Computation and application of generalized linear mixed model derivatives using lme4.

Area of Science:

• Statistics
• Statistical Modeling
• Computational Statistics

Background:

• Maximum likelihood estimation (MLE) of generalized linear mixed models (GLMMs) is computationally intensive due to the marginalization of random effects.
• Derivatives of the GLMM likelihood are not readily available from standard estimation algorithms, hindering further statistical inference.

Purpose of the Study:

• To develop and present efficient methods for computing derivatives of fitted GLMMs.
• To demonstrate the practical applications of these derivatives in statistical inference.

Main Methods:

• Utilized a quadrature method for efficient computation of GLMM derivatives, specifically for models fitted using the lme4 package with a single clustering variable.
• Leveraged psychometric insights from item response models to aid in derivative calculation and validation.

Main Results:

• Established theoretical results for GLMM derivatives.
• Developed and validated a computationally efficient method for obtaining these derivatives.
• Demonstrated the application of derivatives for robust standard errors, score tests, and likelihood ratio tests.

Conclusions:

• The presented methods for computing GLMM derivatives are efficient and accessible.
• These derivative computations offer valuable tools for robust statistical inference and model comparison.
• The methods and applications are implemented in readily available R packages.