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 '
```
Best Answer
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.