Page 52 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

ˆ
ˆ
−1
−1
where b = (X′V X)X′V y, the generalized least square solution (GLS) for b, and k′b
is the best linear unbiased estimator (BLUE) of k′b, given that k′b is estimable. BLUE
is similar in meaning and properties to BLUP but relates to estimates of linear func-
tions of fixed effects. It is an estimator of the estimable functions of fixed effects that
has minimum sampling variance, is unbiased and is based on the linear function of
the data (Henderson, 1984). An outline for the derivation of Eqn 3.3 and the equa-
tion for L′y above are given in Appendix C, Section C.1.
As mentioned in Section C.1, BLUP is equivalent to the selection index with
ˆ
the GLS of b substituted for b in Eqn 3.3. Alternatively, this could simply be illustrated
(W.G. Hill, Edinburgh, 1995, personal communication) by considering the index
to compute breeding values for a group of individuals with relationship matrix A,
which have records with known mean. From Eqn 1.17, the relevant matrices are
then:
2 2 and 2
e
P = Is + As a G = As a
with:
2
2 2 or (1 − h )/h 2
e
a = s /s a
Hence:
I = P Gy = (I + aA ) y
−1
−1 −1
which is similar to the BLUP (Eqn 3.3) assuming fixed effects are absent and
with Z = I.
The solutions for a and b in Eqn 3.3 require V , which is not always computa-
−1
tionally feasible. However, Henderson (1950) presented the mixed model equations
(MME) to estimate solutions b (fixed effects solutions) and predict solutions for
−1
random effects (a) simultaneously without the need for computing V . The proof
that solutions for b and a from MME are the GLS of b and the BLUP of a is given in
Appendix C, Section C.2. The MME for Eqn 3.1 are:
′
′
−1
−1
⎡ X R X X R Z⎤ ⎡ ⎤ ˆ b ⎡ X R y⎤
′
−1
⎢ −1 −1 −1⎥ ⎢ ⎥ = ⎢ −1 ⎥
′
′
′
a ˆ
⎣ Z R X Z R Z + G ⎦ ⎣ ⎦ ⎣ ⎣ X R y ⎦
−1
assuming that R and G are non-singular. Since R is an identity matrix from
the earlier definition of R in this section, it can be factored out from both sides of the
equation to give:
′
′
⎡ XX X Z⎤ ⎡ ⎤ ˆ b ⎡ Xy ′ ⎤
⎢ ′ -1 ⎥ ⎢ ⎥ = ⎢ ⎥ (3.4)
′
a ˆ
⎣ ZX ZZ + A a ⎣ ⎦ ⎣ Zy ′ ⎦
⎦
Note that the MME may not be of full rank, usually due to dependency in the
coefficient matrix for fixed environmental effects. It may be necessary to set certain
levels of fixed effects to zero when there is dependency to obtain solutions to the MME
(see Section 3.6). However, the equations for a (Eqn 3.3) are usually of full rank since
V is usually positive definite and Xb is invariant to the choice of constraint.
Some of the basic assumptions of the linear model for the prediction of breeding
value were given in Section 1.2. The solutions to the MME give the BLUE of k′b and
36 Chapter 3