桥接项目反应理论与因子分析：具有贝叶斯估计的四参数混合二分模型
Bridging Item Response Theory and Factor Analysis: A Four-Parameter Mixture-Dichotomized Model with Bayesian Estimation
AI总结：
提出四参数因子分析模型，通过贝叶斯估计统一项目反应理论与因子分析，处理猜测与疏忽效应，并验证模型等价性。
摘要：
项目反应理论（IRT）和因子分析（FA）是建模潜在特质多项目测量的两大框架。虽然双参数IRT模型与二分FA模型之间的关系已明确建立，但具有额外参数的IRT模型的FA公式较少见。我们聚焦于四参数因子分析（4P FA）模型，该模型通过分层混合公式扩展了传统的二分单因子FA模型，以考虑猜测和疏忽效应。我们分析性地建立了4P FA与4P IRT模型的等价性，将FA-IRT对应关系扩展到双参数情形之外。开发了一种贝叶斯估计程序用于模型估计，以估计四个项目参数、受访者的潜在分数以及经猜测和疏忽效应调整后的分数。所提出的算法在R和Python中实现。一项模拟研究比较了4P模型在FA和IRT公式下的估计，并评估了FA参数化的实际意义。基于入学考试和焦虑量表的实证例子展示了4P FA与4P IRT模型之间的对应关系，并说明了所提出方法的应用。
英文摘要：
Item Response Theory (IRT) and Factor Analysis (FA) are two major frameworks for modeling multi-item measurements of latent traits. While the relationship between two-parameter IRT models and dichotomized FA models is well established, FA formulations for IRT models with additional parameters are less common. We focus on the four-parameter factor-analytic (4P FA) model that extends the traditional dichotomized single-factor FA model through a hierarchical mixture formulation accounting for guessing and inattention effects. We analytically establish the equivalence of the 4P FA and 4P IRT models, extending the FA--IRT correspondence beyond the two-parameter case. A Bayesian estimation procedure is developed for model estimation, to estimate the four item parameters, the respondents' latent scores, and the scores adjusted for guessing and inattention effects. The proposed algorithm is implemented in \texttt{R} and \texttt{Python}. A simulation study compares estimation under the FA and IRT formulations of the 4P model and evaluates the practical implications of the FA parametrization. Empirical examples based on an admission test and an anxiety inventory demonstrate the correspondence between the 4P FA and 4P IRT models and illustrate the application of the proposed methodology.
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