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Discrepancy Functions
For scale-free least squares estimation (SLS), , and are obtained by taking
f to be:
(D5)
where .
For unweighted least squares estimation (ULS), , and are obtained by
taking f to be:
(D6)
The Emulisrel6 method in Amos can be used to replace (D1) with:
(D1a)
F is then calculated as .
When G = 1 and r = 1, (D1) and (D1a) are equivalent, giving:
For maximum likelihood, asymptotically distribution-free, and generalized least
squares estimation, both (D1) and (D1a) have a chi-square distribution for correctly
specified models under appropriate distributional assumptions. Asymptotically, (D1)
and (D1a) are equivalent; however, both formulas can exhibit some inconsistencies in
finite samples.
()
[
]
)()()(
,,
g
kl
g
ij
g
klijklij
g
www −=U
C
SLS
F
SLS
() ()
()
() () ()
()
[]
2
2
1
tr
ggggg
SLS
f Σ−=Σ
−
SDS;
1
D
g()
diag S
g()
()=
C
ULS
F
ULS
() ()
()
() ()
[]
2
2
1
tr
gggg
ULS
f Σ−=Σ SS;
()
()
()
∑
=
−=
G
g
gg
FNC
1
1
FCNG–()⁄=
()
()
()
FNFNC )1(1
11
−=−=