Prediction Theory 1 Introduction 2 General Linear Mixed Model

11.1 B ′ b is an estimable function
If B ′ b represents a set of estimable functions of b in the original model, then
1. the estimability of b is unchanged, and
2. the modified equations above do not have an inverse.
11.2 B ′ b is not an estimable function
If B ′ b represents a set of non-estimable functions of b with (p − r(X)) rows, where p is the
number of columns of X, then
1. b is estimable as if X was full column rank, and
2. the modified equations above have a unique inverse.
11.3 B ′ b is not an estimable function
If B ′ b represents a set of non-estimable functions of b with fewer than (p − r(X)) rows, and if
we let ( ) ( )
P11 P 12 X
=
′ V −1 −
X B
P 21 P 22 B ′ 0
then K ′ b is estimable if
(
K ′ 0
) ( ) (
P 11 P 12 X ′ V −1 X B
P 21 P 22 B ′ 0
=
(
K ′ 0
The modified MME do not have a unique inverse in this situation.
)
.
)
12 Restricted BLUP
BLUP is commonly applied to models to evaluate the genetic merit of livestock in order to make
decisions on culling and breeding of animals. In these cases, an objective of selection might be
to improve the performance of animals for one trait while leaving another trait unchanged. In
matrix notation, we might have two functions,
K ′ 1b + M ′ 1u and K ′ 2b + M ′ 2u,
14