"...to make people happy".

See p.10 here (attached file):

https://books.google.com.br/books?id=AMLN-oKmC_IC&printsec=frontcover&hl=pt-BR&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false

 SEM using AMOS, GFI was devised by Jöreskog and Sörbom (1984) and Tanaka and Huba (1985). as of conditions of goodness of FIT, GFI is the one of the measures that gives a minimum discrepancy function for to  achieve a perfect FIT when analyses under maximum likely hood condition.

if GFI is less than or equal to 1. A value of 1 indicates a perfect fit. now days most of  they are consider as a fitness criteria in their research paper recently.as of my views it will not lost its popularity as fit index.

GFI was evaluated to be insensitive towards misspecified models so it is not useful to detect a wrong model. I cannot remember on the top of my head which paper that reported such evidence.

A healthy perspective on GFI is to view them as providing qualitative or descriptive information about model fit. The value of this information increases when you report values of indexes that as a set assess model fit from different perspectives. The drawback is the potential for obfuscation, or the concealment of evidence about poor fit. This is less likely to happen if you follow a comprehensive approach to assessing model fit that includes taking the model chi-square test seriously and describing patterns of residuals ( Rex B. Kline,2010).

"Popularity" might be the wrong word to characterize an approach model fit. The best overviews I have found on the issue have been from Herb Marsh et al. He gives not only the technical issues, but historical and philosophical perspective on this difficult area.

Marsh, H. W., Hau, K.-T., & Grayson, D. (2005). Goodness of fit in structural equation models. In A. Maydeu-Olivares & J. J. McArdle (Eds.), Contemporary psychometrics: A festschrift for Roderick P. McDonald (pp. 275-340). Mahwah, NJ: Lawrence Erlbaum.

Marsh, H. W., Hau, K.-T., & Wen, Z. (2004). In search of golden rules: Comment on hypothesis-testing approaches to setting cutoff values for fit indexes and dangers in overgeneralizing Hu and Bentler's (1999) findings. Structural Equation Modeling: A Multidisciplinary Journal, 11(3), 320-341. doi: 10.1207/s15328007sem1103_2

Jim Steiger demonstrated that GFI (and thus, AGFI) has bias based on the ratio of sample size to degrees of freedom, and this is why GFI should be avoided.  When sample size is large relative to degrees off freedom, the bias is positive and so fit looks better.  If degrees of freedom are large relative to sample size, then the bias is negative, and fit looks worse.  There is no rationale for incorporating such a bias into fit assessment, though researchers who who understand these issues may find the index useful as a screening tool, to weed out the worst models in the course of a research project.  Below is a long extract from a Semnet posting by Steiger from 6 January 1995:

--Ed Rigdon

> A colleague of mine obtained some

>unusual fit index results, and I thought I would

>ask for comment. The model has 24 measures, a

>fairly sparse structural model, and N = 160.

>The chi-square is about 245 with 237 df (p = .34).

>Naturally, the CFI rounds to 1.00, but the

>PNFI is also high--.75 or so. However, GFI = .89

>and AGFI = .87. These seem anomalously low, given

>the other indices--although RMR for this correlation-

>based analysis is .056, which seems a touch high, too.

> Since N drives both the chi-square and the GFI in

>the same direction--making one look better while making

>the other one look worse--I suspected an N effect, but

>(and this is my question) does N=160 seem like a *small*

>sample size for this kind of model?

> Thanks--Ed Rigdon

>

>

Dear Ed and Terry:

This result is not as surprising as it seems at first glance,

if you know the results of Steiger (1989) on the sampling distribution of

the GFI.

Indeed, it is exactly what you would expect.

Terry, why don't you compare your Monte Carlo results against the

theoretical formulae

given below, and tell us what you find.

These theoretical results are apparently not widely known, and authors of

other structural modeling

programs seem inclined to ignore them, so here they are.

1. Steiger (1989), in the manual to his computer program EZPATH, presented

an estimation rationale

for goodness of fit assessment in structural modeling. In this approach,

sample-based

fit coefficients were justified as estimates of relevant population

quantities. On this basis,

very few fit coefficients qualified as being of much use, since most were

ad hoc and had no supporting

population rationale. Two notable exceptions were the RMSEA index of

Steiger and Lind, and the GFI

family of indices described by Tanaka and Huba, Bentler, and Joreskog and

Sorbom. Tanaka and Huba

described a "coefficient of determination" rationale for the GFI family

of indices. This rationale

obviously applies to a population as well as a sample.

The "population GFI" is simply that GFI that you would obtain if you KNEW

the population

covariance matrix sigma, fitted your model to it, and computed the

Joreskog-Sorbom GFI formula.

Since the "population GFI" is worth noting, the question arises, "Can we

estimate it?"

2. The answer is, "Yes." Not only can we estimate it, we can construct a

confidence interval on it, and

perform STATISTICAL TESTS on it.

But, guess what. The sample GFI described by Joreskog and Sorbom, Tanaka

and Huba, and others, is

NOT the coefficient to use if you wish to estimate the population GFI.

The sample GFI is a

biased estimator of the population GFI. It is, in general, TOO LOW.

