Factorial invariance across multiple populations in discrete and continuous data

Factorial invariance has a long history, with the earliest work questioning whether invariance in factor structure is even possible. Factorial invariance is a limited form of measurement invariance, evaluated within the assumptions of the factor analysis model. To illustrate factorial invariance tests using continuous variables, data from the Early Steps Multisite Study are used. To illustrate invariance tests with discrete data, the home visitor's ratings from the Early Steps study are used. The single-factor model for discrete measures can be viewed as a graded response model within the broad class of item response theory (IRT) models. The multiple-factor model can be viewed as a compensatory model within the class of multidimensional IRT (MIRT) models. Historical distinctions between the factor analysis model and IRT models are becoming less important, given that both modelling traditions are subsumed within a general latent variable framework.