Projective item response theory models for count data and their application as interpretable approximations to black box models of machine learning

项目反应理论（IRT）为潜变量提供了测量模型。个人特定的潜在变量估计及其测量误差可以在各种情况下计算。因此，IRT是心理学和实证教育研究中诊断测试的操作使用和评估的最全面的统计基础。 与二进制数据的项目反应理论方法相比，计数数据的项目反应理论模型还不成熟。许多计数数据IRT模型使用泊松分布，但由此产生的条件等离差假设很少在经验上站得住脚。目前基于康威-麦克斯韦-泊松分布的方法解决了这个问题，但到目前为止还不能考虑因素负荷。在应用中，特别是当使用非结构化指标时，多维潜在变量星座是合理的。利用投射IRT方法可以推导出一个具有局部项目依赖的一维IRT模型，该模型有利于解释和进一步应用。因此，投射IRT模型被推广到计数数据的情况。特别是，多维IRT模型可以在其主维度上进行投影。机器学习方法在心理学的许多领域都被视为强大的数据分析工具，但在教育数据挖掘领域也是如此。许多机器学习方法的结果很难解释。在这个研究项目中开发的投射IRT模型被应用为易于解释的替代模型，在黑盒机器学习模型用于其预测性能或分类准确性的情况下。由于多维甚至高维潜变量星座在数值上是复杂的，本文提出了一种带因子加载的一般计数数据IRT模型的EM算法。将考虑具有过度分散和欠分散的所有常见计数数据分布，以及人员和项目水平的协变量及其相互作用。

Item Response Theory (IRT) provides measurement models for latent variables. Person-specific latent variable estimates and their measurement errors can be calculated in a variety of situations. IRT thus is the most comprehensive statistical basis for the operative use and evaluation of diagnostic testing in psychology and empirical educational research. Compared to IRT methods for binary data, count data IRT models are underdeveloped. Many count data IRT models use the Poisson distribution, but the resulting conditional equidispersion assumption is rarely empirically defensible. Current approaches based on the Conway-Maxwell-Poisson distribution solve this problem, but so far cannot take factor loadings into account. In applications, especially when unstructured indicators are to be used, multidimensional latent variable constellations are plausible. With projective IRT methods it is possible to derive an empirically indistinguishable one-dimensional IRT model with local item dependence, which is favorable for interpretation and further use. Therefore, projective IRT models are generalized to the count data case. In particular, multidimensional IRT models can be projected on their main dimension. Machine learning methods are seen as powerful data analytical tools in many areas of psychology, but also in the field of educational data mining. Results of many machine learning methods are difficult to interpret. The projective IRT models developed in this research project are applied as easily interpretable surrogate models in situations where a black box machine learning model is used for its predictive performance or its classification accuracy. This results in an interpretable approximation to a black box model, which helps to better understand it. Since multi-dimensional and even high-dimensional latent variable constellations are numerically complex, an EM algorithm for a general count data IRT model with factor loadings is developed. All common count data distributions with over- and under-dispersion will be considered, as well as covariates at the person and item level and their interactions.