Mixture multigroup factor analysis for unraveling factor loading noninvariance across many groups

Psychological research often builds on between-group comparisons of (measurements of) latent variables; for instance, to evaluate cross-cultural differences in neuroticism or mindfulness. A critical assumption in such comparative research is that the same latent variable(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 by means of multigroup factor analysis. When the assumption is untenable, one may compare group-specific measurement models to pinpoint sources of noninvariance, but the number of pairwise comparisons exponentially increases with the number of groups. This makes it hard to unravel invariances from noninvariances and for which groups they apply, and it elevates the chances of falsely detecting noninvariance. An intuitive solution is clustering the groups into a few clusters based on the measurement model parameters. Therefore, we present mixture multigroup factor analysis (MMG-FA) which clusters the groups according to a specific level of measurement invariance. Specifically, in this article, clusters of groups with metric invariance (i.e., equal factor loadings) are obtained by making the loadings cluster-specific, whereas other parameters (i.e., intercepts, factor (co)variances, residual variances) are still allowed to differ between groups within a cluster. MMG-FA was found to perform well in an extensive simulation study, but a larger sample size within groups is required for recovering more subtle loading differences. Its empirical value is illustrated for data on the social value of emotions and data on emotional acculturation.