  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, this type of multidimensional latent variable models
may suffer from non-identifiability, which can further lead to inconsistent
parameter estimation and invalid inference. The current work provides a
relatively complete characterization of identifiability for the linear and
dichotomous bifactor models and the linear extended bifactor model with
correlated subdimensions. In addition, similar results for the two-tier models
are also developed. Illustrative examples are provided on checking model
identifiability through inspecting the factor loading structure. Simulation
studies are reported that examine estimation consistency when the
identifiability conditions are/are not satisfied.
