FIT indices behave as if model is saturated, why?

[Sofia Orellana]

Hi everyone, 

I am trying to understand why  the fit indices of my model behave like a saturated model when I still have a degree of freedom. I have some ideas as to why (below) but it is unclear to me if these are correct. I am also struggling with interpretation and would like some guidance. 

For context on my results:

I have a model with one degree of freedom and a very large N (~10.000). 

My robust RMSEA is 0.000, robust CI's [0.000, 0.017]. My CFI and TLI are 1. 

These are the values associated with my chi-square:

For the RMSEA at least, I can intuit that my chi-square is just smaller than my degrees of freedom (which is 1) and so it defaults to zero as the denominator in the sqrt((x^2-df)/(df(N-1)) formula for computing RMSEA is negative. I assume something similar is happening with the CFI and TLI.  Is this correct?

Additionally, it is unclear to me how I should interpret these results. Technically my model is not saturated right? But is it the case that all models where X^2 is smaller than the DFs behave (in their fit indices) like a saturated model?

Finally, I assume the x^2 is small because there is little discrepancy between observed and modelled data, as X^2 is just ~F(N-1) where F should be the maximum likelihood discrepancy between my observed and implied data/var-covar matrices. Additionally this X^2 is non-significant. So:

1) I assume this means that there is good agreement between my model and my data. 

2) But, aren't these results saying my model is overfitting the data? and sorry for the silly question but: can you overfit even when your model is not saturated?

3) Are there circumstances without overfit  where you could get  RMSEA=0, CFI/TLI=1?

Thanks!

[Edward Rigdon]

The formulas for RMSEA and CFI are such that, if chi-square < df, then RMSEA = 0 and CFI = 1.

With 1 df, your data does not disagree with your model, but it has a very limited range to disagree. Especially if the model was specified with an eye to fit, it may be better to say that the model does not disagree with the data used to specify the model, which will not be a surprise.

There is no way to assess "overfit" from the information provided. Your model may be an insightful depiction of a complex system or it may be that the constraints that give rise to your 1 df are trivial and not of interest.

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