Recovering Latent Structures after Variational Bayesian Variable Selection: Fit Assessment and Factor-Number Selection in Partially Exploratory Factor Analysis

2026-07-08

AI Summary

This study addresses the challenges in exploratory factor analysis arising from unknown factor structures and indeterminate numbers of latent factors, which often hinder model identification and evaluation. The authors propose a variational Bayesian variable selection framework that employs spike-and-slab priors to recover the underlying factor structure and introduces a post-selection model fit assessment system. By recasting hard and soft selection strategies as covariance models, the approach facilitates diagnostic evaluation and determination of the number of factors. A novel dimensionless gain rule, combined with multidimensional fit indices—including RMSEA, SRMR, CFI, TLI, AIC, BIC, and ELBO—is introduced to effectively prevent misidentification of factor count. Simulations demonstrate that absolute fit indices sensitively track loading recovery and detect underfactoring, while the gain rule accurately recovers the true dimensionality, with the ELBO variant exhibiting the greatest robustness. Applied to the 100-item PID-5 dataset, the method significantly outperforms a prespecified 25-factor confirmatory model.

Abstract

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.

Problem

• partially exploratory factor analysis
• factor number selection
• model fit assessment
• latent structure recovery
• Bayesian variable selection

Innovation

• variational Bayesian variable selection
• partially exploratory factor analysis
• factor number selection
• post-selection fit assessment
• spike-and-slab priors