Page 132 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

ˆ ù X¢
é b é R X X¢ -1 X¢ R Z X¢ -1 ù -1
-1
-1
ê ú ê -1 - 1 R Z 1 - 1 - 1 12 2 R Z 3 ú
-
-
¢
¢
¢
1
¢
11
1
¢
ê ˆ u pú ê ZR X 1 1 ZR Z + A p g 1 Z R Z + A g 1 ZR Z 3ú
1
1
p
2
ê ú ê - 1 - 1 -1 21 -1 -1 22 -1 ú
¢
¢
¢
¢
+
ê ˆ m ú ê ZR X ZR Z¢ + A p g Z R Z + A g ZR Z 3ú
p
2
2
1
2
2
2
ê ú ê -1 -1 -1 -1 2 ú
¢
¢
¢
¢¢
ë ˆ pe û ë ZR X ZR Z 1 Z R Z 2 Z 3 RZ ¢ + I 1 /s peû
3
3
3
3
é X ¢R yù
-1
ê -1 ú
= ê 1 ¢ ZR y ú (7.7)
ê ¢ ZR y ú
-1
ê 2 ú
- -1
ë ¢ Z R y û
3
where g are the elements of the inverse of G.
ii
As shown in Section 3.5, each block of equations in the MME above can be
expressed as the sum of the contributions from parents’ records and non-parents’
records. Thus:
′
′
−1
−1
−1
X R X = X RX + X RX n
p
p
n
p
′ n
Expressing Eqn 7.7 as shown for the equations for the block of fixed effects above
and multiplying by R gives:
p
é X X p + X R X n XZ p + X R Z n X Z + X R Z 2 XZ + X R n Z ù
-1
-1
-1
-1
¢ p
¢ n
¢ n
¢ p
¢ p
¢ p
¢ n
¢ n
ê n n 2 n 3 3 3 ú
-
-
n ¢
p ¢
2 Z R Z¢
-
ê ZX p + Z R X n Z Z p + Z R Z n + A a 1 Z Z + n -1 2 A a 2 2 ZZ 3 Z R Z¢ n - n 1 3 ú
n ¢
¢
-
p ¢
p ¢
1
+
1
+
1
1
p
n
n
n
ê - 1 - 1 - 1 -1 -1 -1 ú
¢
2 Z RZ¢
¢
Z
ê ZX p + ZR X¢ 2 n n Z Z¢ 2 p + Z R Z¢ 2 n n + A a 2 Z¢ 2 Z + 2 n 2 + A a 3 Z Z 3 + Z RZ 3 ú
¢ 2
2
2
n
ê ë ZX p + ZR X n Z Z + Z¢ 3 R Z n Z ¢ 3 Z + ¢ 3 R Z 3 Z ¢ 3 Z + ¢ 3 R Z 3 Ia 4 ú û
-1
-1
3 ¢
- -1
+
-1
3 Z
3 Z
n
n
¢ 3
¢ 3
n
n
p
ˆ
é b bù é Xy + X R y n ù
-1
¢ n
¢ p
ê ú ê p n ú
ê a ˆ ú ê Z y + ZR y n ú
-1
n
¢ n
¢ p
p
ê ú = ê ú
-1
ê m ˆ ú ê Z Z¢ 2 y p + ZR y ú
¢ 2
n
n
ê ú ê -1 ú
ë pe ˆ ê û ú ë ê Zy p + ZR y ú û
n
¢ 3
n
¢ 3
−1
−1
The a terms are as defined in Eqn 7.2 and R now equals 1/(1 + Da ). The MME
n
for the solutions of b, u, m and pe can therefore be set up as shown above or as in
Eqn 7.7.
7.3.1 An illustration
Example 7.2
The same data set and genetic parameters as in Section 7.2 are used below to demon-
strate the principles for setting up a RAM with maternal effects in the model using
Eqn 7.5. Recollect that:
é 40 10ù é 0.029 - 0.014ù
-1
G = ê ú and G = ê ú
ë 10 20 û ë - 0.014 0.057 û
116 Chapter 7