General structural model – Part 1:
Power of testing, mean-structure, etc.
Psychology 588: Covariance structure and factor models
Mar 10, 2010

Estimation
2
•
Fitting functions
(
F
ML
,
F
GLS
,
F
ULS
)
of a general SE model have
the same forms as those for path modeling of observed variable
only and CFA, but the implied covariance matrix is differently
defined, e.g.,
•
Properties of the ML, GLS and ULS estimators hold essentially
the same
•
Given a converged solution, all estimates must be substantively
sensible
---
Exercise: fit the model explained in p. 334 to the
political democracy data (with and without the equal-loading
constraints in nested modeling approach); which are given in
the data directory (poldemcov.xls)
(
)
(
)
1
ML
ˆ
ˆ
log
tr
log
F
p
q
−
=
+
−
−
+
Σ
S
Σ
S

Power of chi-square tests
3
•
Given a pair of nesting-nested models, we can set up
H
0
and
H
a
as follows:
H
0
---
the constraints that make the only difference between
the two models are correct, such that
θ
a
=
θ
0
,
and
θ
b
contains free parameters for both
H
0
and
H
a
H
a
---
•
The constraints are often
θ
a
=
0
,
though not necessary;
it
equally holds for constraints at nonzero constants
•
If the nesting model (i.e.,
H
a
is true) has
0
df
,
the test is
about goodness of fit of a hypothesized model
[
]
a
b
,
,
′
′
′
=
θ
θ
θ
a
0
≠
θ
θ