Mixture multigroup factor analysis for unraveling factor loading non-invariance across many groups

Psychological research often builds on between-group comparisons of (measurements of) latent constructs; for instance, to evaluate cross-cultural differences in neuroticism or mindfulness. A critical assumption in such comparative research is that the same construct(s) are measured in exactly the same way across all groups (i.e., measurement invariance). Otherwise, one would be comparing apples and oranges. Nowadays, measurement invariance is often tested across a large number of groups. When the assumption is untenable, one may compare group-specific measurement models to pinpoint sources of non-invariance, but the number of pairwise comparisons exponentially increases with the number of groups. This makes it hard to unravel invariances from non-invariances and for which groups they apply, and it elevates the chances of falsely detecting non-invariance. An intuitive solution is clustering the groups into a few clusters based on the measurement model parameters. Not only does this confine the number of comparisons needed to identify non-invariances, but the clustering of the groups is an interesting result in itself and provides clues on how to move forward with the between-group comparisons or further measurement invariance testing. To this aim, we present mixture multigroup factor analysis which accommodates a unique blend of cluster- and group-specific parameters to disentangle different levels of non-invariance and set aside parameter differences that are irrelevant to the measurement invariance problem.