Identification constraints and fit indices in Bayesian latent variable models
It is well known that, in traditional SEM applications, a scale must be set for each latent variable: either the latent variance or a factor loading is typically fixed to one. While this has no impact on the fit metrics in ML estimation, it can potentially lead to varying Bayesian model comparison metrics due to the use of different priors under each parameterization. Using a single-factor CFA as motivation for study, we first show that Bayesian model comparison metrics systematically change depending on constraints used. We then study principled methods for setting the latent variable scale that stabilize the model comparison metrics. These methods involve (i) the placement of priors on ratios of factor loadings, as opposed to individual loadings, and (ii) use of effect coding. We illustrate the methods via simulation and application.