Page 134 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

The matrix Z , which relates records to animal effect is:
1
1 2 3 4 5 6 7 8 9
⎡
5 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0⎤
⎢ ⎥
6 0.000.0 0.00.0 0.01.0 0.00.0 0.0 ⎥
⎢
7 ⎢ 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0⎥
⎢ ⎥
8 0.0 0.000.0 0.00.0 0.00.0 1.00.0 ⎥
⎢
⎢
9 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 ⎥
Z 1 = ⎢ ⎥
10 0.0 0.5 0.55 0.0 0.0 0.0 0.0 0.0 0.0 ⎥
⎢
⎢
11 0.0 0.0 0.5 0.0 0.0 0.0 0.5 0.0 0.0 ⎥ ⎥
⎢
12 0.0 0.0 0.0 0.00 0.0 0.0 0.5 0.5 0.0⎥
⎢
⎢
13 0.0 0.5 0.0 0.0 0.0 0.0 0.0 0.0 0.5 ⎥ ⎥
⎢
⎢
14 0.0 0.0 0.5 0.0 0.00 0.5 0.0 0.0 0.0⎥ ⎦
⎣
The first five rows correspond to animals 5 to 9, which are parents, and each has
one record. The last five rows correspond to the records for animals 10 to 14 (non-
parents), which are related to their parents. The matrices Z and Z are exactly the
2 3
same as W and S in Section 7.2.1, respectively, and the vector of observation, y, is the
same as in Section 7.2.1. Apart from the relationship matrix, all the matrices in the MME
can easily be calculated through matrix multiplication from the design matrices
and vector of observation set up above. The inverse of the relationship matrix is set
−1
up only for parents (A ), i.e. for animals 1 to 9, using the procedure outlined in Chapter 2.
p
−1 11
−1 22
The matrix A g is added to animal equations, A g to the equations for maternal
p p
−1 12
−1 21
genetic effects, A g to the animal by maternal genetic equations, A g to the mater-
p p
2
nal genetic by animal equations and 1/s to the diagonals of the equations for per-
pe
manent environmental effects to obtain the MME. The MME are not presented
because they are too large. Solving the MME by direct inversion with the equation
for the first herd set to zero gives the same solutions as from the animal model
(Example 7.1). However, the number of non-zero elements in the coefficient matrix
was 329 compared with 429 in the animal model, due to the reduced number of
equations, indicating the advantages of the RAM.
BACK-SOLVING FOR NON-PARENTS
The solutions for direct animal and maternal effects for non-parents are back-solved
after the MME have been solved.
BACK-SOLVING FOR DIRECT EFFECTS
Solutions for direct animal effect for the non-parents are obtained from parent aver-
age and an estimate of Mendelian sampling using Eqn 3.27. Thus the solution for
the non-parent i is:
ˆ
u = 0.5(uˆ + uˆ ) + k (y − b − mˆ − p ˆe − 0.5(uˆ + uˆ )) (7.8)
ˆ
i s d i i j d d s d
with:
−1
−1
−1 −1
k = r /(r + d g ) = 1/(1 + d a); 2 2
−1
i a = s /s a
e
118 Chapter 7