An Alternative to Post Hoc Model Modification in Confirmatory Factor Analysis: The Bayesian Lasso
As a commonly used tool for operationalizing measurement models, confirmatory factor analysis (CFA) requires strong assumptions that can lead to a poor fit of the model to real data. The post hoc modification model approach attempts to improve CFA fit through the use of modification indexes for identifying significant correlated residual error terms. We analyzed a 28-item emotion measure collected for n = 175 participants. The post hoc modification approach indicated that 90 item-pair errors were significantly correlated, which demonstrated the challenge in using a modification index, as the error terms must be individually modified as a sequence. Additionally, the post hoc modification approach cannot guarantee a positive definite covariance matrix for the error terms. We propose a method that enables the entire inverse residual covariance matrix to be modeled as a sparse positive definite matrix that contains only a few off-diagonal elements bounded away from zero. This method circumvents the problem of having to handle correlated residual terms sequentially. By assigning a Lasso prior to the inverse covariance matrix, this Bayesian method achieves model parsimony as well as an identifiable model. Both simulated and real data sets were analyzed to evaluate the validity, robustness, and practical usefulness of the proposed procedure.