Page 28 - Linear Models for the Prediction of Animal Breeding Values

2
The b values in Eqn 1.15 are obtained by minimizing (a − I) , which is equivalent
to maximizing r . This is the same procedure employed in obtaining the regression
aI
coefficients in multiple linear regression. Thus the b values could be regarded as
partial regression coefficients of the individual’s breeding value on each measurement.
The minimization results in a set of simultaneous equations similar to the normal
equations of multiple linear regression, which are solved to obtain the b values. The
set of equations to be solved for the b values is:
bp + b p 12 + + b p m1 = g 11
2
111
m
bp + b p 22 + + b p m = g 12
2
121
m 2

bp m1 + b p m2 + bp = 1 g m (1.16)
+
1
2
mmm
2
where p and g are the phenotypic and genetic variances, respectively, for individual
mm mm
or trait m; p and g are the phenotypic and genetic covariances, respectively,
mn mn
between individuals or traits m and n.
In matrix form, Eqn 1.16 is:
Pb = G
and:
−1
b = P G
where P is the variance and covariance matrix for observations, and G is the covariance
matrix between observations and breeding value to be predicted.
Therefore the selection index equation is:
−1
ˆ
I = a = (P G)(y − m) (1.17)
= b(y − m) (1.18)
where m refers to estimates of environmental influences on observations, assumed to
be known without error. The application of the selection index to some data therefore
involves setting up Eqn 1.17. From Eqn 1.18 it is obvious that the previous methods
for predicting breeding values discussed in Sections 1.3 to 1.6 are no different from
a selection index and they could be expressed as in Eqn 1.17.

1.7.1 Accuracy of index

As before, the accuracy (r ) of an index is the correlation between the true breeding
a,I
value and the index. The higher the correlation, the better the index as a predictor of
breeding value. It provides a means of evaluating different indices based on different
observations, to find out, for instance, whether a particular observation is worth
including in an index or not.
From the definition above:
r = cov(a, I)/(s s )
a,I a I
First we need to calculate s and cov(a, I) in the above equation. Using the formula
2
I
for the variance of predicted breeding value in Section 1.3.1:
2
s = var(b y + var(b y + ... + 2b b cov(y , y ) + ...
I 1 1 2 2 1 2 1 2
12 Chapter 1