Three groups of goodness of fit (GIF) measures are available: absolute fit indices, incremental fit indices, and parsimony fit indices. However, it is not expected that all of these would be used or reported. This rather depends on the purpose for which the work is intended. For example, different journals may have different requirements regarding what indices to be reported.

Absolute Fit Indices assess how well a researcher’s theoretical model fits the observed data they have collected. The only statistically based measure of fit is that of the Likelihood Ratio Test, better known as the “Chi-square” (CMIN) statistic and its associated “probability” or p-value - which should not be statistically significant if there is a good model fit. However, the x² statistic is very sensitive to sample size and is no longer relied upon as a basis for acceptance or rejection (Schlermelleh-Engel et al. 2003, Vandenberg 2006).

As a result, the use of multiple fit indexes has developed to provide a more holistic view of GOF, taking account not only of sample size but also of model complexity and other relevant issues of the study.

• Schermelleh-Engel, K., Moosbrugger, H., and Müller, H. (2003), “Evaluating the Fit of Structural Equation Models: Tests of Significance and Descriptive Goodnessof-Fit Measures”, Methods of Psychological Research Online, Vol. 8, No. 2, pp.23-74.
• Vandenberg, R. J. (2006), “Statistical and Methodological Myths and Urban Legends”, Organizational Research Methods, Vol. 9, No. 2, pp. 194-201.

Hello there, chi square test is a test in which your data is compared with what is going on in general population (e.g. in social sciences). Hence it is a very crude test. We are supposed to report this test and then ignore it as it is highly sensitive to a sample size (look at Tanaka, 1987). It is also often inflated with non-normal data. In a nut shell, non significant value for this test is not mandatory. CFIs, TLIs, AICs, and BICs are better fit indices. 

Hope it helps a bit

Hi Ramendra,

In fact there is a grate debate about this issue. According to Barret (2007) "the chi-square exact-fit test is the only substantive test of fit for SEM, but, its sensitivity to discrepancies from expected values at increasing sample sizes can be highly problematic if those discrepancies are considered trivial from an explanatory-theory perspective". As the exact fit test might fail, new indices were "created" to indicate the degree to which a model might be ‘‘discrepant’’, rather than a binary NHST fit/no-fit decision. 

I suggest you check for multivariate normality and explory your data better after test your model fit. Check ouliers, correlation, multicolinearity and others.

See:

Barrett, P. (2007). Structural equation modeling: Judging model fit. Personality and Individual Differences, 42(5), 815-824.

Bentler, P. M., & Bonett, D. G. (1980). Significance tests and goodness of fit in the analysis of covariance structures. Psychological Bulletin, 88(3), 588-606.

Gefen, D., Straub, D. W., & Boudreau, M. C. (2000). Structural equation modeling and regression: Guideline for research practice. Communications of the Association for Information Systems, 4(1), 1-77.

Hu, L., & Bentler, M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6(1), 1-55.

Three groups of goodness of fit (GIF) measures are available: absolute fit indices, incremental fit indices, and parsimony fit indices. However, it is not expected that all of these would be used or reported. This rather depends on the purpose for which the work is intended. For example, different journals may have different requirements regarding what indices to be reported.

Absolute Fit Indices assess how well a researcher’s theoretical model fits the observed data they have collected. The only statistically based measure of fit is that of the Likelihood Ratio Test, better known as the “Chi-square” (CMIN) statistic and its associated “probability” or p-value - which should not be statistically significant if there is a good model fit. However, the x² statistic is very sensitive to sample size and is no longer relied upon as a basis for acceptance or rejection (Schlermelleh-Engel et al. 2003, Vandenberg 2006).

As a result, the use of multiple fit indexes has developed to provide a more holistic view of GOF, taking account not only of sample size but also of model complexity and other relevant issues of the study.

• Schermelleh-Engel, K., Moosbrugger, H., and Müller, H. (2003), “Evaluating the Fit of Structural Equation Models: Tests of Significance and Descriptive Goodnessof-Fit Measures”, Methods of Psychological Research Online, Vol. 8, No. 2, pp.23-74.
• Vandenberg, R. J. (2006), “Statistical and Methodological Myths and Urban Legends”, Organizational Research Methods, Vol. 9, No. 2, pp. 194-201.

AMOS is a good program, but many variables are not normally distributed, so the chi-square can be distorted. Keep in mind that depends on the sample size. So, a good alternative is the EQS program, which has a fix for non-normal data (chi-square Satorra-Bentler) and robust statistics for categorical variables.

Best regards.

Sergio.

Dear Ramendra,

You need to understand that Chi Square is a non parametric statistics, and it is very sensitive to sample size. If the sample size is large (normally more than 200 is already considered large), its p-value will tend to be very small and deem to be significant and reject the status quo of the test (null hypothesis) that the sample follows normal distribution.

Thus, in my book I urge the researcher to ignore the Chi Square p-value when reporting the fitness of the measurement model. Please read the attachment.

Regarding p-values, a p-value is not a complete statistic by itself, and is generally overused.  Standard deviations and sample sizes matter.  With large samples, 0.05 is often an unreasonable level to set for "significance."  Actually, the word "significance" is a misnomer. 

Hello Ramendra,

Note that the Chi-square test is used to study if the observed values are similar to the expected values by the model you got. It is important issue to test the Model goodness of fit. But Note that it is sensitive to the sample size, If n > 200 then chi-square p-value will be significant.

So I suggest to use other goodness of fit indices such as:

- CMIN/DF ===> should be less than 5

- TLI ========> should be more than 0.9

- CFI ========> should be more than 0.9

- GIF ========> should be more than 0.9

- NFI ========> should be more than 0.9

- RFI ========> should be more than 0.9

- RMSEA ===> should be less than 0.08

I hope my reply helps you a bit

Nader Mohamed Hello, I am very interested by your answer. Could you please provide me with the article where you got the information that " If n > 200 then chi-square p-value will be significant. " It would be very usefull to me!

Thanks in advance for your precious help!

I suggest to read "A Brief Guide to Structural Equation Modeling" written by Rebecca Weston in "THE COUNSELING PSYCHOLOGIST" 2006. There are enough information to analysis by SEM.

Hello, Is it possible to show the value of the “Chi-square” zero in the structural model?

Hi Ramendra, as Alvin has indicated the chi-square is used as a test of exact fit but we know that all our models are wrong, otherwise they wouldn't be models. As Box noted, "all models are wrong but some are useful', so I suppose we should be using fit statistics that recognize that close fit, rather than exact fit, is what we want to test for. Indices like the RMSEA can do this for us.

So we would want the chi-square to be non-significant but it is not a necessary prerequisite for judging a model to be acceptable. People may not agree with this position, see the papers in Personality and Individual Differences in 2007 for a great discussion on this issue.

Mark