Exact Exploratory Bi-factor Analysis: A Constraint-based Optimisation Approach
Jiawei Qiao, Yunxiao Chen, Zhiliang Ying
TL;DR
This paper tackles exploratory bi-factor analysis when the group-factor structure is unknown by formulating it as a constrained optimization problem that enforces an exact bi-factor loading structure through equality constraints. It introduces an augmented Lagrangian method (ALM) with a reparameterization of the factor correlation matrix $\Phi$ to solve the nonconvex problem, and uses a BIC-based criterion to select the number of group factors $G$. The authors provide identifiability conditions under mild assumptions, demonstrate superior recovery of the true loading structure and factor count in simulations (including exact and approximate bi-factor data), and apply the method to IPIP Extraversion data to obtain an interpretable seven-group-factor bi-factor solution with good fit. The approach offers a principled alternative to rotation-based exploratory methods, with potential extensions to hierarchical and non-linear factor models and the option to relax constraints for approximate bi-factor structures. The work advances exact loading structure discovery in psychometric measurement and related domains, enabling more precise interpretation and factor-detection in practice.
Abstract
Bi-factor analysis is a form of confirmatory factor analysis widely used in psychological and educational measurement. The use of a bi-factor model requires the specification of an explicit bi-factor structure on the relationship between the observed variables and the group factors. In practice, the bi-factor structure is sometimes unknown, in which case an exploratory form of bi-factor analysis is needed to find the bi-factor structure. Unfortunately, there are few methods for exploratory bi-factor analysis, with the exception of a rotation-based method proposed in Jennrich and Bentler (2011, 2012). However, this method only finds approximate bi-factor structures, as it does not yield an exact bi-factor loading structure, even after applying hard thresholding. In this paper, we propose a constraint-based optimisation method that learns an exact bi-factor loading structure from data, overcoming the issue with the rotation-based method. The key to the proposed method is a mathematical characterisation of the bi-factor loading structure as a set of equality constraints, which allows us to formulate the exploratory bi-factor analysis problem as a constrained optimisation problem in a continuous domain and solve the optimisation problem with an augmented Lagrangian method. The power of the proposed method is shown via simulation studies and a real data example. Extending the proposed method to exploratory hierarchical factor analysis is also discussed. The codes are available on ``https://anonymous.4open.science/r/Bifactor-ALM-method-757D".