How to interpret the results when the models are constrained to be invariant in multi-group modeling

I have constrained two models to be equal, and based on the $$\chi^2$$ test there is no difference between the models, they are invariant. These two models contain exactly the same path coefficients. However, the first model uses the data obtained at the 2nd month, and the second model uses the data obtained at the 4th month from the same participants.

I obtained the path coefficients in the Mplus results for each model separately, and they are slightly different across the models. However, I get only one set of model fit indices.

How can I obtain two sets of results?

Also, when I run only one model instead of multi-group modeling, the path coefficients are different from what I have obtained for that specific model when doing multi-group model analysis.

Can anyone explain the differences between these models to me? Why do I get different results when they are constrained to be invariant in multi-group modeling? Should I report path coefficients values for two models?

If the $$\chi^2$$ difference test shows no significant difference across the models, does it make sense to combine all data and run a single model?

• Can you clarify some things? You say you've constrained two models, but then you say you have two sets of coefficients, and one set of fit indices. – Jeremy Miles Mar 24 '16 at 18:48
• You say that your coefficients differ when you use two groups with everything free. That shouldn't happen - there are probably some constraints that Mplus has included by default. Perhaps intercepts or thresholds. Put tech1 in output and check the parameters are all different. (I guess it could be missing data too - full information ML in Mplus is model based, so changing your model will change your data (as it were). Put sampstat in the output section and make sure the sample stats are the same. – Jeremy Miles Mar 24 '16 at 18:52
• – renakre Mar 26 '16 at 21:35
• Table 3 in this paper is how we solved it on one occasion: ncbi.nlm.nih.gov/pubmed/26678071 – Jeremy Miles Mar 29 '16 at 19:35
• @JeremyMiles Ohh, after I constrained everything, I just noticed that chi-square difference became significant. I think I will constrain only the path coefficients, and live with it :) Thanks for all the help, I really appreciate it! – renakre Mar 30 '16 at 19:00