3. Steiger (1989) showed how to obtain a debiased estimator of the

population GFI, AND how to compute

confidence intervals on the population GFI. More than a year later, Maiti

and Mukerjee (1990) published

an (independent) derivation of the bias of the GFI in Psychometrika.

Several of their results are

identical to those of Steiger (1989), who was not a reviewer of their

paper. Larry Hubert was Editor

of Psychometrika at the time, and attended my colloquim presentation on

the topic.

He sent me the Maiti and Mukerjee paper, which was in press at the time,

and which I urged him to publish. Maiti and Mukerjee cited me in a

footnote. They did not present a confidence interval procedure.

Here are the fundamental results from Steiger (1989) that are relevant to

the interesting case you

have presented.

a. Let FPop be the "population fit function," the value of the ML fit

function you would obtain if

you knew sigma, and fitted your model to it.

b. Suppose your sample size is N, your sample fit function if FSamp, and

your degrees of freedom for your

model are v. Then the sample chi-square statistic

X = (N-1) FSamp

has an asymptotic chi-square distribution with v degrees of freedom, and

noncentrality parameter

L = (N-1) FPop

c. The expected value of a noncentral chi-square variate is

E(X) = v + L

It follows that the sample discrepancy function is a rather biased

estimator of the population

discrepancy function.

Specifically

E(FSamp) = 1/(N-1) E(X) = 1/(N-1) L = FPop + v/(N-1)

Hence, a "debiased" estimate of FPop, the "population noncentrality

parameter" is

FHat = FSamp - v/(N-1)

d. Tanaka and Huba, among others, have shown that the GFI, for models that

are ICSF

(invariant under a constant scaling factor), can be computed directly

from FSamp as

GFI = p/(p + 2FSamp)

The population GFI is defined by Steiger (1989) as

Gamma = p/(p + 2FPop)

At large samples, then, the expected value of GFI is APPROXIMATELY

p/(p + 2FPop + 2v/(N-1)), which can be CONSIDERABLY SMALLER than

Gamma!!!!

GFI can be a BADLY BIASED ESTIMATE OF GAMMA. The formula above tells

you that when v/(N-1)

is large, and p is not huge, you are in trouble using GFI. There is no

need to do a Monte

Carlo study to verify this, although it does help to confirm the

essential accuracy of

the asymptotic results.

4. Gamma is the quantity of essential interest, and GFI is not a good

estimator of it. A reasonably

good estimate is obtained by simply substituting the debiased estimate of

FPop (i.e., FHat)

in the GFI formula

GammaHat = p/(p + 2FHat)

My feeling is that such an estimate should only be used in conjunction

with a confidence interval.

My programs EzPATH and SEPATH compute such a confidence interval, as well

as the corrected point estimate.

5. NUMERICAL EXAMPLE. Your numerical example is an excellent example of how

badly the

Joreskog-Sorbom-Tanaka-Huba-Bentler GFI can perform. Specifically, you have

p=24, N = 160, X = 245, v = 237. Note that v is large relative to N, so

there is an immediate problem.

Indeed, if FPop is zero in your case, Gamma is 1.00,

and the expected value of GFI is approximately .89!!

I am assuming your model is ICSF. Most are. My programs check this

assumption numerically.

Notice, FSamp = 1.541, but FHat = 0.05!!! Remember, a CENTRAL chi square

has an expected value of v, and

your chi-square is barely larger than v!!! The best estimate is that

there is very little population

noncentrality for your model.

GFI = 24/(24 + (2)(1.541) = 24/27.08 = .89

But we already know this is a badly biased estimate.

The corrected estimate for population gamma is

GammaHat = 24/(24 + 2(.05) ) = .996.

Thanks everybody for clarification. Now I understand that GFI is highly dependent on sample size. I can find research papers using CFI, RMSEA and chi square/degrees of freedom as measures of fit.

To overcome deficiencies of each method use a range of measures when reporting reporting model fit including GFI....here is an excerpt from my results for a business simulation experiential learning model...the goodness-of-fit measures, for the latent variable model using partially aggregated parcels (Coffman & Macallum, 2005) met requirements as followsX2/ df = .897 where < 5, preferable < 3; Goodness-of-Fit Index (GFI) = .907 where > 0.90 is ideal; Adjusted Goodness-of-Fit Index (AGFI) = .850 where > 0.80 is ideal; Comparative Fit Index (CFI)= 1.000 where > 0.90 is ideal; Root Mean Square Residuals (RMSR) .094 where < 0.1, and closer to 0 is ideal; Root Mean Square Error of Approximation (RMSEA) = 0.000 where less than .080 is ideal; Normed Fit Index (NFI)= .962 where > 0.90 is ideal; and Parsimony Normed Fit Index (PNFI) = .727 where > 0.60 is ideal (Garson, 2007; Screiber, et al., 2006).

"...to make people happy".

See p.10 here (attached file):

https://books.google.com.br/books?id=AMLN-oKmC_IC&printsec=frontcover&hl=pt-BR&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false

Diogenes--

Yes, exactly. In the1969 Psychometrika paper which launched the chi-square statistic, Joreskog warned that, of course, plausible models will fail this test with substantial sample sizes. But people still gave CFA grief for rejecting their models according to this strict criterion. So Joreskog gave them something to make them happy. (There is substantial dispute over who actually invented GFI and AGFI.)