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A Variational Maximization-Maximization Algorithm for Generalized Linear Mixed Models with Crossed Random Effects.

Minjeong Jeon1, Frank Rijmen2, Sophia Rabe-Hesketh3

  • 1Department of Education, University of California, Los Angeles, 405 Hilgard Avenue, Los Angeles, CA, 90095 , USA. mjjeon@ucla.edu.

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Summary
This summary is machine-generated.

This study introduces a new variational algorithm for estimating generalized linear mixed models with complex random effects. The method improves computational efficiency and accuracy, especially in small sample scenarios.

Keywords:
EM algorithmGLMMKullback–Leibler divergenceVMM algorithmadaptive quadraturecrossed random effectslower boundvariational approximation

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Area of Science:

  • Statistics
  • Machine Learning
  • Psychometrics

Background:

  • Generalized linear mixed models (GLMMs) are widely used but estimating them with crossed random effects is computationally challenging.
  • Existing methods like Laplace approximation struggle with high-dimensional integrals and accuracy in small samples.

Purpose of the Study:

  • To develop a novel variational maximization-maximization algorithm for approximate maximum likelihood estimation of GLMMs with crossed random effects.
  • To enhance computational efficiency and accuracy compared to existing methods, particularly under small sample conditions.

Main Methods:

  • A factorized variational approximation of the latent variable distribution is employed to create a lower bound of the log marginal likelihood.
  • This lower bound is maximized with respect to both the factorized distributions and model parameters.
  • Adaptive Gauss-Hermite quadrature is integrated to further improve computational efficiency, reducing a high-dimensional integration to a two-dimensional problem.

Main Results:

  • The proposed variational algorithm effectively handles GLMMs with crossed random effects.
  • Numerical studies demonstrate superior performance over the Laplace approximation, especially under small sample size conditions.
  • The method successfully transforms intractable high-dimensional integration into a manageable two-dimensional problem.

Conclusions:

  • The developed variational maximization-maximization algorithm offers a more accurate and efficient approach for estimating GLMMs with crossed random effects.
  • This method provides a valuable alternative to the Laplace approximation, particularly in challenging small sample data scenarios.
  • The integration of adaptive Gauss-Hermite quadrature significantly boosts computational efficiency.