Is a MIMIC model or multigroup SEM model better?

[Kelly Lam]

I was told that a multigroup model is good because it's more customizable, and you can compare the coefficients between groups (assuming the appropriate assumptions are met). Whereas a MIMIC model is better when you have a lot of groups. Is this an accurate summary or is there something I'm missing?

I'm currently practicing with the European Social Survey (ESS), and I have a SEM model with 16 independent variables (many are dummy variables for the endogenous variables). I'm focusing on 6 countries, so 5 of these included variables are my country variables. When I practiced using just 2 countries, I was told that while MIMIC was an option, multigroup would be better. I feel more comfortable with MIMIC but don't know if there's any good justification to use it in my case (6 countries). I think it would be interesting to know how the different countries compare to one another, but I find testing for measurement invariance and scalar invariance a bit advanced for me right now. I was also told that interpreting the regression results from the lavaan output, assuming the country dummy variables are used in the MIMIC model, is that the coefficient is relative to the reference variable (country).

If you have any insight as model types, I would really appreciate your input! Thank you.

[Edward Rigdon]

Kelly-

     A fundamental assumption in SEM is that respondents within a group are homogeneous. If you think that tehre are differences across countries, then the MIMIc approach will violate this assumption and make your results suspect. I think that the nature of your dataset strongly argues for a multiple group approach.

--Ed Rigdon

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[Nickname]

Kelly,
  If it helps, here is another way to think about what Ed said.  When an loading differs between groups in a multi-group analysis, this represents an interaction between the latent variable and the group variable in determining the indicator value.  The effect of the latent variable differs between groups. In contrast, your MIMIC model only includes main effects for group membership on the latent variable, not interactions between the group variable and the latent variable.  That is like a multi-group model in which the intercepts of the latent variable can vary by group, but the loadings are constrained to equality across groups.  Adding a direct effect from group to an indicator in the MIMIC model would be like adding an individual intercept for the indicator to the multi-group model and allowing that intercept to vary across groups.

Keith
------------------------
Keith A. Markus
John Jay College of Criminal Justice, CUNY
http://jjcweb.jjay.cuny.edu/kmarkus
Frontiers of Test Validity Theory: Measurement, Causation and Meaning.
http://www.routledge.com/books/details/9781841692203/

[Terrence Jorgensen]

Hi Kelly,

Ed and Keith correctly described the assumptions of the standard MIMIC model (as well as the statistically equivalent restricted factor analysis [RFA] model, in which the grouping variable merely covaries with the factor rather than predicts it).  However, you can additionally test for differences in factor loadings by using a product-indicator approach to model an interaction between the latent factor and the grouping variable.  Oddly enough, the covariance between the latent interaction factor and the common factor captures information about heteroskedasticity of common factor variances across groups, so it is robust to "violating" that traditional assumption of MIMIC/RFA models.  

https://doi.org/10.3758/s13428-018-1151-3 (simulation study)

https://doi.org/10.1007/978-3-319-77249-3_20 (tutorial with example lavaan syntax)

Heteroskedasticity of residual variances is not captured (unless residual covariances with product-indicators are modeled, which would render this model underidentified), but my grad student has recently found that results are quite robust to violation of that assumption as well (only minor Type I error inflation in cases of severe residual heteroskedasticity).  Paper is still under review.  A limitation of the product-indicator approach for latent interactions is that it is only applicable to continuous indicators ("products" of ordinal variables make no sense).

The advantage of MIMIC/RFA over MG-CFA is that single-group models are more stable in smaller samples because they estimate fewer parameters (e.g., all parameters can differ across groups in MG-CFA, except for arbitrary identification constraints, which requires more observations).  However, with 5 dummy codes to represent 6 countries, you would have to calculate product indicators with each dummy code to define a latent interaction between each dummy code and the common factor(s).  So that model might get so large that it loses its small-sample advantage.

If you use MG-CFA, you still get the interpretation "relative to the reference country" because the latent mean is fixed to zero in the first (reference) country.  Thus, estimated latent means in other countries are merely differences from the reference country.  In the semTools package, the measEq.syntax() function could be helpful for writing lavaan syntax to test invariance using MG-CFA, since you are unfamiliar with it.  But I recommend reading as many textbooks, slides, or tutorials as you can find.

Terrence D. Jorgensen

Assistant Professor, Methods and Statistics

Research Institute for Child Development and Education, the University of Amsterdam

http://www.uva.nl/profile/t.d.jorgensen