Page 148 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

where y is the vector of TD yields, b is a vector of solutions for HTD and fixed regres-
sions, and u and pe are vectors of animal additive genetic and permanent environ-
mental effects, respectively. The variances of u and pe are as defined in Eqn 4.1. The
matrices X, Q and Z are incidence matrices and are described in detail in the next
2
section, which illustrates the application of the model. It is assumed that var(u) = As ,
u
2
2
and var(pe) = Is , and var(e) = Is = R. The MME for Eqn 9.1 are:
p e
′
⎛ XX X Q X Z⎞ ⎛ ˆ ⎞ ⎞ ⎛ Xy′ ⎞
′
′
b
⎜ − 1 ⎟ ⎜ ⎟ ⎜ ⎟
′
′
⎜ QX Q Q + A a 1 Q Z ′ ⎟ ⎜ u ˆ ⎟ = Qy′ ⎟
⎜
⎜ ⎝ ZX Z Q Z Z + a 2 ⎟ ⎜ p ˆ e ⎟ ⎠ ⎝Zy′ ⎠
′
′
′
⎠ ⎝
2 2 2 2
1
e
2
e
with a = s /s u and a = s /s p.
9.2.1 An illustration
Example 9.1
Given in Table 9.1 are the test day fat yields of five cows in a herd with details of
HTD and days in milk (DIM). The aim is to estimate solutions for HTD effects,
regression coefficients for a fixed lactation curve fitting Legendre polynomials of
order 4, solutions for permanent environmental effects and breeding values for ani-
mal effects using Eqn 9.1. Assume that the estimated variances for additive genetic
2
effects, permanent environmental effects and residual variances were 5.521 kg ,
2
2
8.470 kg and 3.710 kg , respectively. Then:
2
2
a = s /s = 3.710/5.521 = 0.672
1 e u
and:
2
2
a = s /s = 3.710/8.470 = 0.438
2 e p
Table 9.1. Test day fat yields (TDY) for some cows in a herd.
Animals
4 5 6 7 8
DIM HTD TDY HTD TDY HTD TDY HTD TDY HTD TDY
4 1 17.0 1 23.0 6 10.4 4 22.8 1 22.2
38 2 18.6 2 21.0 7 12.3 5 22.4 2 20.0
72 3 24.0 3 18.0 8 13.2 6 21.4 3 21.0
106 4 20.0 4 17.0 9 11.6 7 18.8 4 23.0
140 5 20.0 5 16.2 10 8.4 8 18.3 5 16.8
174 6 15.6 6 14.0 9 16.2 6 11.0
208 7 16.0 7 14.2 10 15.0 7 13.0
242 8 13.0 8 13.4 8 17.0
276 9 8.2 9 11.8 9 13.0
310 10 8.0 10 11.4 10 12.6
DIM, days in milk; HTD, herd–test–day.

132 Chapter 9