Bayesian structural equation modeling: A more flexible representation of substantive theory.

TLDR

The Bayesian approach is especially useful when adding parameters to a conventional model creates a nonidentified model under maximum‑likelihood estimation, and it is applicable to measurement aspects of latent variable modeling such as confirmatory factor analysis and structural equation modeling. The authors propose a Bayesian factor analysis and structural equation modeling framework that replaces exact zero constraints with approximate zeros via informative, small‑variance priors, aiming to better reflect substantive theories and address nonidentification. The method employs informative small‑variance priors to approximate zeros, incorporates posterior predictive checking, estimation, and modification, and is implemented in Mplus, with performance evaluated through Monte Carlo simulations and real data from the Holzinger–Swineford, Big Five, and NELS studies. An example using a full structural equation model demonstrates that the approach efficiently identifies model misspecification.

Abstract

This article proposes a new approach to factor analysis and structural equation modeling using Bayesian analysis. The new approach replaces parameter specifications of exact zeros with approximate zeros based on informative, small-variance priors. It is argued that this produces an analysis that better reflects substantive theories. The proposed Bayesian approach is particularly beneficial in applications where parameters are added to a conventional model such that a nonidentified model is obtained if maximum-likelihood estimation is applied. This approach is useful for measurement aspects of latent variable modeling, such as with confirmatory factor analysis, and the measurement part of structural equation modeling. Two application areas are studied, cross-loadings and residual correlations in confirmatory factor analysis. An example using a full structural equation model is also presented, showing an efficient way to find model misspecification. The approach encompasses 3 elements: model testing using posterior predictive checking, model estimation, and model modification. Monte Carlo simulations and real data are analyzed using Mplus. The real-data analyses use data from Holzinger and Swineford's (1939) classic mental abilities study, Big Five personality factor data from a British survey, and science achievement data from the National Educational Longitudinal Study of 1988.