Partial metric invariance
Message/Author
 Jason Prenoveau posted on Friday, July 18, 2008 - 8:35 am
I have a longitudinal measurement model. I am attempting to demonstrate metric invariance of factor loadings and intercepts for two reasons: to enable use of the latent variables as outcomes in a latent trajectory model, and also to simply compare the mean of the latent variable at the four time points. The model demonstrates metric invariance of the factor loadings, however, applying intercept equality constraints results in significant decrement in model fit. I am using modification indices to identify intercept constraints causing model strain so that I can relax these constraints and have a model which is NOT significantly different than my configurally invariant model (partial metric invariance).

When I am looking to identify constraints to relax, should I relax the latent variable means BEFORE I look for these constraints, or AFTER I look for and relax these constraints. I have done it both ways, and the results are SLIGHTLY different. In one case, there is relaxation of 2 constraints and in the other, there is only one constraint. I assume that I relax the latent variable means BEFORE looking at the other constraints, but could not find anything that provided a protocol for this situation.

If I have partial metric invariance of intercepts, is it still acceptable to compare the means of the latent variables? It seems that this has been done in several papers I have seen in good journals, so I assume it is ok, but wanted to verify.
 Bengt O. Muthen posted on Friday, July 18, 2008 - 5:00 pm
I think your second paragraph is talking about intercept constraints. To use MIs to pick intercept constraints to relax, you should have your usual factor mean structure for longitudinal factor analysis, say with 3 time points:

[f1@0 f2 f3].

This assumes that not all intercepts at a given time point are relaxed because that would lead to non identification.

But I am not sure this is what you are asking.

 Jason Prenoveau posted on Saturday, July 19, 2008 - 12:10 pm
Thank you for the reply. Yes, I am talking about intercept constraints. So, it sounds like you are saying that I relax the factor means ([f2 f3] or [f1@0 f2 f3]) FIRST and then use MIs to identify intercept equality constraints to relax. This makes sense to me, since we are assuming the intercepts are equal, not necessarily the factor means.

The other way (which gives slightly different answers) would be to hold the means equal (they are all fixed at 0 as the default) and to identify and relax constraints BEFORE allowing the means to vary. Which did not make as much sense to me. But I was uncertain how it is typically done.

Thank you again!