Page 23 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

assuming there is no environmental covariance between the half-sib records and t, the
2
1
2
1 2
intra-class correlation between half-sibs, is s /s = h .
4 a y 4
Therefore:
1 2 2
b = s a /[t + (1 − t)/n]s y
2
1 2 2 1 2 1 2 2
= h s y /[ h + (1 − h )/n]s y
2 4 4
2
2
2
= 2nh /(nh + (4 − h ))
2
2
= 2n/(n + (4 − h )/h )
= 2n/n + k
with:
2
k = (4 − h )/h 2
The term k is constant for any assumed heritability. The weight (b) depends on the herit-
ability and number of progeny and approaches 2 as the number of daughters increases.
The accuracy of the EBV is:
,
y))
r = cov( a) var(
a y) / (var(
,
ay
From the above calculations, this could be expressed as:
1 2 2 1 h
r = 2 h s y = 2
ay , 1 2 1 2
4 4
2 2 ⎛ 1 2 (1 − 4 h )⎞ 2 1 2 (1 − h )
h s y ⎜ 4 h + ⎟ s y 4 h +
⎝ n ⎠ n
nh 2
=
2
nh 2 +(4 − h )
n
=
n +k
which approaches unity (1) as the number of daughters becomes large. Reliability of
the predicted breeding value therefore equals n/(n + k).
The equation for expected response when selection is based on the mean of half-
sibs is the same as that given in Section 1.3.2 for the mean of repeated records but
with t now referring to the intra-class correlation between half-sibs.
The performance of any future daughters of the sire can be predicted from the
mean performance of the present daughters. The breeding value of a future daughter
(a ˆ ) of the sire can be predicted as:
daugh.
a ˆ = b(y − m)
˜
daugh.
˜
with y and m as defined in Eqn 1.8, respectively, and:
˜
b = cov(a , y)/var(y)
˜
daugh.
Now:
1
˜
cov(a , y) = cov( a + a , a + a + Σe/n)
1
1
1
daugh. 2 s 2 d* 2 s 2 d
where the subscript d* refers to the dam of the future daughter, which is assumed to
be unrelated to dams (d) of present daughters. Therefore:
cov(a ˜ 1 1 1 1 2
daugh. , y) = cov( a , a ) = cov(a , a ) = s a
s
s
s
s
2
2
4
4
Genetic Evaluation with Different Sources of Records 7