论文标题
因子建模中的扩展四四分析：与多维依赖性结构的可分离性和不确定性
Extended Tetrad Analysis in Factor Modelling: Separability and Uncertainty from Multidimensional Dependence Structures
论文作者
论文摘要
几何表示提供了一个原则上的框架，用于构建潜在构造的描述，并阐明其维度表征的不确定性来源。我们通过配对矩阵跨越的两个子空间引入了因子模型的新几何表示，其中确定表达式明确量化了不同维度子集对因子结构的贡献。该公式优化了与理解因子得分不确定性相关的基于等级的条件以及非独立性在仪器变量估计中对过度识别模型的含义。 通过加权这些多维贡献以编码对它们变化的敏感性的敏感性，我们将四型的定义扩展到一个代数过程中，该程序建立了识别可归因于各个维度的可变性成分的条件。关注一个因素编码结构信息的情况，我们通过图平面度得出了最小的条件，以确保这种特定于维度的可识别性。这些证明产生了正式的验证工具和建设性方法，用于生成这些条件失败的反例。这些反描述揭示了一种歧义性的，称为上下文，其中维度贡献的比较取决于选择剩余参考维度的选择，违反了有序理论一致性的公理。我们将这些发现与心理测量文献中研究的特定形式的不确定性形式联系起来。
Geometric representations provide a principled framework for structuring the description of latent constructs and clarifying sources of uncertainty in their dimensional characterisation. We introduce a novel geometric representation of factor models via two subspaces spanned by paired matrices, where determinantal expressions explicitly quantify the contributions of different dimension subsets to the factor structure. This formulation refines rank-based conditions relevant to understanding factor score indeterminacy and the implications of non-uniqueness in instrumental variable estimation for over-identified models. By weighting these multidimensional contributions to encode sensitivity to their variation, we extend the definition of tetrads into an algebraic procedure that establishes conditions for identifying variability components attributable to individual dimensions. Focusing on cases where one factor encodes structural information, we derive minimal conditions-expressed through graph planarity-that ensure such dimension-specific identifiability. The proofs yield both formal verification tools and constructive methods for generating counterexamples where these conditions fail. These counterexamples reveal a type of ambiguity, termed contextuality, in which the comparison of dimensional contributions depends on the choice of remaining reference dimensions, violating well-established axioms of order-theoretic consistency. We relate these findings to specific forms of uncertainty examined in the psychometric literature.
