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-C1E6".

翻译：双因子分析是一种广泛应用于心理与教育测量学的验证性因子分析形式。使用双因子模型需要明确指定观测变量与群组因子之间关系的显式双因子结构。实践中，双因子结构有时未知，此时需要通过探索性双因子分析来发现双因子结构。遗憾的是，除Jennrich和Bentler（2011, 2012）提出的基于旋转的方法外，目前鲜有探索性双因子分析方法。然而，该方法仅能获得近似双因子结构，即使经过硬阈值处理后仍无法得到精确的双因子载荷结构。本文提出一种基于约束的优化方法，可从数据中学习精确的双因子载荷结构，从而克服基于旋转方法的缺陷。该方法的核心理念是将双因子载荷结构数学表征为一组等式约束，从而将探索性双因子分析问题转化为连续域中的约束优化问题，并通过增广拉格朗日法求解。通过模拟研究和实际数据案例验证了所提方法的有效性。本文还探讨了将该方法扩展至探索性层次因子分析的可行性。相关代码已发布于``https://anonymous.4open.science/r/Bifactor-ALM-C1E6"。