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QUANTITATIVE
METHODS
IN
PSYCHOLOGY
Comparative
Fit
Indexes
in
Structural Models
P.
M.
Bentler
University
of
California,
Los
Angeles
Normed
and
non
normed
fit
indexes
are
frequently
used
as
adjuncts
to
chi-square
statistics
for
evalu-
ating
the fit of a
structural
model.
A
drawback
of
existing
indexes
is
that they estimate
no
known
population
parameters.
A new
coefficient
is
proposed
to
summarize
the
relative
reduction
in the
noncentrality
parameters
of two
nested models.
Two
estimators
of the
coefficient
yield
new
normed
(CFI)
and
nonnormed
(Fl)
fit
indexes.
CFI
avoids
the
underestimation
of fit
often
noted
in
small
samples
for
Bentler
and
Bonett's
(1980)
normed
fit
index
(NFI).
FI
is a
linear function
of
Bentler
and
Bonett's
non-normed
fit
index
(NNFI)
that avoids
the
extreme underestimation
and
overestima-
tion
often
found
in
NNFI. Asymptotically, CFI,
FI,
NFI,
and a new
index developed
by
Bollen
are
equivalent measures
of
comparative
fit,
whereas NNFI
measures
relative
fit by
comparing
noncen-
trality
per
degree
of
freedom.
All of the
indexes
are
generalized
to
permit
use of
Wald
and
Lagrange
multiplier
statistics.
An
example illustrates
the
behavior
of
these indexes under conditions
of
correct
specification
and
misspccification.
The new fit
indexes perform very
well
at all
Sample sizes.
As
is
well
known,
the
goodness-of-fit
test
statistic
T
used
in
evaluating
the
adequacy
of a
structural model
is
typically
re-
ferred
to the
chi-square distribution
to
determine acceptance
or
rejection
of a
specific
null
hypothesis,
S =
2(0).
In the
context
of
covariance
structure analysis,
S is the
population
covariance
matrix
and 0 is a
vector
of
more
basic
parameters,
for
example,
the
factor
loadings
and
intercorrelations
and
unique
variances
in
a
confirmatory
factor
analysis.
The
statistic
T
reflects
the
closeness
of 2 =
S(0),
based
on the
estimator
8, to
the
sample
matrix
S, the
sample covariance matrix
in
covariance structure
analysis,
in the
chi-square metric. Acceptance
or
rejection
of
the
null hypothesis
via a
test based
on T may be
inappropriate
or
incomplete
in
model evaluation
for
several reasons:
1.
Some
basic
assumptions
underlying
Tmay
be
false
and
the
distribution
of the
statistic
may not be
robust
to
violation
of
these assumptions.
2. No
specific model
S(0)
may be
assumed
to
exist
in the
population,
and T is
intended
to
provide
a
summary regarding
closeness
of S to
S,
but not
necessarily
a
test
of S
=
2(0).
3. In
small
samples,
T may not be
chi-square distributed;
hence,
the
probability values used
to
evaluate
the
null
hypothe-
sis
may not be
correct.
This research
was
supported
in
part
by
United States Public Health
Service
Grants
DA01070
and
DA00017
and is
based
on a
February
1988
technical report
and a
paperprescnted
at the
Psychometric Society
meetings,
June
1988,
Los
Angeles.
Helpful
discussions with
J. de
Leeuw,
R. I.
Jennrich,
T. A. B.
Snijders,
and J. A.
Woodward;
the
eomputer assistance
of
Shinn-Tzong
Wu; and
the
production assistance
of
Julie
Speckart
are
gratefully
acknowl-
edged.
Correspondence concerning this article should
be
addressed
to P. M.
Bentler, Department
of
Psychology, University
of
California,
Los
Ange-
les,
California
90024-1563.
4.
In
large samples,
any a
priori hypothesis
2 =
S(0),
al-
though only trivially false,
may be
rejected.
As
a
consequence,
the
statistic
T
may
not be
clearly
interpret-
able,
and
transformations
of T
designed
to map it
into
a
more
interpretable
0-1,
or
approximate
0-1,
range have been devel-
oped.
Those
indexes
are
usually called goodness-of-fit indexes
(e.g.,
Bentler, 1983,
p.
507;
Joreskog
&
Sorbom,
1984,
p.
1.40).
A
related class
of
indexes, here called comparative goodness-of-
fit
indexes,
assess
T in
relation
to the fit of a
more restrictive
model. These comparative
fit
indexes, formalized
by
Bentler
and
Bonett
(1980),
are
very
widely
used (Bentler
&
Bonett,
1987)
and are the
sole object
of
this article. Alternative
ap-
proaches
to
evaluating model adequacy
are
reviewed elsewhere
(e.g.,
Bollen
&
Liang, 1988; Bozdogan, 1987;
LaDu
&
Tanaka,
in
press;
Wheaton,
1987).
Although covariance structure analy-
sis is
emphasized,
the
methods
developed here hold
for
any
type
of
structural model,
including,
for
example,
mean-covariance
structures
and
log-linear models.
Although more than
30 fit
indexes have been reported
and
their
empirical
behavior
studied
(Marsh,
Balla,
&
McDonald,
1988),
and
although
new
ones continue
to be
developed (Bollen,
1989),
it is
surprising
to
note
that
they have been developed
as
purely
descriptive statistics. Apparently,
no
population parame-
ter has
been
defined
that
is
being estimated
by any of the
exist-
ing
indexes.
In
this article,
I
define
an
explicit population
com-
parative
fit
coefficient,
provide
two
alternative
estimators
of the
coefficient,
and
investigate
the
asymptotic relations between
the
new
and
previously defined comparative
fit
indexes. Further-
more,
new
indexes based
on
Wald
and
Lagrange multiplier sta-
tistics
are
developed.
Nested
Models
and
Comparative
Fit
In
evaluating comparative model
fit, it is
helpful
to
focus
on
more than
one
pair
of
models. Consider
a
series
of
nested
models,
Psychological
Bulletin,
1990,
Vol.
107,
No
2.238-246
Copyright
1990
by (he
American
Psychological
Association,
Inc.
0033-2909/90/$00.P;5
238
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