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

Author

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  • Minjeong Jeon

    (University of California, Los Angeles)

  • Frank Rijmen

    (American Institutes for Research)

  • Sophia Rabe-Hesketh

    (University of California, Berkeley)

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.

Suggested Citation

  • Minjeong Jeon & Frank Rijmen & Sophia Rabe-Hesketh, 2017. "A Variational Maximization–Maximization Algorithm for Generalized Linear Mixed Models with Crossed Random Effects," Psychometrika, Springer;The Psychometric Society, vol. 82(3), pages 693-716, September.
    • Handle: RePEc:spr:psycho:v:82:y:2017:i:3:d:10.1007_s11336-017-9555-z
    DOI: 10.1007/s11336-017-9555-z
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    3. Brian Gin & Nicholas Sim & Anders Skrondal & Sophia Rabe-Hesketh, 2020. "A Dyadic IRT Model," Psychometrika, Springer;The Psychometric Society, vol. 85(3), pages 815-836, September.
    4. Jiming Jiang & Matt P. Wand & Aishwarya Bhaskaran, 2022. "Usable and precise asymptotics for generalized linear mixed model analysis and design," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(1), pages 55-82, February.
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    7. Kazuhiro Yamaguchi & Kensuke Okada, 2020. "Variational Bayes Inference Algorithm for the Saturated Diagnostic Classification Model," Psychometrika, Springer;The Psychometric Society, vol. 85(4), pages 973-995, December.

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