When Is a General Factor Distinguishable? Non-Proportionality, Stable Structure, and the Bifactor Decision
Abstract
Whether an added general dimension is necessary beyond correlated first-order factors is a property of population covariance, not an estimator. A bifactor structure is covariance-equivalent to correlated factors when general and group loadings are proportional within every cluster. When every cluster violates proportionality, at least three indicators per cluster and mild conditions rule out exact reproduction by a K-factor model with diagonal uniquenesses. Mixed configurations remain only partly characterized, motivating graded distinguishability. We develop a two-step procedure that delivers a stable first-order structure only when it persists across direct wider-count comparisons, then compares oblique and bifactor representations conditionally on delivery. A preliminary simulation study provides tentative design evidence. A major study develops three non-nested persistence profiles, designates .80/r2 as the practical default, and evaluates the frozen family on held-out replications from the same generators. Within the studied scenarios, anchor-only persistence outperforms anchor-zero, minimum congruence modestly outperforms maximum RMSD, and depth four adds no precision at the fixed .70 cutoff. Held-out results also expose stable-yet-wrong delivery under a weak core and show that the conditional comparison is highly accurate after correct recovery. Four empirical applications illustrate agreement, adjacent-count uncertainty, and non-delivery. Persistence, not count alone, carries the structural decision.
Related
- Are Good Generators Good Decision-Makers? Policy Learning for General Interventions via Retargeted Counterfactual Generation
- On generalized k-token graphs with distinguishable tokens and a maximum capacity of s tokens per vertex
- Tolerance Proportionality and Computational Stability in Adaptive Parallel-in-Time Runge-Kutta Methods
- Proportionality as a Property of Families of Objects: A Shift in the Level of Modelling
- Distinguishable Category Groupings in Complex Surveys: Which Categories Can the Design Tell Apart?
BibTeX
@misc{chen2026when,
title = {{When Is a General Factor Distinguishable? Non-Proportionality, Stable Structure, and the Bifactor Decision}},
author = {Jinsong Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.10731},
eprint = {2608.10731},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.10731},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - When Is a General Factor Distinguishable? Non-Proportionality, Stable Structure, and the Bifactor Decision
AU - Jinsong Chen
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2608.10731
UR - https://arxiv.org/abs/2608.10731
AB - Whether an added general dimension is necessary beyond correlated first-order factors is a property of population covariance, not an estimator. A bifactor structure is covariance-equivalent to correlated factors when general and group loadings are proportional within every cluster. When every cluster violates proportionality, at least three indicators per cluster and mild conditions rule out exact reproduction by a K-factor model with diagonal uniquenesses. Mixed configurations remain only partly characterized, motivating graded distinguishability. We develop a two-step procedure that delivers a stable first-order structure only when it persists across direct wider-count comparisons, then compares oblique and bifactor representations conditionally on delivery. A preliminary simulation study provides tentative design evidence. A major study develops three non-nested persistence profiles, designates .80/r2 as the practical default, and evaluates the frozen family on held-out replications from the same generators. Within the studied scenarios, anchor-only persistence outperforms anchor-zero, minimum congruence modestly outperforms maximum RMSD, and depth four adds no precision at the fixed .70 cutoff. Held-out results also expose stable-yet-wrong delivery under a weak core and show that the conditional comparison is highly accurate after correct recovery. Four empirical applications illustrate agreement, adjacent-count uncertainty, and non-delivery. Persistence, not count alone, carries the structural decision.
ER -