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REML estimation for binary data in GLMMs


  • Noh, Maengseok
  • Lee, Youngjo


The restricted maximum likelihood (REML) procedure is useful for inferences about variance components in mixed linear models. However, its extension to hierarchical generalized linear models (HGLMs) is often hampered by analytically intractable integrals. Numerical integration such as Gauss-Hermite quadrature (GHQ) is generally not recommended when the dimensionality of the integral is high. With binary data various extensions of the REML method have been suggested, but they have had unsatisfactory biases in estimation. In this paper we propose a statistically and computationally efficient REML procedure for the analysis of binary data, which is applicable over a wide class of models and design structures. We propose a bias-correction method for models such as binary matched pairs and discuss how the REML estimating equations for mixed linear models can be modified to implement more general models.

Suggested Citation

  • Noh, Maengseok & Lee, Youngjo, 2007. "REML estimation for binary data in GLMMs," Journal of Multivariate Analysis, Elsevier, vol. 98(5), pages 896-915, May.
  • Handle: RePEc:eee:jmvana:v:98:y:2007:i:5:p:896-915

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    References listed on IDEAS

    1. Thomas R. Ten Have & A. Russell Localio, 1999. "Empirical Bayes Estimation of Random Effects Parameters in Mixed Effects Logistic Regression Models," Biometrics, The International Biometric Society, vol. 55(4), pages 1022-1029, December.
    2. Yun, Sungcheol & Lee, Youngjo, 2004. "Comparison of hierarchical and marginal likelihood estimators for binary outcomes," Computational Statistics & Data Analysis, Elsevier, vol. 45(3), pages 639-650, April.
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    Cited by:

    1. Meza, Cristian & Jaffrézic, Florence & Foulley, Jean-Louis, 2009. "Estimation in the probit normal model for binary outcomes using the SAEM algorithm," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 1350-1360, February.
    2. Håvard Rue & Sara Martino & Nicolas Chopin, 2009. "Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 319-392.
    3. Noh, Maengseok & Lee, Youngjo, 2008. "Hierarchical-likelihood approach for nonlinear mixed-effects models," Computational Statistics & Data Analysis, Elsevier, vol. 52(7), pages 3517-3527, March.
    4. Noh, Maengseok & Wu, Lang & Lee, Youngjo, 2012. "Hierarchical likelihood methods for nonlinear and generalized linear mixed models with missing data and measurement errors in covariates," Journal of Multivariate Analysis, Elsevier, vol. 109(C), pages 42-51.
    5. Sumanta Adhya & Tathagata Banerjee & Gaurangadeb Chattopadhyay, 2012. "Inference on finite population categorical response: nonparametric regression-based predictive approach," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 96(1), pages 69-98, January.
    6. Lee, Woojoo & Shi, Jian Qing & Lee, Youngjo, 2010. "Approximate conditional inference in mixed-effects models with binary data," Computational Statistics & Data Analysis, Elsevier, vol. 54(1), pages 173-184, January.
    7. Youngjo Lee & Myoungjin Jang & Woojoo Lee, 2011. "Prediction interval for disease mapping using hierarchical likelihood," Computational Statistics, Springer, vol. 26(1), pages 159-179, March.
    8. Chan, Moon-tong & Yu, Dalei & Yau, Kelvin K.W., 2015. "Multilevel cumulative logistic regression model with random effects: Application to British social attitudes panel survey data," Computational Statistics & Data Analysis, Elsevier, vol. 88(C), pages 173-186.
    9. Yu, Dalei & Yau, Kelvin K.W., 2012. "Conditional Akaike information criterion for generalized linear mixed models," Computational Statistics & Data Analysis, Elsevier, vol. 56(3), pages 629-644.
    10. Joe, Harry, 2008. "Accuracy of Laplace approximation for discrete response mixed models," Computational Statistics & Data Analysis, Elsevier, vol. 52(12), pages 5066-5074, August.
    11. Sun-Joo Cho & Paul Boeck & Susan Embretson & Sophia Rabe-Hesketh, 2014. "Additive Multilevel Item Structure Models with Random Residuals: Item Modeling for Explanation and Item Generation," Psychometrika, Springer;The Psychometric Society, vol. 79(1), pages 84-104, January.
    12. Cho, S.-J. & Rabe-Hesketh, S., 2011. "Alternating imputation posterior estimation of models with crossed random effects," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 12-25, January.


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