Adapting Fit Indices for Bayesian Structural Equation Modeling: Comparison to Maximum Likelihood

In a frequentist framework, the exact fit of a structural equation model (SEM) is typically evaluated with the chi-square test and at least one index of approximate fit. Current Bayesian SEM (BSEM) software provides one measure of overall fit: the posterior predictive p value (PPP χ2 ). Because of the noted limitations of PPP χ2 , common practice for evaluating Bayesian model fit instead focuses on model comparison, using information criteria or Bayes factors. Fit indices developed under maximum-likelihood estimation have not been incorporated into software for BSEM. We propose adapting 7 chi-square-based approximate fit indices for BSEM, using a Bayesian analog of the chi-square model-fit statistic. Simulation results show that the sampling distributions of the posterior means of these fit indices are similar to their frequentist counterparts across sample sizes, model types, and levels of misspecification when BSEMs are estimated with noninformative priors. The proposed fit indices therefore allow overall model-fit evaluation using familiar metrics of the original indices, with an accompanying interval to quantify their uncertainty. Illustrative examples with real data raise some important issues about the proposed fit indices' application to models specified with informative priors, when Bayesian and frequentist estimation methods might not yield similar results. Translational Abstract As Bayesian structural equation modeling (BSEM) has become more accessible with user-friendly software, there is still a need to research and present BSEM in a familiar setting for applied researchers. A basic step of SEM is the test of model fit, which in frequentist SEM is done with an exact fit chi-square test and at least one index of approximate fit. However, current BSEM model fit testing is limited. For this reason, the present study develops and tests 7 chi-square-based approximate fit indices (RMSEA, Gamma-Hat, adjusted-Gamma-Hat, Mc, CFI, TLI, NFI) for BSEM. These approximate fit indices are shown to be in the same metric as the frequentist counterparts, which allows one to have the same familiar interpretation of model-fit evaluation. The proposed BSEM approximate fit indices have an added advantage over their frequentist counterparts in that they allow quantifying uncertainty for each one with the posterior standard deviation and credible interval.