I compared the chi-square value from the model with all parameters allowed to be unequal across groups (e.g., parameters are set free) to the chi-square from the model where paths at time-point 1 and time-point 2 were constrained to be equal. I used chi-square difference test. According to the results of the chi-square difference test, no significant difference was found across time suggesting that the model is invariant across different time points.

So, with the constrained model, I obtain more power calculating the path coefficients, is that right? I get a single value for the path coefficients at two time points, which is what I expected.

However, the r-square values are different in two models. I do not understand this since I thought I use all data to calculate r-square when constraining the models to be equal. I am sure I am missing something here. Can someone help? Why can I get the same path coefficient values but different r-square values? What does constraining do actually?

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    $\begingroup$ If you constrained all the parameters, then the R-squared will be equal. You probably missed something - I usually do. Check the list of parameters. $\endgroup$ – Jeremy Miles May 1 '16 at 23:40

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