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This is a question seeking to follow up on this post.

I have a multigroup SEM with a mix of observed and latent variables.

In the measurement model to inspect latent variables, metric invariance holds (loadings), but scalar invariance (intercepts) does not hold.

The regression is one manifest variable (Std_LC) and one latent variable predicting a latent outcome.

I wish to determine if the regression coefficients significantly different across three groups. If I hold "loadings" and "regressions" invariant, there are no sig. differences across groups. But if intercepts are not also constrained to test for differences, aren't the regressions coefficients representing different values?

Basically, I understand configural, metric, and scalar invariance, but I don't understand if "intercepts" being constrained here refers to intercepts of indicators loading onto latent factors, or also intercepts of variables modelled in the regression.

Many thanks in advance for any advice you could offer!

modelx <- '

FluentWR =~ Std_WI +Std_ORF + Raw_RANN
RC =~ Std_PC + Std_WC
RC ~ FluentWR + Std_LC 
FluentWR ~~ Std_LC ' 
```
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2 Answers 2

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The regression slope coefficients in your structural (latent variable) model only involve the covariance structure (latent variances and covariances). Therefore, loading (metric/weak) invariance is sufficient to meaningfully test whether the regression slope coefficients in the structural model are equal across groups.

The intercepts in the structural regression model involve the mean structure of the latent variables. To meaningfully compare structural intercepts, you need to have at least strong/scalar invariance in your measurement model (i.e., equal factor loadings and equal indicator intercepts). Otherwise, the mean structure of the latent variables is not comparable across groups.

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  • $\begingroup$ Thank you Christian! This is very helpful. I should have mentioned that I expect the regression model coefficients to vary across groups, especially for the single manifest variable. I guess I am wondering ... my model is maybe more like a path analysis with latent variables, and less like a proper SEM since I am not using item-level indicators, but sum scores. Is a measurement model even necessary? Is it enough to hold "intercepts" and "regression" constrained like this post describes? jslefche.github.io/sem_book/multigroup-analysis.html $\endgroup$ Commented May 28, 2023 at 0:55
  • $\begingroup$ Or perhaps I am going about it backwards and need to first see if a measurement model can be developed that indeed meets scalar invariance, so I can test the regression model coefficients. I will try both, but I am not sure how they are nested. Thanks again for the response! $\endgroup$ Commented May 28, 2023 at 0:58
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Yes, probably the most appropriate way to approach this would be to test an item-level multigroup CFA (measurement model) first to see what level of measurement equivalence holds across the groups. Depending on the level of measurement equivalence, you can then test the equivalence of the structural regression coefficients across groups using multigroup SEM or path analysis.

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  • $\begingroup$ Thank you so much! $\endgroup$ Commented May 29, 2023 at 12:05

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