Hi, how do I test measurement invariance of a direct hierarchical (higher-order) CFA model across two groups in AMOS?

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Melissa Ong

Thanks a lot, Daniel! The check list is indeed helpful.

Unfortunately I am actually looking for a stepwise check list to test for MI in a second-order CFA model (I have four first-order latent variables and one second-order latent variable).

I'm not as familiar with AMOS as I would like to be in terms of applying constraints in the model. Any insight is very much appreciated!

Miguel A Ruiz

Melissa, AMOS has a built-in menu-based procedure available to test multi-group invariance. Barbara Byrne explains how to use it with detail in her SEM manual. You can also follow her 2004 article "Testing for Multigroup Invariance Using AMOS Graphics: A Road Less Traveled", STRUCTURAL EQUATION MODELING, 11(2), 272–300.

Thanks Miguel, that helped a lot!

Although I am still unsure of what are the extra steps for testing MI in a second-order CFA model.. would you by any chance know of any articles that explains the procedure for that?

Thank you!

Hi Melissa,

The AMOS built-in procedure is not designed to test 2nd-order structures but rather SEM full-models. This is not a problem though. In order to test measurement invariance it should be enough to test the hierarchical steps until testing Structural covariances equality (which correspond to first order factor variances), since structural residuals (second order factor error terms) and Measurement residuals (first order factor measurement errors) are seldom equal between groups.

If you have only one second-order factor, there will be no Structural covariances in the model, since they are explained (substituted) in the model by the factor loadings of the first order factors in the second-order factor. If you have more than one second-order factor, then you will be able to test is the covariances between second-order factors are also equal.

Although the approach is slightly different, because the baseline null model is slightly different, you can find a more detailed explanation in Wang J, Wang X (2012). Structural Equation Modeling. Applications Using Mplus. John Wiley.

Hi Prof @Miguel,

This is so delayed but I just want to say thank you very much for your clear and concise answer! This helped me so much, thank you.