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

Published online by Cambridge University Press:  01 January 2025

Minjeong Jeon ,
Frank Rijmen  and
Sophia Rabe-Hesketh
Minjeong Jeon*
Affiliation:
University of California, Los Angeles
Frank Rijmen
Affiliation:
American Institutes for Research
Sophia Rabe-Hesketh
Affiliation:
University of California, Berkeley
*
Correspondence should be made to Minjeong Jeon, Department of Education, University of California, Los Angeles, 405 Hilgard Avenue, Los Angeles, CA 90095, USA. Email: mjjeon@ucla.edu
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Abstract

We present a variational maximization–maximization algorithm for approximate maximum likelihood estimation of generalized linear mixed models with crossed random effects (e.g., item response models with random items, random raters, or random occasion-specific effects). The method is based on a factorized variational approximation of the latent variable distribution given observed variables, which creates a lower bound of the log marginal likelihood. The lower bound is maximized with respect to the factorized distributions as well as model parameters. With the proposed algorithm, a high-dimensional intractable integration is translated into a two-dimensional integration problem. We incorporate an adaptive Gauss–Hermite quadrature method in conjunction with the variational method in order to increase computational efficiency. Numerical studies show that under the small sample size conditions that are considered the proposed algorithm outperforms the Laplace approximation.

Keywords

variational approximationlower boundKullback–Leibler divergenceEM algorithmVMM algorithmadaptive quadratureGLMMcrossed random effects
Type
Original paper
Information
Psychometrika , Volume 82 , Issue 3 , September 2017 , pp. 693 - 716
Copyright
Copyright © 2017 The Psychometric Society

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Footnotes

Electronic supplementary material The online version of this article (doi:10.1007/s11336-017-9555-z) contains supplementary material, which is available to authorized users.

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