CFA: chi-square value is 0 but with degrees of freedom

Question

I want to do a SEM analysis with an actor-partner interdependence model in Mplus. I managed to calculate it and everything seems right if I look at the means, SD's, effects, etc. But I have a problem with the model fit. The fit indices are weird, and too good... So there must be something wrong with it. I read that if one has a chi-square value of 0 as well as 0 degrees of freedom, the model is saturated and therefore a model fit test isn't done. But even though my chi-square value is 0, my df is 12. Also all other model fit indices are perfect. This can't be right.. Does anyone know what the problem might be, or what this tells me? Because to be quiet frank, I have no plan. And I can't seem to find any information on it...

Please, please, please help me!

Best, Annie

This is the output:

MODEL FIT INFORMATION

Number of Free Parameters 15

Loglikelihood

      H0 Value                        -927.087
      H1 Value                        -927.087

Information Criteria

      Akaike (AIC)                    1884.173
      Bayesian (BIC)                  1928.082
      Sample-Size Adjusted BIC        1880.627
        (n* = (n + 2) / 24)

Chi-Square Test of Model Fit

      Value                              0.000
      Degrees of Freedom                    12
      P-Value                           1.0000

RMSEA (Root Mean Square Error Of Approximation)

      Estimate                           0.000
      90 Percent C.I.                    0.000  0.000
      Probability RMSEA <= .05           1.000

CFI/TLI

      CFI                                1.000
      TLI                                1.000

Chi-Square Test of Model Fit for the Baseline Model

      Value                             90.763
      Degrees of Freedom                    14
      P-Value                           0.0000

SRMR (Standardized Root Mean Square Residual)

      Value                              0.000

Answer 1

You are correct to conclude that a $\chi^2$ value of 0 is odd when df > 0, though that does not mean it is impossible. From your description of the model in the comments, your model does appear to be quite complex. I would first check your mplus syntax to see if a mistake was made, and if not, I would consider a simpler model. My reasoning for this is that it is possible your model is misspecified. Try freeing or fixing paths you believe are not central to your research question and see how it changes your results. Additionally, you could check if any of your sample variables have constant values (i.e., everyone in your dataset has the same value for a particular variable), as this can often lead to odd results (this has happened to me!). Finally, you could try to use Bayesian estimation, which is easy to implement in mplus. If you do this, be cautious. First, because the use of Bayesian methods does not "fix" any potential issues related to misspecification, and second, because the mplus default priors$^1$ may not be appropriate for your data (e.g., MacCallum, Edwards, & Cai, 2012).

$^1$ Note that naive use of default priors is not only an issue with mplus but with all Bayesian software packages that don't require explicit prior specification.

References

MacCallum, R. C., Edwards, M. C., & Cai, L. (2012). Hopes and cautions in implementing Bayesian structural equation modeling.