I am new to CFA / SEM, so I apologize in advance if this question is poorly worded. I really appreciate your help!
I have achieved scalar variance on a measure with two factors (Supportive and Nonsupportive) with categorical indicators across three ethnic groups. I have also found there are mean-level differences between the groups, and I want to see if three different indicators of poverty-related risk explain these mean-level differences. The three risk indicators are also correlated, so I want to see how they each may uniquely explain some of the differences between groups. How would I do this in a CFA/SEM framework? Would it be some sort of SEM mediational model? I know I would do an ANCOVA if these were not modeled, latent values, so I wasn't sure how to translate this to CFA/SEM.
Once you have established scalar invariance, you could turn to a single-group analysis where ethnicity is represented by 2 dummy variables that influence the factors and not the indicators directly. That is saying the same thing as your 3-group analysis.
Then you can add your risk variables to predict the factors and ask regular linear regression questions. For instance, you can see if a risk variable has an effect on a factor holding ethnic group constant.