Page 159 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

The actual yield deviation at 305 DIM for cow 6 using Eqn 9.2 with uˆ replaced with
YD is −1086.6450.
6
The equation for the partitioning of random regression coefficients for animals
to contributions for parent average, yield deviations and progeny is:
u = W PA + W (YD) + W PC (9.4)
ˆ
anim 1 2 3
with:
PC = å a prog ( 2u ˆ prog - u ˆ mate ) å a prog and W + W + W = I
/
1
2
3
This is the same equation as Eqn 5.8, which partitioned breeding values under the
multivariate model. The weights W , W and W are as defined in Eqn 5.8, but
1 2 3
here W is of the order of orthogonal polynomials for animal effects. Illustrating
i
with cow 6, the weights on parent average (W ) and yield deviation (W ) can be
1 2
calculated as:
−
1
⎡ 2.1520 − 0.9957 2.0986⎤ ⎡ 1.4781 − 0.3473 1.9313⎤
⎢
W = − 0.9957 3.2921 − 0.3580 ⎥ ⎢ − 0.3473 2.4685 0.2326 ⎥
1 ⎢ ⎥ ⎢ ⎥
⎢ ⎣ 2.0986 − 0.3580 5.7284⎥ ⎢ 1.99313 0.2326 4.7107⎥ ⎦
0
⎦ ⎣
′
−1
(QR Q −1 ) −1 −1 )
6 2 ( G a par
+ G a
⎡ 0.66156 0.1940 0.2987⎤ ⎤
⎢
= 0.0935 0.8107 0.2402 ⎥ ⎥
⎢
⎢ ⎣ 0.1175 0.0202 0.7279⎥ ⎦
W1
and:
−1
⎡ 2.1520 −0.9957 2.0986⎤ ⎡ 0.6738 − 0.6484 0.1674⎤
⎢ ⎥ ⎢ ⎥
W = −0.9957 3.2921 −0.3580 − 0.6484 0.8235 − 0.5906
2 ⎢ ⎥ ⎢ ⎥
⎢ ⎣ 2.0986 −0.3580 5.7284⎥ ⎢ 0..1674 − 0.5906 1.0177⎥ ⎦
⎦ ⎣
0
−1
−1
′
′
(QR Q + G a 6 ) −1 (Q R −1 Q )
⎡ 0.33844 − 0.1940 − 0.2987⎤
⎢
= − 0.0935 0.1893 − 0.2402 ⎥ ⎥
⎢
1
⎣ − ⎢ 0.1175 − 0.0202 0.2721⎥ ⎦
W2
The contributions from PA and YD to the random regression coefficients for cow
6 are:
é 0 ˆ u ù - é 0.2560ù - - é 5.0004 ù é -0.1221ù - é 0.4265 ù é -0.5485ù
ê ú ê ú ê ú ê 0.0057 + ê ú ê ú
ú
ú ê
ú
ê
ê 1 ˆ u ú = W 1 ê 0.0016 + W 2 -4.6419 = ú ê 0.0674 = 0.0730 ú
ú ê
ê 2 ˆ u ë ú û ê ë 0.1178ú û ê ë -1.9931 ú ê ê 0.0557ú û ê 0.1389 ú ê 0.19946ú û
û ë
ë
û ë
Analysis of Longitudinal Data 143