Bayesian factor selection in a hybrid approach to confirmatory factor analysis

To investigate latent structures of measured variables, various factor structures are used for confirmatory factor analysis, including higher-order models and more flexible bifactor models. In practice, measured variables may also have relatively small or moderate non-zero loadings on multiple group factors, which form cross loadings. The selection of correct and 'identifiable' latent structures is important to evaluate an impact of constructs of interest in the confirmatory factor analysis model. Herein, we first discuss the identifiability condition that allows several cross loadings of the models with underlying bifactor structures. Then, we implement Bayesian variable selection allowing cross loadings on bifactor structures using the spike and slab prior. Our approaches evaluate the inclusion probability for all group factor loadings and utilize known underlying structural information, making our approaches not entirely exploratory.