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As far as I understand, a just-identified model has zero degrees of freedom and its model fit indices do not make much sense. On the contrary, overidentified models have a meaningful chi-square, CFI, etc. Some researchers compare just-identified models to overidentified ones using these fit indices.

Is it justifiable to compare fit indices (such as CFI, root mean square error of approximation (RMSEA), standardised root mean square residual (SRMR), and chi-square) between just-identified and overidentified models given these models are nested?

I know I can do it using AIC and BIC. But what about CFI and other ones?

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  • $\begingroup$ Could you give an example of "Some researchers compare just-identified models to overidentified ones using these fit indices." $\endgroup$
    – Kuku
    Commented Jun 5, 2021 at 17:28

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Most model fit indices for just-identified SEMs are not useful, such as CFI and TLI of 1.0 or $\chi^2$ and RMSEA of 0.0, so obviously for those measures it makes no sense at all to use them for comparing a just-identitied model to an over-identified model.

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  • $\begingroup$ Thank you. Do you know by any chance any paper I can refer to in this regard? $\endgroup$
    – Macky Mincer
    Commented Jun 4, 2021 at 21:05
  • $\begingroup$ You're welcome. Any book on SEM should go into this. Apart from books a good resource is the Stata documentation for sem/gsem which is availble for free. Also, David Kenny's website has lots of good info, such as this $\endgroup$
    – Robert Long
    Commented Jun 4, 2021 at 21:08
  • $\begingroup$ Thanks, I checked several books as well as Kenny's website, they never discussed this exact issue of comparing just-identified and overidentified models. $\endgroup$
    – Macky Mincer
    Commented Jun 4, 2021 at 21:40
  • $\begingroup$ Does this answer your question ? If so please consider marking it as the accepted answer, and if you haven't already please consider upvoting it. If not, please let us know why ? $\endgroup$
    – Robert Long
    Commented Jun 26, 2021 at 12:17
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The just identified model has 0 df - it is also called the saturated model.

It has a chi-square of zero, SRMR of zero and RMSEA of zero. Chi-square explicitly compares the fitted model with the just identified model.

It can make sense to compare the AIC and BIC of the fitted and saturated models. If they are not higher in the saturated model, you have a problem.

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