595
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
gklijklij
gwww −=U
CSLS
FSLS
() ()
()
() () ()
()
[]
2
2
1tr ggggg
SLS
fΣ−=Σ −SDS; 1
Dg() diag Sg()
()=
CULS
() ()
()
() ()
[]
2
2
1tr gggg
ULS
fΣ−=Σ SS;
()
()
()
∑
=
−= G
g
gg FNC
1
1
FCNG–()⁄=
()
()
()
FNFNC )1(1 11 −=−=