Hello Alizera,

Model-data fit determination is rightly a part of both structural model and measurement model appraisal. The challenge is that there are so many different indices that have been developed over time, and many folks appear to have their personal favorites, eschewing all others. I like David Kenny's quick overview of the families of fit indicators: http://davidakenny.net/cm/fit.htm

Good luck with your work.

Hello Dear David Morse

Thank you for your reply.

however, I would like to know, once we confirm acceptable model fit indices for our model why we need to do it again for Structural model?

Thanks

Dear Alireza shabani shojaei

At first, let me point out that the model fit measures either for the measurement or structural models are typically applied within the covariance-based structural equation modelling shortly known as (CB-SEM). This leads to necessary of briefly mentioning that there is another structural equation modelling approach which is called partial least square structural equation modelling shortly known as (PLS-SEM).

I would like to draw your attention to the big picture of the fit measures. Typically every study has a set of (items/indicators/questions), which altogether can have a correlation or/and covariance matrix. In addition, the researcher so often specifies a research model to validate and further analyze its findings.

If(CB-SEM) is applied then the researchers are able to test to fit measures since it requires normal-distributed input data. This means that there is a benchmark that you can compare the model outcomes with. Therefore, the difference between the data (correlation/covariance) matrix and the (correlation/covariance) matrix implied by the model specification is where the fit measures come from.

There are multiple approaches to classify the fit indices, however, based on what previously stated, two main fit indices groups can be clearly generated, which are the following:

1. Goodness of fit statistics – want large values, compares reproduced correlation matrix to real correlation matrix e.g. (CFI, NFI, TLI)

2. Residual statistics – want small values, look at the residual matrix (i.e. reproduced – real correlation table e.g. (RMSEA, RMSR).

Regarding why the measurement and structural model have to be checked, well, I think the answer lies as the models are not the same. I used a CFA model diagram and SEM model for the same measurement model to illustrate the difference I am willing to mention. One main difference, unlike the measurement model, the structural model has to assess the significance of the hypothesised relationships.

Finally, I have added a table with a set of the fit indices that might help.

Sources for the Fit Indicies:

• Bollen, K., Lennox, R., 1991. Conventional wisdom on measurement: A structural equation perspective. Psychological Bulletin 110, 305–314. https://doi.org/10.1037/0033-2909.110.2.305
• Hair, J.F., 2010. Multivariate data analysis : a global perspective. Pearson Education, Upper Saddle River, N.J.; London.
• Mackinnon, D.P., Lockwood, C.M., Williams, J., 2004. Confidence Limits for the Indirect Effect: Distribution of the Product and Resampling Methods. Multivariate Behav Res 39, 99–99. https://doi.org/10.1207/s15327906mbr3901_4

Source of CFA and CB-SEM model:

• Hair, J.F., 2010. Multivariate data analysis : a global perspective. Pearson Education, Upper Saddle River, N.J.; London.

Hello again, Alizera,

The reason is that the structural model adds more parameters (representing your hypothesized relationships); imprecision of model-data fit is contributed to by both the structural portion and the measurement portion. Hence, it makes sense to quantify the quality of that fit for the full model.

I agree with Belal Edries that the measurement model alone and testing the full (structural & measurement) model represent two different creatures altogether.

Good luck with your work.

David Morse and Belal Edries

Thank you for the detailed and informative answers.

Best regards,

Alireza