Sufficient and Necessary Conditions for the Identifiability of the Q-matrix
Sufficient and Necessary Conditions for the Identifiability of the Q-matrix
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DOI：
10.5705/ss.202018.0410
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发表时间：
2018-10
影响因子：
1.4
通讯作者：
Yuqi Gu;Gongjun Xu
中科院分区：
文献类型：
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作者：
Yuqi Gu;Gongjun Xu
限制潜在类模型（RLCMs）最近在教育评估，精神病评估和医疗诊断中获得了突出地位。与传统的潜在类模型不同，设计矩阵对RLCM模型参数施加限制，以尊重从业者的科学假设。设计矩阵，在认知诊断文献中称为$Q$-矩阵，通常由从业者和领域专家构建，但它是主观的，可能会被错误指定。为了解决这个问题，研究人员提出了估计设计$Q$-矩阵的数据。另一方面，$Q$-矩阵和模型参数的基本可学习性问题仍然没有得到充分的探索，现有的研究往往施加比需要的更强，甚至不切实际的条件。本文给出了Q-矩阵和RLCM模型参数联合可辨识的充要条件。所提出的可辨识性条件仅依赖于设计矩阵，因此易于在实践中验证。
Restricted latent class models (RLCMs) have recently gained prominence in educational assessment, psychiatric evaluation, and medical diagnosis. Different from conventional latent class models, restrictions on RLCM model parameters are imposed by a design matrix to respect practitioners' scientific assumptions. The design matrix, called the $Q$-matrix in cognitive diagnosis literature, is usually constructed by practitioners and domain experts, yet it is subjective and could be misspecified. To address this problem, researchers have proposed to estimate the design $Q$-matrix from the data. On the other hand, the fundamental learnability issue of the $Q$-matrix and model parameters remains underexplored and existing studies often impose stronger than needed or even impractical conditions. This paper proposes the sufficient and necessary conditions for the joint identifiability of the $Q$-matrix and RLCM model parameters. The developed identifiability conditions only depend on the design matrix and therefore is easy to verify in practice.