Page 161 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

where V = DW , with D being a diagonal matrix such that d = t , with t being the
i i ii i i
element of the row vector t in Eqn 9.2, PA* = V PA, YD* = V YD and PC* = V PC.
1 2 3
However, V + V + V ≠ I. Thus the estimated BV at 305 days (BV ) from
1 2 3 (305)anim
Eqn 9.5 is:
nr
nr
i å
i å
=
=
*
*
BV anim å u ˆ anim å PA + nr YD + nr PC *
(305 ) (305 ) i
=
=
=
i 1 i 1 i 1 i=1 1
where the contributions to the EBV at 305 days from PA, YD and PC are:
nr nr nr
i ∑
∑ PA , YD , and ∑ PC , respec tively.
*
*
*
i
i
i=1 i=1 i=1
Using Eqn 9.5, the contributions from various sources of information can be calcu-
lated for EBV at days or ages j to n along the longitudinal scale, and this could be
plotted to examine how the contributions vary with days or age.
Using cow 6 in Example 9.2, the matrix D used in calculating the V terms in
Eqn 9.5 is:
D = diag(215.6655, 2.4414, −1.5561)
Using the W and W calculated earlier for cow 6:
1 2
æ 132 7637 41 8391 64 4193ö
.
.
.
V = DW = ç 0 2283 1 9792 0 5864 ÷
.
.
.
1 1 ç ÷
ç ÷
.
.
1 1327
è - 0 1828 - 0 03314 - . ø
æ 82 9018 - 41 8391 - 64 4193ö
.
.
.
ç
.
.
0 2283
V 2 = DW 2 = - . 0 0 4622 - 0 5864 ÷ ÷ and
ç
ç ÷
.
.
è 0 1828 0 0314 - 0 4234 ø
.
æ -26 3320. ö æ -91 9351. ö
+
.
.
u ˆ ( 305 6 = VPA + V YD = ç ç 0 0138 ÷ ç 0 1649 ÷ ÷
A
÷ ç
2
1
)
ç ÷ ç ÷
.
è 0 1178 ø è - 021. 6 60 ø
Therefore, contributions from PA and YD are −26.4049 and −91.9862, respectively,
and:
BV = −26.4049 + 91.9862 = −118.3911
(305)6
Thus contribution from parent average is about 22% of the EBV at 305 days. The
EBV at 305 days calculated above is slightly different from the value of −118.4265
shown earlier, due to rounding.
9.3.3 Calculating daughter yield deviations
The equation for calculating daughter yield deviation under an RRM is the same as
Eqn 5.12 presented for the multivariate models. However, with the RRM, DYD in
Eqn 5.12 is a vector of random regression coefficients and the weights M , M and
1 2
Analysis of Longitudinal Data 145