Page 144 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

where:
æ PC1ö æ2 ˆ u D prog - ˆ u D mate ö
=
ç ÷ å a prog ç ç ÷ å a prog
÷ ÷
è PC2 ø è 2 ˆ u S prog - ˆ u S mate ø
The weights W , W and W = I, with:
1 2 3
⎛ 2 ⎞⎞ ⎛ Z′ Z 0⎞
− 1 ⎛a 1 a − 1 i D iD
W = ( DIAG) 2a , W = ( DIAG) ⎜ ⎟
⎟⎟
⎠
1 ⎜ ⎝ par ⎜ ⎝a 2 a 3 ⎠ 2 ⎝ 0 Z′′ Z ⎠
iS
iS
− 1 ⎛a 1 a 2 ⎞
.
and W = ( DIAG) 05 Σ a
3 ⎜ ⎟ ⎠ prog
⎝a 2 a 3
Equation 8.9 is illustrated below using pig 7 in Example 8.1. For pig 7:
ˆ
yd = (y - b - uˆ - uˆ - cˆ - gˆ ) = (5.50 - 6.006 - (-0.098)
1 7 1 s8 s9 1 1
- 0.010 - 0.325 - (-0.274) =-0.469 and
ˆ
ˆ
ˆ
yd = 1/(n - 1)((y + y ) -2b - uˆ - u ˆ - u - u - c - c - 2g )
ˆ
ˆ
ˆ
2 8 9 1 D8 D9 s8 s9 1 2 1
1
-
= ( (9 8. + 4 9 - (8 241. ) 0 527. - - ( 0 878 - - )
.
) 2
) ( 0 098.
.
2
.
–
- 0.010 - 0 325 – - ( 005 4 ) 2- - ( . 0 274 ) ) =-0 . 03 8
.
Since both parents are known:
⎞ ⎛10 ⎞ ⎛ 4 991 − 2 494⎞ ⎞
.
.
DIAG= 2 ⎛a 1 a 2 ⎟ + ⎜ ⎟ = ⎜ ⎟
⎜
.
.
⎝a 2 a 3 ⎠ ⎝02 ⎠ ⎝ −2 494 30 492⎠
Therefore:
⎞ ⎛ 0 .791 − .034 ⎞
0
−1
WT = ( DIAG) 2 ⎛a 1 a 2 ⎟ = ⎜ ⎟ and
⎜
1
0
⎝a 2 a 3 ⎠ ⎝ − .017 . 0 932 ⎠
⎛10 ⎞ ⎛ . 0 209 . 0 034 ⎞
WT = ( DIAG) − 1 ⎜ ⎝0 ⎟ ⎠ 2 = ⎜ ⎝ .0 017 . 0 068 ⎟ ⎠
2
From Eqn 8.9:
⎛ ˆ u D7 ⎞ ⎡ PA ⎤ ⎛ yd ⎞ ⎡ . 0 298⎤ ⎛ − . ⎛ 0 129⎞
0 469⎞
.
1
1
⎜ ⎝ ˆ u ⎠ ⎟ = WT 1 ⎢ ⎣ PA 2 ⎦ ⎥ + WT 2 ⎜ ⎝ yd ⎠ ⎟ = WT 1 ⎢ ⎣ − . 0 0044 ⎥ ⎦ + WT 2 ⎜ ⎝ − . ⎟ = ⎜ ⎝ − . ⎟
0 308⎠
0 075⎠
S7
2
The weights indicate that the relative emphasis on parent contribution was higher
for the SBV compared to the DBV. This might be due to the lower genetic variance
for associative effects in the model.
8.4 Analysis Using Correlated Error Structure
The analysis of the same data using Eqn 8.7 gave the same solutions obtained from
Eqn 8.6. Since the major difference is the structure of the residual covariance, R, this
section has only focused on illustrating the structure of R, for this example. Although
the number of equations using Eqn 8.7 were three less compared to Eqn 8.6, the num-
ber of non-zero elements was higher (481 compared with 462 for Eqn 8.6). This is due
128 Chapter 8