Page 25 - Linear Models for the Prediction of Animal Breeding Values

1.5 Breeding Value Prediction from Pedigree

When an animal has no record, its breeding value can be predicted from the evaluations
of its sire (s) and dam (d). Each parent contributes half of its genes to their progeny, and
so the predicted breeding value of progeny (o) is:
ˆ
a = (aˆ + aˆ )/2 (1.9)
o s d
ˆ
Let f = (a + aˆ )/2, then the accuracy of the predicted breeding value is:
s d
1
cov( a , ˆ a + 1 a ˆ )
s
d
r ˆ a o f , = 2 o 2 2
1
2
d
s
s a var( ˆ a + 1 2 a ˆ )
Now:
1
1
1
cov(a , aˆ + aˆ ) = cov(a , aˆ ) + cov(a , aˆ )
1
o 2 s 2 d o 2 s o 2 d
1
1
1
1
1
1
= cov( a + a , aˆ ) + cov( a + a , aˆ )
2 s
2 d
2 s
2 s
2 d
2 d
Assuming sire and dam are unrelated:
1
1
cov(a , aˆ + aˆ ) = cov(a , aˆ ) + cov(a , aˆ )
1
1
o 2 s 2 d 4 s s 4 d d
1
= var(aˆ ) + var(aˆ )
1
4 s 4 d
Substituting the solution for the variance of EBV in Eqn 1.4:
1 1 1 2 2 2
cov(a , aˆ + aˆ ) = (r + r )s a
s
o 2 s 2 d 4 d
1
1
From the calculation above, the term var( a + a ) in the denominator of Eqn 1.9 is
ˆ
ˆ
2 s 2 d
2
2
2
1
also equal to (r + r )s , assuming random mating and the absence of joint informa-
4 s d a
tion in the sire and dam proofs. Therefore:
1 ( 2 2 2
d
2
r
ˆ r o a f , = 4 s r + r )s a = s f = 1 ( 2 s r + r )
d
2 1 ( 2 2 2 2
a 4 d a s a
s s r + r )s
where:
d )
s = évar ( 1 2 a + 1 2 a ù û
ë
s
f
From the above equation, the upper limit for r when prediction is from pedigree is
.
1 2 = 0 7 ; that is, the accuracy of the proof of each parent is unity. Note that when
2
the prediction is only from the sire proof, for instance, then:
r , 1 2 a ˆ = 1 2 r s 2 = 1 2 n n / + k
a o ˆ
s
the accuracy of the predicted breeding value of a future daughter of the sire as shown
in Section 1.4.
Expected response to selection on the basis of average proof of parents is:
R = ir
ˆ
ao, f s a
Substituting s /s for r:
f a
f
R = is
Genetic Evaluation with Different Sources of Records 9