Measurement Invariance Analysis for Hierarchical Factor Modeling with Many Groups: Comparing Multi-Group Confirmatory Factor Analysis and Alignment Approaches

The analysis of measurement invariance is important in the confirmatory factor analysis to test the psychometric property of an instrument. The traditional measurement invariance analysis is based on multiple-group confirmatory factor analysis with specific constraints that considers three levels of measurement invariance: configural equivalence, metric equivalence, and scalar equivalence. Since the traditional multigroup CFA often fails for the large number of groups, a new method - alignment method was proposed by Asparouhov and Muthen (2014). One limitation to this methodology is that cross-loadings are not allowed in the model. To overcome this limitation, the extensions have been developed to apply alignment methods to a more general situation, such as BSEM-based alignment with approximate measurement invariance and alignment within CFA methodology. This paper proposed two potential methodologies, alignment within CFA and alignment within factor scores. Using Monte Carlo simulations, this paper evaluated how they performed in the context of second-order model. The results showed that neither of them was a good choice