Recovering Latent Structures after Variational Bayesian Variable Selection: Fit Assessment and Factor-Number Selection in Partially Exploratory Factor Analysis
For researchers using partially exploratory factor analysis, this provides a principled way to assess model fit and select factor numbers, addressing a known bottleneck in post-selection inference.
This paper introduces a post-selection assessment framework for partially exploratory factor analysis (PEFA) using variational Bayesian variable selection, enabling fit diagnostics (RMSEA, SRMR, CFI, TLI) and factor-number selection via a scale-free gain rule. Simulations show the gain rule accurately recovers true dimensionality, with the ELBO variant being most robust, and a 100-item example demonstrates better fit than a confirmatory model.
In partially exploratory factor analysis (PEFA), the loading structure and factor numbers are weakly specified. The regularized variational approximation for partially confirmatory factor analysis (PCFA VA) recovers this structure via Bayesian variable selection, using spike and slab priors to assign inclusion probabilities to unspecified loadings. This research introduces a post selection assessment framework for this approach. We convert converged solutions into covariance models using either hard selection (thresholding probabilities into a sparse pattern) or soft selection (retaining them as weights for effective parameter counts). We derive the resulting degrees of freedom, absolute fit diagnostics (RMSEA, SRMR, CFI, TLI), and relative criteria (AIC, BIC, ELBO). To determine factor numbers, we propose a scale free gain rule with a sustained drop guard. Simulations show absolute indices successfully track loading recovery and flag under factoring. While raw criteria over factor, our gain rule accurately recovers true dimensionality, with the ELBO variant proving most robust. Finally, a 100 item PID 5 example demonstrates that our model fits better than a confirmatory 25 facet model and concordantly recovers major structures across disjoint specifications.