Faster Mixing Markov Chain Monte Carlo for Multidimensional IRT and Cognitive Diagnosis Models

拟合复杂项目反应理论(IRT)模型在教育评估、心理评估和患者报告的健康评估中变得越来越重要。近年来，马尔可夫链蒙特卡罗(MCMC)方法在这些模型中的应用激增，特别是在多维IRT和认知诊断模型中，这些模型很难应用最大似然估计等标准方法。不幸的是，马尔可夫链的缓慢混合通常是一个严重的问题，限制了研究人员从MCMC输出中得出有效推断的能力。该项目将开发适用于复杂IRT模型的快速混合MCMC算法，包括多维IRT模型和认知诊断/诊断分类模型。具体地说，改进的算法将包括重新缩放和重新居中的方法，如Meng和van Dyk的条件和边际增强方法，使用梯度信息的方法，以及自适应MCMC方法。该项目将扩展心理测量学文献中先进的MCMC方法的分析范围，将心理测量学社区的注意力集中在MCMC混合上，并为在复杂模型中有效地实施MCMC提供基础。有效、准确的IRT模型拟合具有重要意义。大多数教育评估的高风险性质要求有效的评估程序。因此，缓慢的混合导致MCMC估计过程的缓慢或不收敛，可能会深刻地影响推断，导致对个人的不正确排名。该项目开发的方法将解决心理测量学文献中MCMC应用的一个关键问题。该项目还将开发用于实施新方法的自由、开放源码软件。

Fitting complex item-response theory (IRT) models has become increasingly important in educational assessment, psychological assessment, and patient-reported health assessment. The use of Markov Chain Monte Carlo (MCMC) methods for fitting these models has surged in recent years, particularly for multidimensional IRT and cognitive diagnosis models where standard methods such as maximum likelihood estimation are difficult to apply. Unfortunately, slow mixing of the Markov chains is often a severe problem, limiting the ability of researchers to draw valid inferences from the MCMC output. This project will develop fast-mixing MCMC algorithms applied to complex IRT models including multidimensional IRT models and cognitive diagnosis/diagnostic classification models. Specifically, the improved algorithms will include rescaling and re-centering approaches such as Meng and van Dyk's conditional and marginal augmentation, methods using gradient information, and adaptive MCMC methods. The project will extend the analytical reach of advanced MCMC methods in the psychometric literature, focus the attention of the psychometric community on MCMC mixing, and provide the basis for effective MCMC implementation in complex models. Efficient and accurate fitting of IRT models is of profound importance. The high-stakes nature of most educational assessments demands valid estimation procedures. Therefore, slow mixing, leading to either slow or non-convergence of MCMC estimation procedures, can profoundly affect inference, leading to incorrect ranking of individuals. The methods developed by this project will address a critical problem in applications of MCMC in the psychometric literature. The project also will develop free, open-source software with which to implement the new methods.