Page 184 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

′
′
′
′ ⎤
⎡XX XZ XW ⎤ ⎡ ˆ ⎤ ⎡Xy
b
⎢ −1 ⎥ ⎢ ⎥ ⎢ ⎥
′
′
′
⎢ ZX ZZ a 1 Z W ⎥ ⎢ ⎥ = Zy ⎥ (10.20)
u ˆ
⎢
′ + A a
⎢ −1 ⎥ ⎢ ⎥ ⎢Wy
′ ⎥
q ˆ
′
′
′
⎣ WX WZ WW + A a 2 ⎦ ⎣ ⎦ ⎣ ⎦
⎣
v
with:
2 2 and 2 2
e
2
e
1
a = s /s u a = s /s q
10.7.1 An illustration
Example 10.3
Using the same data set as in Example 10.1 and the same genetic parameters, the pre-
diction of additive genetic effects breeding values at the QTL not linked to the MQTL,
and combined additive genetic effect of the MQTL at the animal level, is illustrated.
From the parameters, a = 0.6/0.3 = 2 and a = 0.6/0.10 = 6. The design matrices
1 2
X and Z are as defined in Example 10.2 and W is now equal to Z. The MME is too
−1
−1
large to show but the matrix W′R W + A a is:
v 2
é 30.796 1.716 -0.888 -16.338 -8.292ù
ê ú
ê 1.716 10.114 -6..408 0.078 1.494 ú
WR W ++ A a = ê - 0.888 - 6.408 14.470 - 1.788 - 4.392ú
¢
-1
-1
v 2
ê ú
ê - 16.338 0.078 -11.788 36.868 - 17.826 ú
ê ë - 8.292 1.494 - 4.392 - 17.826 30.016 ú û û
Solving the MME gave the following solutions:
Effects Solutions
Sex
Male 7.356
Female 5.529
Animal Additive genetic effects not linked to MQTL
1 0.091
2 −0.091
3 0.341
4 0.329
5 0.515
Animal Combined additive genetic effects at the MQTL
1 0.076
2 −0.076
3 0.079
4 0.104
5 0.130
The solutions for the additive effect at the MQTL are the same as the sum of
estimated effects in Examples 10.1 and 10.2. The application of this model may be

168 Chapter 10