A Penalized Likelihood Method for Structural Equation Modeling.

A penalized likelihood (PL) method for structural equation modeling (SEM) was proposed as a methodology for exploring the underlying relations among both observed and latent variables. Compared to the usual likelihood method, PL includes a penalty term to control the complexity of the hypothesized model. When the penalty level is appropriately chosen, the PL can yield an SEM model that balances the model goodness-of-fit and model complexity. In addition, the PL results in a sparse estimate that enhances the interpretability of the final model. The proposed method is especially useful when limited substantive knowledge is available for model specifications. The PL method can be also understood as a methodology that links the traditional SEM to the exploratory SEM (Asparouhov & Muthén in Struct Equ Model Multidiscipl J 16:397-438, 2009). An expectation-conditional maximization algorithm was developed to maximize the PL criterion. The asymptotic properties of the proposed PL were also derived. The performance of PL was evaluated through a numerical experiment, and two real data illustrations were presented to demonstrate its utility in psychological research.