Electronic Journal of Statistics

A unifying approach to the estimation of the conditional Akaike information in generalized linear mixed models

Benjamin Saefken, Thomas Kneib, Clara-Sophie van Waveren, and Sonja Greven

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The conditional Akaike information criterion, AIC, has been frequently used for model selection in linear mixed models. We develop a general framework for the calculation of the conditional AIC for different exponential family distributions. This unified framework incorporates the conditional AIC for the Gaussian case, gives a new justification for Poisson distributed data and yields a new conditional AIC for exponentially distributed responses but cannot be applied to the binomial and gamma distributions. The proposed conditional Akaike information criteria are unbiased for finite samples, do not rely on a particular estimation method and do not assume that the variance-covariance matrix of the random effects is known. The theoretical results are investigated in a simulation study. The practical use of the method is illustrated by application to a data set on tree growth.

Article information

Electron. J. Statist., Volume 8, Number 1 (2014), 201-225.

First available in Project Euclid: 27 February 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J12: Generalized linear models
Secondary: 62J07: Ridge regression; shrinkage estimators

Conditional Akaike information criterion Kullback-Leibler distance model selection random effects generalized linear mixed models


Saefken, Benjamin; Kneib, Thomas; van Waveren, Clara-Sophie; Greven, Sonja. A unifying approach to the estimation of the conditional Akaike information in generalized linear mixed models. Electron. J. Statist. 8 (2014), no. 1, 201--225. doi:10.1214/14-EJS881. https://projecteuclid.org/euclid.ejs/1393510264

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  • Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. 2nd International Symposium on Information Theory 267–281.
  • Boehner, J., McCloy, K. R. & Strobl, J. (2006). SAGA – Analysis and Modelling Applications. Goettinger Geographische Abhandlungen 115, 130.
  • Breslow, N. E. & Clayton, D. G. (1993). Approximate inference in generalized linear mixed models. Journal of the American Statistical Association 88, 9–25.
  • Butler-Manning, D. (2008). Stand structure, gap dynamics and regeneration of a semi-natural mixed beech forest on limestone in central Europe: A case study. PhD thesis, Universität Freiburg, Forstwissenschaftliche Fakultät.
  • Chen, L. H. Y. (1975). Poisson approximation for dependent trials. The Annals of Probability 3, 534–545.
  • Donohue, M. C., Overholser, R., Xu, R. & Vaida, F. (2010). Conditional Akaike information under generalized linear and proportional hazards mixed models. Biometrika 98, 685–700.
  • Eilers, P. H. C. & Marx, B. D. (1996). Flexible smoothing with B-splines and penalties. Statistical Science 2, 89–121.
  • Greven, S. & Kneib, T. (2010). On the behaviour of marginal and conditional AIC in linear mixed models. Biometrika 97, 773–789.
  • Hodges, J. S. & Sargent, D. J. (2001). Counting degrees of freedom in hierarchical and other richly-parameterised models. Biometrika 88, 367–379.
  • Hudson, H. M. (1978). A natural identity for exponential families with applications in multiparameter estimation. The Annals of Statistics 6, 473–484.
  • Hurvich, C. M. & Sargent, C.-L. (1989). Regression and time series model selection in small samples. Biometrika 76, 297–307.
  • Lian, H. (2011). A note on conditional Akaike information for Poisson regression with random effects. Electronic Journal of Statistics 6, 1–9.
  • Liang, H., Wu, H. & Zou, G. (2008). A note on conditional AIC for linear mixed-effects models. Biometrika 95, 773–778.
  • McGilchrist, C. A. (1994). Estimation in generalized mixed models. Journal of the Royal Statistical Society. Series B (Methodological) 56, 61–69.
  • Nelder, J. A. & Wedderburn, R. W. M. (1972). Generalized linear models. Journal of the Royal Statistical Society, Series A, General 135, 370–384.
  • Ruppert, D., Wand, M. & Carroll, R. (2003). Semiparametric Regression. Cambridge University Press: New York, 2003
  • Saefken, B., Kneib, T., van Waveren, C.-S. & Greven, S. (2014). Supplements to “A unifying approach to the estimation of the conditional Akaike information in generalized linear mixed models.” DOIs: 10.1214/14-EJS881SUPPA, 10.1214/14-EJS881SUPPB.
  • Shen, X. & Huang, H.-C. (2006). Optimal model assessment, selection, and combination. Journal of the American Statistical Association 474, 554–568.
  • Stein, C. (1972). A Bound for the Error in the Normal Approximation to the Distribution of a Sum of Dependent Random Variables. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability 2, 586–602.
  • Vaida, F. & Blanchard, S. (2005). Conditional Akaike information for mixed-effects models. Biometrika 92, 351–370.
  • Wood, S. N. (2006). Generalized Additive Models: An Introduction with R. Chapman and Hall/CRC
  • Yu, D. & Yau, K. K. W. (2012). Conditional Akaike information criterion for generalized linear mixed models. Computational Statistics & Data Analysis 56, 629–644.

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