Identification of Confirmatory Factor Analysis Models of Different Levels of Invariance for Ordered Categorical Outcomes

This article considers the identification conditions of confirmatory factor analysis (CFA) models for ordered categorical outcomes with invariance of different types of parameters across groups. The current practice of invariance testing is to first identify a model with only configural invariance and then test the invariance of parameters based on this identified baseline model. This approach is not optimal because different identification conditions on this baseline model identify the scales of latent continuous responses in different ways. Once an invariance condition is imposed on a parameter, these identification conditions may become restrictions and define statistically non-equivalent models, leading to different conclusions. By analyzing the transformation that leaves the model-implied probabilities of response patterns unchanged, we give identification conditions for models with invariance of different types of parameters without referring to a specific parametrization of the baseline model. Tests based on this approach have the advantage that they do not depend on the specific identification condition chosen for the baseline model.