Satorra-Bentler Scaled Chi-Square
Message/Author
 Fred B. Bryant posted on Saturday, September 11, 2010 - 7:03 pm
From available documentation, I see that Mplus defines the Satorra-Bentler (SB) scaled chi-square for a given model as the maximum-likelihood (ML) chi-square value for that model divided by the scaling correction factor (c). When specifying MLM (the SB scaled chi-square) as the estimator, Mplus automatically includes the value of c in the output under the section labeled “Tests of Model Fit.” However, when I multiply the MLM (SB) chi-square value for a particular model times c for that model, the result is not exactly equal to the ML chi-square value for the model.

For example, using Mplus 6, I have a CFA model for which MLM = 36.214, c = 1.188, and ML = 43.072. For this model, (MLM)(c) = (36.214)(1.188) = 43.022 which is smaller than the ML value reported for the model in the output by 0.055. Also, dividing ML by c: 43.072 / 1.188 = 36.256, which is larger than the MLM value reported for the model in the output by 0.042. Can you explain the source of these apparent discrepancies? Are they due to rounding error?

Related to this question, is there any way to obtain more than three decimals in the values reported in the Mplus output?
 Linda K. Muthen posted on Sunday, September 12, 2010 - 9:55 am
The differences are due to using less decimals than Mplus uses. You can use the RESULTS option to save the results with more decimals. I don't see that the differences are large enough to worry about.
 Fred B. Bryant posted on Sunday, September 12, 2010 - 1:26 pm
Thanks for clarifying, Linda. It's great to have the option of using more decimals, if the scaled chi-square value happens to fall just barely below the critical value at p = .05. I appreciate your help with this.
 Fred B. Bryant posted on Sunday, September 12, 2010 - 11:20 pm
Using the RESULTS option with the SAVEDATA command provides up to 8 decimals for each estimated model parameter. But it does not display more decimals for the model's goodness-of-fit chi-square value. Is there a way to obtain more than 3 decimals for the latter?
 Linda K. Muthen posted on Monday, September 13, 2010 - 5:27 am
All fit statistics are saved in the same way as the parameter estimates. If you can't find them in the file, please send the saved results file, your output, and your license number to support@statmodel.com.
 Fred B. Bryant posted on Monday, September 13, 2010 - 9:38 am
In carefully matching each element in the RESULTS output to the estimates in the initial output, I now see that the former contains everything that Mplus estimated, including the model's goodness-of-fit chi-square. The goodness-of-fit statistics are at the very end of the RESULTS output. This is excellent, and is just what I need. Thanks for clarifying this for me, Linda. What a great resource this forum provides in conducting SEM with this powerful and versatile program!
 Linda K. Muthen posted on Monday, September 13, 2010 - 9:50 am
If you look at the end of the output, it lists what is saved and the order in which it is saved.
 Fred B. Bryant posted on Wednesday, September 15, 2010 - 9:36 am
Yes, at the end of the analysis output file I see the ordered listing of each specific estimate that's stored in the RESULTS output. WOW -- Mplus is a very well-designed program indeed! Very impressive! Thanks for your helpful feedback, Linda.
 Steve posted on Sunday, September 22, 2013 - 3:25 am
Hello,

I am doing SEM model comparisons and using MLR. In the case of Chi-square difference testing I am using Satorra-Bentler scaled chi-square difference test as provided on your website. However, for non-nested situations I am comparing fit indices (CFI, RMSEA, SRMR) - and also Chi-square to degrees of freedom ratio which I have been advised to include by reviewers.

However, while I have seen this done in the literature, I just wanted to check with you to see if Chi-square/df ratio can still be done with MLR by simply using the chi-square value provided in the MLR output?

Or - if some kind of scaling correction method (like in difference testing) needs to be employed to produce an accurate Chi-square/df ratio?

Thank you very much for your help.
 Linda K. Muthen posted on Monday, September 23, 2013 - 2:15 pm
We do not recommend using a chi-square to degrees of freedom approach. Instead use RMSEA.
 Terry Ng-Knight posted on Friday, May 09, 2014 - 10:39 am
Hello,

I am comparing a two-factor second-order CFA solution (with five 1st order factors) to the measurement model with only the five 1st order factors - using the WLSMV estimator.

The difftest suggests a significant difference (23.708, df=4, p<.0001), but I am a little concerned that the measurement model has a higher chi-square (S-B chi-sq(142)=724.926, p<0.001) than the model with the second-order factors (S-B chi-sq(146)= 686.125, p<.0001).

This seems counter-intuitive that the more constrained model appears to fit better, is it something to be concerned about?

Many thanks.
 Linda K. Muthen posted on Friday, May 09, 2014 - 2:44 pm
Chi-square with WLSMV In Mplus does not preserve the ordering of the least restrictive model getting the lowest chi-square. So this type of comparison should not be done. This is why the DIFFTEST option was developed to obtain a correct chi-square difference test using the derivatives from the two models not the chi-square values.