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?
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.
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.
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.
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?