Page 183 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

10.7 Directly Predicting the Additive Genetic Merit at the MQTL

Another approach to reduce the number of equations in the MME is to directly
predict the combined additive genetic effects for the paternal and maternal alleles at
the MQTL of an individual. The number of equations per animal would therefore
be two: one for the additive genetic effects not linked to the MQTL and the other
for MQTL. This implies predicting the additive genetic effects at the MQTL at the
animal level; therefore, a covariance matrix (A ) for the MQTL at the animal level
v
1
is needed. The covariance matrix A can be obtained from G as A = 2 BG B′; where
v v v v
B = I Ä [1 1], with n being the number of animals, and Ä denotes the Kronecker
n
product. For Example 10.1, the matrix B = W in Section 10.5 and A is:
v
é 1.000 0.000 0.500 0.950 0.945ù
ê ú
ê 0.000 1.000 0.500 0.050 0.055 ú
ê
A v = 0.500 00.500 1.000 0.590 0.631ú
ê ú
ê 0.950 0.050 0.590 1.162 1.072 ú
ê ë 0.945 0.055 0.6631 1.072 1.228 ú û
Equation 10.11 can be used to obtain the inverse of A . However, the vector s
v i
containing the contributions from ancestors is needed and this can be computed using
−1
Eqn 10.10. The vector s for the ith animal needed to calculate A is shown in Table 10.1.
i v
The inverse of A is:
v
é 4.966 0.286 - 0.148 - 2.723 - 1.382ù
ê 1.519 - ú
ê 0.286 1.068 0.013 0.2249 ú
ê
-1
A v = - 0.148 - 1.068 2.245 - 0.298 - 0.732ú
ê ú
ê - 2.723 0.013 - 0.298 5.978 - 2.9971 ú
ê ë - 1.382 0.249 - 0.732 - 2.971 4.836 ú û
The model for the prediction now becomes:
y = Xb + Zu + Wq + e (10.19)
where all terms are as defined in Eqn 10.17 except that W is now identical to Z and
relates additive genetic effects at the MQTL to animals. Both matrices Z and W are
identity matrices and are of the order of animals. The vector q is the vector of additive
genetic effects at the MQTL and is equal to the sum of the additive genetic effects
of the paternal and maternal alleles for the animal. The variance–covariance matrix
2
2
2
of q = 2A s = A s , since s = 2s . The MME for the above model are:
2
v v v q q v
Table 10.1. Vector (s ) with contributions at the MQTL from ancestors
i
(animals 1 to 4) to animals 2 to 5 using the pedigree in Example 10.1.
Elements in s relating to animal
i
Animal 1 2 3 4
2 0.0000
3 0.5000 0.5000
4 0.8600 −0.0400 0.1800
5 0.2857 –0.0514 0.1514 0.6143

Use of Genetic Markers in Breeding Value Prediction 167