Maximum likelihood estimation of generalized linear mixed models(GLMMs) is difficult due to marginalization of the random effects. Computing derivatives of a fitted GLMM's likelihood (with respect to model parameters) is also difficult, especially because the derivatives are not by-products of popular estimation algorithms. In this paper, we describe GLMM derivatives along with a quadrature method to efficiently compute them, focusing on lme4 models with a single clustering variable. We describe how psychometric results related to IRT are helpful for obtaining these derivatives, as well as for verifying the derivatives' accuracies. After describing the derivative computation methods, we illustrate the many possible uses of these derivatives, including robust standard errors, score tests of fixed effect parameters, and likelihood ratio tests of non-nested models. The derivative computation methods and applications described in the paper are all available in easily-obtained R packages.

翻译：由于随机效应的边缘化,很难对通用线性混合模型(GLMMs)进行最大可能性估计,因为随机效应的边缘化,因此很难对通用线性混合模型(GLMMs)的最大可能性进行估计,安装的GLMMs可能性的计算机衍生物(就模型参数而言)也是困难的,特别是因为衍生物不是大众估计算法的副产品。在本文中,我们描述了GLMM衍生物以及有效计算这些衍生物的二次分析方法,重点是具有单一组群变量的lme4模型。我们描述了与IRT有关的心理测量结果如何有助于获取这些衍生物,以及验证衍生物的灵敏度。在描述衍生物计算方法之后,我们举例说明了这些衍生物的许多可能用途,包括严格的标准错误、固定效应参数的评分测试和非废弃模型的可能性比测试。本文中描述的衍生物计算方法和应用都存在于容易获取的R包中。