Page 33 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

where w is the vector of economic weights and all other terms are as defined in Eqn 1.17.
The equations to be solved to get the weights (b values) to be used in the index are:
bp 11 + b p 12 + + b p m1 = w g 11 + w g 12 + w g m1
1
2
m
2
1
m
bp + b p 22 + + b p m = wwg + w g 22 + + w g
m 2
mm1
2
121
121
2

1 + 2 + + b p mm = 1 + 2 + + g
bp m1 b p m2 m w g m1 w g m2 w mmmm
In matrix notation these equations are:
Pb = Gw
−1
b = P Gw
It should be noted that it is possible there are some traits in the index that are not
in the aggregate breeding value but may be correlated with other traits in H. Conversely,
some traits in the aggregate breeding value may be difficult to measure or occur late in
life and may therefore not be in the index. Such traits may be replaced in the index with
other highly correlated traits that are easily measurable or occur early in life. Con-
sequently, the vector of economic weights may not necessarily be of the same dimension
as traits in the index, as indicated in the equations for b above. Each trait in the index
is weighted by the economic weight relevant to the breeding value of the trait it is pre-
dicting in the aggregate breeding value.
The index calculated using Eqn 1.22 implies that the same economic weights are
applied to the traits in the aggregate genotype across the whole population. A change
in the economic weight for one of the traits would imply recalculating the index. An
alternative formulation of Eqn 1.22 involves calculating a sub-index for each trait in
H without the economic weights. The final index in Eqn 1.23 is obtained by summing
the sub-indices for each trait weighted by their respective economic weights. Thus:
m
I = S w i (1.23)
I i
i=1
−1
where I = P G (y − m), the sub-index for trait i in H and w = economic weight for
i i i
trait i.
With Eqn 1.23, a change in the economic weights of any of the traits in the index
can easily be implemented without recalculating the index.
To demonstrate that Eqns 1.22 and 1.23 are equivalent, assume that there are
two traits in H, then Eqn 1.23 becomes:

I = I w + I w
1 1 2 2
−1
−1
= P G w (y − m) + P G w (y − m)
1 1 2 2
where G is the covariance matrix between trait i and all traits in the index. Thus:
i
−1
I = P (G w + G w )(y − m)
1 1 2 2
−1
= P Gw(y − m)
which is the same as Eqn 1.22.
Example 1.9
Assume the economic weights for ADG and LP are £1.5 and £0.5 per an increase of
1 kg in ADG and 1% increase in LP, respectively. Using the genetic parameters

Genetic Evaluation with Different Sources of Records 17