基于新兰开斯特-切舍信息矩阵检验的认知诊断模型误设检测
Detection of Cognitive Diagnostic Model Misspecification using New Lancaster-Chesher Information Matrix Tests
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中文总结 AI 辅助
本文扩展前期工作，将新方法应用于认知诊断模型，推导了仅需1或2个自由度的误设检验，无需复杂模拟，模拟显示其表现良好。
中文摘要 AI 辅助
模型设定检验在评估概率模型是否适用于估计和推断中发挥关键作用。现有的模型误设检测方法，如卡方拟合优度（GOF）检验，以及近年提出的M2统计量[Maydeu-Olivares等，2005；Maydeu和Joe，2014]，对于参数数量较多的模型，往往会产生自由度过大的检验统计量。一种替代方法基于信息矩阵（IM）等式，该等式断言若概率模型设定正确，最大似然估计量的渐近协方差矩阵可通过基于对数似然函数一阶或二阶导数的方法渐近估计。利用逆否论证，White（1982）提出了一种基于比较这两种替代协方差矩阵估计量的误设检验方法。Presnell和Boos（2004）扩展了该工作，证明了如何开发一种无论模型或数据复杂度如何都仅需1个自由度的误设检验。本文中，我们扩展了前期工作，额外应用Golden等人（2013、2016）的方法，推导并评估了认知诊断模型（CDM）的误设检验，无论模型或数据复杂度如何，这些检验仅需1或2个自由度。我们推导了检验的解析公式，使其无需计算密集的自助法模拟即可应用。模拟研究表明，这些渐近统计检验对实际中可能遇到的CDM模型和数据具有良好的水平（1类错误）和功效表现。
英文摘要
Model specification tests play a crucial role in evaluating the appropriateness of probability models for estimation and inference. Existing methods for the detection of model misspecification such as the chi-square goodness-of-fit (GOF) tests and more recently the M2 statistic \autocite{MaydeuOlivares2005,MaydeuJoe2014} tend to result in test statistics with excessive degrees of freedom for models with larger numbers of parameters. An alternative approach is based upon the Information Matrix (IM) equality. The IM equality asserts that if a probability model is correctly specified, the asymptotic covariance matrix of the maximum likelihood estimators can be asymptotically estimated using a methodology based upon either the first or second derivatives of the log-likelihood function. Using a contrapositive argument, White (1982) \nocite{Wh82} proposed a misspecification test methodology based upon comparing these two alternative covariance matrix estimators. Extending this work, Presnell and Boos (2004) \nocite{Presnell2004} showed how to develop a misspecification test which only requires one degree of freedom regardless of the complexity of the model or data. In this paper, we extend prior work and additionally apply methods of Golden et al. (2013, 2016) \nocite{golden2013}\nocite{Golden2016} to derive and evaluate misspecification tests for Cognitive Diagnostic Models (CDMs) which only require 1 or 2 degrees of freedom regardless of model or data complexity. Analytic formulas for the tests are derived so they can be applied without requiring computationally intensive bootstrap simulation methods. Our simulation studies show the asymptotic statistical tests have good level (type 1 error) and power performance for CDM models and data which might be encountered in practice.
