Prediction Theory 1 Introduction 2 General Linear Mixed Model

and
( ) ( )
X
−
′ R −1 X X ′ R −1 Z
θ
Z ′ R −1 X Z ′ R −1 Z + G −1 S
=
(
K
M
Let a solution to these equations be obtained by computing a generalized inverse of
(
X ′ R −1 X X ′ R −1 Z
Z ′ R −1 X Z ′ R −1 Z + G −1 )
denoted as ( )
Cxx C xz
,
C zx C zz
then the solutions are
Therefore, the predictor is
L ′ y =
=
(
θ
S
(
(
)
= −
(
Cxx C xz
C zx C zz
) (
K
M
) ( ) (
K ′ M ′ C xx C xz X ′ R −1 y
C zx C zz Z ′ R −1 y
) ( ) ˆb
K ′ M ′ ,
û
)
.
)
)
.
where ˆb and û are solutions to
( ) (
X ′ R −1 X X ′ R −1 Z
ˆb
Z ′ R −1 X Z ′ R −1 Z + G −1 û
)
=
(
X ′ R −1 y
Z ′ R −1 y
)
.
The equations are known as Henderson’s Mixed Model Equations or MME. The equations are
of order equal to the number of elements in b and u, which is usually much less than the number
of elements in y, and therefore, are more practical to solve. Also, these equations require the
inverse of R rather than V, both of which are of the same order, but R is usually diagonal or
has a more simple structure than V. Also, the inverse of G is needed, which is of order equal
to the number of elements in u. The ability to compute the inverse of G depends on the model
and the definition of u.
The MME are a useful computing algorithm for obtaining BLUP of K ′ b + M ′ u. Please keep
in mind that BLUP is a statistical procedure such that if the conditions for BLUP are met,
then the predictor has the smallest mean squared error of all linear, unbiased predictors. The
conditions are that the model is the true model and the variance-covariance matrices of the
random variables are known without error.
In the strictest sense, all models approximate an unknown true model, and the variancecovariance
parameters are usually guessed, so that there is never a truly BLUP analysis of data,
except possibly in simulation studies.
8