A Bayesian approach to confirmatory factor analysis: Moving beyond dichotomous thinking

In recent years there has been a growing interest in Bayesian inference in numerous scientific disciplines. Structural equation models (SEM) are an important tool in the social and behavioural sciences to evaluate the structure of a model with latent and observed variables. However, the use of a Bayesian approach (BA) in this field is still underexplored. In this work, we illustrate the advantages of using the BA in a relevant SEM sub- model, i.e., confirmatory factor analysis (CFA). Specifically, the goals are to (1) compare the traditional maximum likelihood approach (MLA) with the BA in terms of parameter estimation and fit indices; (2) show how the BA allows to estimate further models that may result unidentified via the classical approach, but may better reflect the underlying psychological theory; and (3) present BA-based techniques for model diagnostic in terms of the distribution of estimated parameters as well as single case influence. To address the first aim, a simulation study was performed. Starting from a baseline two-correlated factor model, we manipulated sample size and effect size of a potential cross-loading for a specific item. For each condition, we estimated 2 bayesian CFAs (one including and one excluding the cross- loading) and 2 ML CFAs. Next, a bayesian CFA on a real case-study is presented. The magnitude of cross-loadings and residual correlations were simultaneously evaluated according to different theoretical models through different parameter prior specifications. The most plausible model was selected using the Deviance Information Criteria (DIC). Parameter posterior distributions and predictive posterior distribution of the observed data were used to examine model fit. All analyses were conducted using free software (i.e., R in combination with JAGS). Differences and similarities between the BA and MLA will be discussed. Overall, the formalization of model parameters in terms of prior probability distributions, instead of the less realistic parameters presence-or-absence formalization, provided a more flexible and informative evaluation of the latent structure of the observed data. To conclude, we will briefly review potential applications of the BA to the broader context of structural equation models.