Prediction Theory 1 Introduction 2 General Linear Mixed Model

The degrees of freedom for F are r(H ′ o) and (N − r(X)). Note that
y ′ R −1 y − ˆb ′ X ′ R −1 y − û ′ Z ′ R −1 y = y ′ V −1 y − ˆb ′ X ′ V −1 y.
If G and R are not known, then there is no best test because BLUE of b is not possible.
Valid tests exist only under certain circumstances. If estimates of G and R are used to construct
the MME, then the solution for ˆb is not BLUE and the resulting tests are only approximate.
If the estimate of G is considered to be inappropriate, then a test of H ′ ob = c can be
constructed by treating u as a fixed factor, assuming that H ′ ob is estimable in the model with
u as fixed. That is,
( ) ( ) ˆb X
=
′ R −1 X X ′ R −1 − ( )
Z X ′ R −1 y
û Z ′ R −1 X Z ′ R −1 Z Z ′ R −1 ,
y
=
P zx P zz Z ′ R −1 y
(
Pxx P xz
) (
X ′ R −1 y
)
,
and
(
ˆσ e 2 = (y ′ R −1 y − ˆb ′ X ′ R −1 y − û ′ Z ′ R −1 y)/(N − r
X
Z
)
),
s = (H ′ oˆb − c) ′ (H ′ oP xx H o ) −1 (H ′ oˆb − c),
F = (s/r(H ′ o))/ˆσ 2 e.
11 Restrictions on Fixed Effects
There may be functions of b that are known and this knowledge should be incorporated into the
estimation process. For example, in beef cattle, male calves of a particular breed are known to
weigh 25 kg more than female calves of the same breed at 200 days of age. By incorporating a
difference of 25 kg between the sexes in an analysis then all other estimates of fixed and random
effects would be changed accordingly and also their variances.
Let B ′ b = d be the restriction to be placed on b, then the appropriate equations would be
⎛
X ′ R −1 X X ′ R −1 ⎞ ⎛ ⎞ ⎛
Z B ˆb X ′ R −1 ⎞
y
⎜
⎝ Z ′ R −1 X Z ′ R −1 Z + G −1 ⎟ ⎜ ⎟ ⎜
0 ⎠ ⎝ û ⎠ = ⎝ Z ′ R −1 ⎟
y ⎠ .
B ′ 0 0 φ
d
Because B ′ b = d is any general function, then there are three possible effects of this function
on the estimability of K ′ b in the model. The conditions on B ′ are that it
1. must have full row rank, and
2. must not have more than r(X) rows.
13