When Is a General Factor Distinguishable? Non-Proportionality, Stable Structure, and the Bifactor Decision

2026-08-11

AI Summary

This study investigates whether a general factor is necessary to adequately account for the covariance structure when the underlying correlated first-order factor structure is known. By analyzing the algebraic properties of the population covariance matrix, it rigorously establishes, for the first time, the boundary conditions under which general-factor and correlated-factor models are covariance-equivalent, revealing that model distinguishability is a continuous rather than binary property. The work demonstrates that local dependencies are frequently misattributed to a general factor—a misattribution exacerbated with larger sample sizes. Theoretically, it proves that a K-factor model cannot reproduce the true covariance structure when each cluster contains exactly three items with non-proportional loadings. Building on these insights, the authors propose a two-stage partially exploratory factor analysis procedure that only retains solutions when the number of cross-loading factors remains stable across replications. Simulations show that conventional stability metrics may obscure structural misspecifications, and analyses of four empirical datasets corroborate both the method’s efficacy and its capacity for nuanced model discrimination.

Abstract

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.

Problem

• bifactor model
• general factor
• factor distinguishability
• covariance equivalence
• factor analysis

Innovation

• bifactor model
• factor indeterminacy
• covariance equivalence
• proportionality condition
• partially exploratory factor analysis