Prediction Theory 1 Introduction 2 General Linear Mixed Model

Another alternative might be to partition R and y into a full rank subset and analyze that part
ignoring the linearly dependent subset. However, the solutions for ˆb and û may be dependent
on the subsets that are chosen, unless X and Z may be partitioned in the same manner as R.
Singular R matrices do not occur frequently with continuously distributed observations, but
do occur with categorical data where the probabilities of observations belonging to each category
must sum to one.
15 When u and e are correlated
Nearly all applications of BLUP have been conducted assuming that Cov(u, e) = 0, but suppose
that Cov(u, e) = T so that
V ar(y) = ZGZ ′ + R + ZT ′ + TZ ′ .
A solution to this problem is to use an equivalent model where
y = Xb + Wu + ɛ
for
and
(
u
V ar
ɛ
W = Z + TG −1
)
=
(
G 0
0 B
)
where B = R − TG −1 T ′ , and consequently,
V ar(y) = WGW ′ + B
= (Z + TG −1 )G(Z ′ + G −1 T ′ ) + (R − TG −1 T ′ )
= ZGZ ′ + ZT ′ + TZ ′ + R
The appropriate MME for the equivalent model are
( ) (
X ′ B −1 X X ′ B −1 W
ˆb
W ′ B −1 X W ′ B −1 W + G −1 û
The inverse of B can be written as
)
=
(
X ′ B −1 y
W ′ B −1 y
)
.
but this form may not be readily computable.
B −1 = R −1 − R −1 T(G − T ′ R −1 T) −1 T ′ R −1 ,
The biggest difficulty with this type of problem is to define T = Cov(u, e), and then to
estimate the values that should go into T. A model with a non-zero variance- covariance matrix
between u and e can be re-parameterized into an equivalent model containing u and ɛ which
are uncorrelated.
17