When factor scores replace true latent scores in nonlinear prediction, measurement error attenuates the recoverable variance of any $k$th-order component of the regression function by $ρ^k$ -- the $k$th power of the score's coefficient of determination -- for any linear score type. This study derives the bound via Hermite polynomial expansion and proposes PV-ANN -- plausible values (posterior draws preserving latent variance) combined with artificial neural networks (learning functional form without prespecification). The bound governs recovery of the latent-scale function, not prediction of the outcome from observed indicators, for which factor scores are already sufficient; the two metrics are therefore predicted to dissociate. An 18-condition simulation supports both predictions: in the nonlinear low-reliability conditions PV-ANN closes about four fifths of the function-shape recovery gap between a factor-score learner and one given the true latent values, and the margin widens as reliability falls, while predictive accuracy is not improved, as the theory requires. A Big Five application illustrates the intended exploratory workflow and delineates boundary conditions under weak signal and measurement model misspecification.
Whether an additional general dimension is necessary beyond correlated first-order factors is a property of the population covariance matrix, not of any estimator or design. This research establishes when that property can be decided. Where the general and group loadings are proportional within every cluster the bifactor structure is covariance-equivalent to correlated factors, so no sample size separates them (Proposition 1); where that proportionality fails in every cluster, three items per cluster and some mild regularities leave no $K$-factor model with diagonal uniquenesses able to reproduce the covariance matrix (Theorem 1); and between them lies a mixed boundary, located numerically here and turning on cluster resistance. Distinguishability is therefore graded, measured by the population distance to the $K$-factor class. Because that question is conditional on a first-order structure which is itself uncertain, a two-step procedure is developed within partially exploratory factor analysis, delivering a structure only when it reproduces across adjacent counts and treating non-delivery as legitimate. Simulation shows that a unanimous count can accompany a structure that fails to reproduce, and that absorbed local dependence can imitate a general factor, the error growing with sample size while stability indicators stay clean. Four empirical datasets illustrate the possible outcomes.
A mixture of spatial factor analyzers (MSFA) is introduced to address the challenges of clustering high-dimensional spatial data. By leveraging the underlying coordinate system, the proposed framework incorporates a flexible, spline-based spatial decay covariance structure that prevents parameter inflation as dimensionality increases. To model non-spatial dependence, matrix variate factor analyzers are employed for further dimensionality reduction. Parameter estimation is conducted via a variant of the expectation-maximization algorithm combined with a generalized least squares estimator. The proposed models are explored in the context of tensor-variate data analysis, where simulation studies and applications to Raman spectroscopy and hyperspectral texture databases demonstrate their capacity to accurately infer and differentiate distinct spatial patterns.
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.
Jinyuan Chang, Guanglin Huang, Qiwei Yao +1stat.ME econ.EM stat.ML
We adopt the canonical polyadic (CP) decomposition to model high-dimensional tensor time series. Our primary goal is to identify and estimate the factor loadings in the CP decomposition. We propose a one-pass estimation procedure through standard eigen-analysis for a matrix constructed based on the serial dependence structure of the data. The asymptotic properties of the proposed estimator are established under a general setting as long as the factor loading vectors are linearly independent, allowing the factors to be correlated and the factor loading vectors to be not nearly orthogonal. The procedure adapts to the sparsity of the factor loading vectors, accommodates weak factors, and demonstrates strong performance across a wide range of scenarios. To further reduce estimation errors, we also introduce an iterative algorithm based on a novel double projection approach. We theoretically justify the improved convergence rate of the iterative estimator, and derive the associated limiting distribution. A consistent estimator of the asymptotic variance is also provided, which plays a key role in the related inference problems. All results are validated through extensive simulations and two real data applications.