In diagnostic classification models (DCMs), the Q matrix encodes in which attributes are required for each item. The Q matrix is usually predetermined by the researcher but may in practice be misspecified which yields incorrect statistical inference. Instead of using a predetermined Q matrix, it is possible to estimate it simultaneously with the item and structural parameters of the DCM. Unfortunately, current methods are computationally intensive when there are many attributes and items. In addition, the identification constraints necessary for DCMs are not always enforced in the estimation algorithms which can lead to non-identified models being considered. We address these problems by simultaneously estimating the item, structural and Q matrix parameters of the Deterministic Input Noisy "And" gate model using a constrained Metropolis-Hastings Robbins-Monro algorithm. Simulations show that the new method is computationally efficient and can outperform previously proposed Bayesian Markov chain Monte-Carlo algorithms in terms of Q matrix recovery, and item and structural parameter estimation. We also illustrate our approach using Tatsuoka's fraction-subtraction data and Certificate of Proficiency in English data.

中文翻译：

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DINA 模型中 Q 矩阵估计的约束 Metropolis-Hastings Robbins-Monro 算法


在诊断分类模型 (DCM) 中，Q 矩阵对每个项目所需的属性进行编码。 Q 矩阵通常由研究人员预先确定，但实际上可能会被错误指定，从而产生不正确的统计推断。代替使用预定的Q矩阵，可以与DCM的项目和结构参数同时估计它。不幸的是，当存在许多属性和项目时，当前的方法是计算密集型的。此外，DCM 所需的识别约束并不总是在估计算法中强制执行，这可能导致考虑未识别的模型。我们通过使用约束 Metropolis-Hastings Robbins-Monro 算法同时估计确定性输入噪声“与”门模型的项目、结构和 Q 矩阵参数来解决这些问题。仿真表明，新方法计算效率高，并且在 Q 矩阵恢复、项目和结构参数估计方面优于先前提出的贝叶斯马尔可夫链蒙特卡罗算法。我们还使用 Tatsuoka 的分数减法数据和英语熟练程度证书数据来说明我们的方法。