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 CRG posted on Thursday, June 02, 2016 - 4:54 am
Dear professor(s) Muthen,

I am writing an article comparing multiple group LCA and CFA, using Mplus to conduct the analyses. Part of this comparison is the testing of measurement equivalence (invariance). As for the measurement equivalence testing with LCA I am largely basing myself on several articles by Kankaras (2009/2010).

He makes an interesting suggestion to test a specific type of partial measurement invariance in LCA by holding the slopes of the model indicators equal across group, but allowing the intercepts to vary across group (in logistic parameterization). In this case, they argue, there are no interactions between the grouping variable and the latent variable in the model, allowing for meaningful comparisons across groups.

While this is an interesting conceptual question, my question is mainly practical. Thus far, I have only managed to constrain indicator thresholds across groups (with quite some extensive syntax, considering that I have 5 groups and 18 categorical indicators with each three categories and considering the default Mplus setting is heterogeneity of conditional probabilities across groups). If I am correct, this constrains conditional probabilities to be equal across group (both intercepts and slopes). Is there a way in Mplus to constrain the slopes but not the intercepts in a multiple group LCA?

Many thanks in advance for a reply.

 Bengt O. Muthen posted on Friday, June 03, 2016 - 5:55 pm
I don't see how slopes enter the picture here. In LCA you have a categorical latent variable so a slope is not relevant because the intercept/threshold is the only parameter describing the relationship between the indicator and the latent variable. Perhaps this person works with Factor Mixture models.
 CRG posted on Saturday, June 04, 2016 - 4:42 am
Dear professor Muthen,

Thanks for your reply.

The argument is indeed that multiple group latent class models in logistic parameterization differentiate between group-specific indicator slopes (in log-linear formulation, interaction effects between the grouping variable and the latent variable on the manifest variables) and group-specific indicator intercepts (direct effects of the grouping variable on the manifest variables in log-linear formulation).

Kankaras (together with Jeroen Vermunt and Guy Moors) describe how the different levels of measurement invariance in LCA can be compared to configural invariance, metric invariance and scalar invariance in CFA when seen this way. They do not use factor mixture models.

Here is a link to the PhD dissertation:

The above is explained in chapter 3, p.71-72, & p78-81.

From your response I understand that test
 CRG posted on Saturday, June 04, 2016 - 4:43 am
From your response I understand that testing this is currently not possible in Mplus, correct?
 Bengt O. Muthen posted on Sunday, June 05, 2016 - 1:49 pm
It sounds like what you want is possible in Mplus now that I see what it is you want to achieve.

I assume that when you say latent variable you are referring to the latent class variable. By group-specific intercepts I think you mean differences in the item parameters that are constant across classes. By group-specific slopes I think you mean differences in the item parameters that vary across classes (an interaction between group and class). Assuming that identification is achieved by some zero or otherwise restricted parameters, this is straightforward in Mplus.
 CRG posted on Monday, June 06, 2016 - 1:14 am
Dear professor Muthen,

That seems to make sense. All those parameters are assumed to be free across groups in the Mplus default for MLCA, correct?

Could you tell me how I would achieve constraining the slopes and allowing the intercepts to vary?
 Bengt O. Muthen posted on Monday, June 06, 2016 - 5:41 am
You specify group as Knownclass and then you use the "dot" approach to have full freedom to apply any equality constraint you want:


You find an example of this in UG ex8.8.
 CRG posted on Monday, June 06, 2016 - 9:00 am
Yes, I'm aware of the existence of this option. My current input identifies the thresholds/intercepts for every combination of group*class and constrains them across groups this way.

I believe these to be my group-specific intercepts. However, the above restrictions constrain the conditional probabilities to be equal across groups completely (as evidenced by the output), suggesting this model does not have any slope variation (nor intercept variation, but these I specifically constrained) across groups.

It seems I would have to add an interaction between the latent variable and grouping variable to estimate a slope parameter and estimate a model of partial homogeneity. (At first I thought it was the Mplus default to estimate this interaction freely but I no longer think this is the case.). Do you know how I would do this?
 Bengt O. Muthen posted on Monday, June 06, 2016 - 9:56 am
The input you show gives class-specific thresholds, that is, they are the same across groups (equalities go over groups).

If you want the thresholds to vary across groups but be constant across classes you would use the numbering

 CRG posted on Wednesday, June 08, 2016 - 6:31 am
What I want to do is model the effect of my predictors on class membership. However, I cannot use the statement "cu on welf11;" as Mplus gives the following error:

The CATEGORICAL option is used for dependent variables only.
The following variable is an independent variable in the model.
Problem with: WELF11

Do you know how I could model this?

(I believe this would allow me to have the slopes of the model to be freely estimated across groups.)
 Bengt O. Muthen posted on Wednesday, June 08, 2016 - 11:13 am
Your welf11 variable is an indicator of your latent class variable as you show by statements like

Therefore, it cannot also be a predictor of the latent class variable - if it were you would have reciprocal interaction.

But we have already discussed how you would get what you wanted.
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