Page 60 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

As VanRaden and Wiggans (1991) indicated, w is always less than unity and therefore
3
less than n , which reflects that PYD or DYD is an unregressed measure of progeny perfor-
3
mance. Note that, for bulls with granddaughters, PYD or DYD does not include informa-
tion from these granddaughters. Also, in the dairy cattle situation, the information from
sons is not included in the calculation of DYD.

Illustrating the calculation of PYD or DYD

The computation of DYD is usually carried out in dairy cattle evaluations and it is
illustrated later for a dairy data set in Example 4.1. Using the beef data in Example 3.1,
the calculation of PYD is briefly illustrated for animal 3, using information on both
female and male progeny, since observations are available on both sexes.
First, the YDs for both progeny of sire 3 are calculated:
YD = (y − b ) = (2.9 − 3.404) = −0.504
5 5 2
YD = (y − b ) = (5.0 − 4.358) = 0.642
8 8 1
Therefore, using Eqn 3.12:
PYD = n u (2YD − a ) + n u (2YD − a )/(n u + n u )
ˆ
ˆ
3 2(5) (5) 5 2 2(8) (8) 8 6 2(5) (5) 2(8) (8)
= 0.2(1)(−1.008 − (−0.019) + 0.2(1)(1.284 − 0.177)/(0.2(1) + 0.2(1))
= 0.059
where n and u are the n and u for the jth progeny. Note that in calculating n ,
2(j) (j) 2 2(j)
it has been assumed that progeny j has no offspring. Thus n − 1/(1 + 2a(1)) = 0.2.
2(5)
Using Eqn 3.12 to calculate the breeding value of sire 3 gives the value of 0.0098,
with w = 0.833 and w = 0.167. This is different from the breeding value reported
1 2
from solving the MME as the granddaughter information (calf 7) has not been included.
3.3.3 Accuracy of evaluations

The accuracy (r) of predictions is the correlation between true and predicted breeding
values. However, in dairy cattle evaluations, the accuracy of evaluations is usually
expressed in terms of reliability, which is the squared correlation between true and
2
2
predicted breeding values (r ). The calculation for r or r requires the diagonal ele-
ments of the inverse of the MME, as shown by Henderson (1975).
If the coefficient matrix of the MME in Eqn 3.4 is represented as:
⎡ C 11 C ⎤ ⎡ C 11 C ⎤
12
12
⎢ ⎥ and a generalized inverse of the coefficiient matrix as: ⎢ 22 ⎥
⎣ C 21 C 22⎦ ⎣ C 21 C ⎦
Henderson (1975) showed that:
22 2
PEV = var(a − aˆ) = C s e (3.14)
Thus the diagonal elements of the coefficient matrix for animal equations are needed
to calculate PEV for animals. The PEV could be regarded as the fraction of additive
genetic variance not accounted for by the prediction. Therefore, for animal i, it could
be expressed as:
22 2 2 2
e
i
i
PEV = C s = (1 − r )s a
44 Chapter 3