部分掌握认知诊断模型的可识别性

中文摘要 AI 辅助

部分掌握（PM）认知诊断模型（CDMs）通过用多个潜在属性的连续掌握分数取代二元潜在属性掌握指标来扩展传统CDMs。在PM - CDMs中，每个受试者由一个固定的连续潜在掌握向量表征，从中独立生成特定项目的二元属性概况。此公式在经典CDMs和连续潜在变量模型之间架起桥梁。尽管对PM - CDMs兴趣日增，但其可识别性属性尚未探索。本文为PM - CDMs建立了首个可识别性结果。推导了与传统CDMs既定条件直接类似的可识别性充分条件。通过对一个有五个项目和两个潜在属性的最小示例进行符号计算表明模型参数化的雅可比行列式一般非零。结合实分析和代数统计工具，证明此局部属性意味着项目参数和相关潜在属性的边际分布一般有限到一可识别。还表明若Q矩阵对所有属性对都包含此类可识别局部结构，可识别性扩展到整个PM - CDM。这些发现为部分掌握认知诊断模型的估计和推断提供了严格理论基础。

英文摘要

Partial-mastery (PM) cognitive diagnostic models (CDMs) extend traditional CDMs by replacing binary latent attribute mastery indicators with continuous mastery scores for multiple latent attributes. In PM-CDMs, each subject is characterized by a fixed continuous latent mastery vector, from which item-specific binary attribute profiles are independently generated. This formulation provides a bridge between classical CDMs and continuous latent variable models. Despite growing interest in PM-CDMs, their identifiability properties remain unexplored. In this work, we establish the first identifiability results for PM-CDMs. We derive sufficient conditions for identifiability that are direct analogues of established conditions for traditional CDMs. To develop the main argument, we use symbolic computation on a minimal example with five items and two latent attributes to show that the Jacobian of the model parameterization is generically nonzero. Combining tools from real analysis and algebraic statistics, we prove that this local property implies generic finite-to-one identifiability of the item parameters and the marginal distributions of the relevant latent attributes. We further show that if the $Q$-matrix contains such identifiable local structures for all attribute pairs, identifiability extends to the full PM-CDM. These findings provide a rigorous theoretical foundation for estimation and inference in partial-mastery cognitive diagnostic models.

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30 pages, 1 figure, 1 table