y is the mean of the first two lactations for cow i. Since all five cows each have
2i
two records like Zena:
2
2
1
s = (p − s a )
W 11 4
and:
2
2
1/n(s ) = 1/n(p − s a )
1
W 11 4
Therefore:
p = s a + 1/n(p − s a )
1
2
2
1
22 4 11 4
1
1
= (289) + ( )(867 − (289)) = 231.2
1
4 5 4
The index equations are:
− 1
⎡ 1 b ⎤ ⎡ 867 72.25⎤ ⎡ 289 ⎤
=
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
b ⎣ 2⎦ ⎣ 72.25 231.2 ⎦ ⎣ 72.25 ⎦
The solutions are b = 0.316 and b = 0.213 and the index is:
1 2
I = 0.316(230 − 250) + 0.213(300 − 250)
The accuracy of the index is:
r = ( é 0 316. (289 ) + 0 213. (72 5. ) 289/ ) ù = 0 608.
û
ë
1.7.3 Prediction of aggregate genotype
At times, the aim is not just to predict the breeding value of a single trait but that of
a composite of several traits evaluated in economic terms. The aggregate breeding
value (H) or merit for such several or m traits can be represented as:
H = w a + w a + ...+ w a
1 1 2 2 m m
where a is the breeding value of the ith trait and w the weighting factor, which
i i
expresses the relative economic importance associated with the ith trait. The con-
struction of an index to predict or improve H is based on the same principles as those
discussed earlier except that it includes the relative economic weight for each trait.
Thus:
−1
I = P Gw(y − m) (1.22)
16 Chapter 1