Prediction Theory 1 Introduction 2 General Linear Mixed Model
Prediction Theory 1 Introduction 2 General Linear Mixed Model
Prediction Theory 1 Introduction 2 General Linear Mixed Model
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17.2.4 MME and Inverse Coefficient Matrix<br />
The left hand side of the MME (LHS) is<br />
( ) (<br />
X ′ R −1 X X ′ R −1 Z<br />
ˆb<br />
Z ′ R −1 X Z ′ R −1 Z + G −1 û<br />
and the right hand side of the MME (RHS) is<br />
(<br />
X ′ R −1 y<br />
Z ′ R −1 y<br />
)<br />
.<br />
)<br />
,<br />
and<br />
Numerically,<br />
LHS =<br />
⎛<br />
⎜<br />
⎝<br />
21 12 9 9 7 5<br />
12 18 0 3 4 5<br />
9 0 15 6 3 0<br />
9 3 6 20.5 0 0<br />
7 4 3 0 18.5 0<br />
5 5 0 0 0 16.5<br />
RHS =<br />
⎛<br />
⎜<br />
⎝<br />
78<br />
41<br />
37<br />
30<br />
34<br />
14<br />
⎞<br />
.<br />
⎟<br />
⎠<br />
⎞ ⎛ ⎞<br />
ˆµ<br />
Ĉ 1<br />
Ĉ 2<br />
⎟<br />
Ŝ ,<br />
A<br />
⎠ ⎜ ⎟<br />
⎝ ŜB ⎠<br />
ŜC<br />
The inverse of LHS coefficient matrix is<br />
⎛<br />
⎞<br />
.1621 −.0895 −.0772 −.0355 −.0295 −.0220<br />
−.0895 .1161 .0506 .0075 .0006 −.0081<br />
−.0772 .0506 .1161 −.0075 −.0006 .0081<br />
C =<br />
.<br />
−.0355 .0075 −.0075 .0655 .0130 .0085<br />
⎜<br />
⎟<br />
⎝ −.0295 .0006 −.0006 .0130 .0652 .0088 ⎠<br />
−.0220 −.0081 .0081 .0085 .0088 .0697<br />
C has some interesting properties.<br />
• Add elements (1,2) and (1,3) = -.1667, which is the negative of the ratio of σ 2 c /σ 2 e.<br />
• Add elements (1,4), (1,5), and (1,6) = -.08696, which is the negative of the ratio of σ 2 s/σ 2 e.<br />
• Add elements (2,2) and (2,3), or (3,2) plus (3,3) = .1667, ratio of contemporary group<br />
variance to residual variance.<br />
• Add elements (4,4) plus (4,5) plus (4,6) = .08696, ratio of sire variance to residual variance.<br />
Also, ( (5,4)+(5,5)+(5,6) = (6,4)+(6,5)+(6,6) ).<br />
• The sum of ((4,2)+(5,2)+(6,2)) = ((4,3)+(5,3)+(6,3)) = 0.<br />
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