Abstract: The bifactor model and its extensions are multidimensional latent variable models, under which each item measures up to one subdimension on top of the primary dimension(s). Despite their wide applications to educational and psychological assessments, these multidimensional latent variable models may suffer from nonidentiability, which can further lead to inconsistent parameter estimation and invalid inference. The current work provides a relatively complete characterization of identiability for linear and dichotomous bifactor models and the linear extended bifactor model with correlated subdimensions. In addition, similar results for the two-tier models are developed. Illustrative examples on checking model identifiability by inspecting the factor loading structure are provided. Simulation studies examine the estimation consistency when the identifiability conditions are/are not satisfied.

Key words and phrases: Bifactor model, educational and psychological measurement, identifiability, item factor analysis, multidimensional item response theory, testlet, two-tier model..