Page 29 - Linear Models for the Prediction of Animal Breeding Values

2
2
= b var(y ) + b var(y )+ ... + 2b b cov(y , y ) + ...
1
2
1
2
1
2
1 2
2
2
2
s = b p + b p + ... + 2b b p +...
I 1 11 2 22 1 2 12
or in general:
m ⎛ m m ⎞
ii ∑∑
I ∑ b p + bb pi ; ≠ ⎟
2
2
=
s i ⎜ i j ij j
i=1 ⎝ i= j=1 ⎠
1
where m is the number of traits or individuals in the index.
In matrix notation:
2
s = b′Pb
I
−1
Now b = P G; substituting this value for b:
−1
s = G′P G (1.19)
2
I
The covariance between the true breeding value for trait or individual i and index is:
cov(a , I) = cov(a , b y ) + cov(a , b y ) + ... + cov(a , b y )
i i 1 1 i 2 2 i j j
= b cov(a , y ) + b cov(a , y ) + ... + b cov(a , y )
1 i 1 2 i 2 j i j
or in general:
m
cov( i a ,I ) = ∑ j b g ij (1.20)
j= 1
where g is the genetic covariance between traits or individuals i and j, and m is
ij
the number of traits or individuals in the index.
In matrix notation:
cov(a , I) = b′G
i
−1
Substituting P G for b:
−1
cov(a , I) = G′P G
i
2
I
= s
Thus, as previously, the regression of breeding value on predicted breeding values is
unity. Therefore:
2
a,I I a I I a
r = s /(s s ) = s /s
For calculation purposes, r is better expressed as:
m
∑ j b g ij
j=1
r a,I = 2 (1.21)
a
s
Response to selection on the basis of an index is:
a,I a
R = ir s
= is I
Genetic Evaluation with Different Sources of Records 13