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Usable and precise asymptotics for generalized linear mixed model analysis and design

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  • Jiming Jiang
  • Matt P. Wand
  • Aishwarya Bhaskaran

Abstract

We derive precise asymptotic results that are directly usable for confidence intervals and Wald hypothesis tests for likelihood‐based generalized linear mixed model analysis. The essence of our approach is to derive the exact leading term behaviour of the Fisher information matrix when both the number of groups and number of observations within each group diverge. This leads to asymptotic normality results with simple studentizable forms. Similar analyses result in tractable leading term forms for the determination of approximate locally D‐optimal designs.

Suggested Citation

  • 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.
    • Handle: RePEc:bla:jorssb:v:84:y:2022:i:1:p:55-82
    DOI: 10.1111/rssb.12473
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    1. 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.
    2. Yoichi Miyata, 2004. "Fully Exponential Laplace Approximations Using Asymptotic Modes," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 1037-1049, December.
    3. T. W. Waite & D. C. Woods, 2015. "Designs for generalized linear models with random block effects via information matrix approximations," Biometrika, Biometrika Trust, vol. 102(3), pages 677-693.
    4. Magnus, J.R. & Neudecker, H., 1979. "The commutation matrix : Some properties and applications," Other publications TiSEM d0b1e779-7795-4676-ac98-1, Tilburg University, School of Economics and Management.
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    1. Bhaskaran, Aishwarya & Wand, Matt P., 2023. "Dispersion parameter extension of precise generalized linear mixed model asymptotics," Statistics & Probability Letters, Elsevier, vol. 193(C).

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