In statistics, a generalized linear mixed model (GLMM) is an extension to the generalized linear model (GLM) in which the linear predictor contains random effects in addition to the usual fixed effects.[1][2][3] They also inherit from GLMs the idea of extending linear mixed models to non-normal data.
GLMMs provide a broad range of models for the analysis of grouped data, since the differences between groups can be modelled as a random effect. These models are useful in the analysis of many kinds of data, including longitudinal data.[4]
Model
GLMMs are generally defined as such that conditioned on the random effects, u , the dependent variable, y , is distributed according to an exponential family.[5]
\( {\displaystyle \ln {p}(y\vert u)=\sum {\frac {y_{i}\theta _{i}-b(\theta _{i})}{\phi }}+c(y_{i},\phi )} \)
\( {\displaystyle E[y\vert u]=\mu } \)
\( {\displaystyle var[y\vert u]=\phi V(\mu )} \)
\( {\displaystyle g(\mu )=X\beta +Zu} \)
Where X and \( \beta \) are the fixed effects design matrix, and fixed effects; Z and u are the random effects design matrix and random effects.
The complete likelihood,
\( {\displaystyle \ln {p}(y,u)=\ln \int p(y\vert u)p(u)du} \) \)
has no general closed form, and integrating over the random effects is usually extremely computationally intensive. In addition to numerically approximating this integral(e.g. via Gauss–Hermite quadrature), methods motivated by Laplace approximation have been proposed.[6] For example, the penalized quasi-likelihood method, which essentially involves repeatedly fitting (i.e. doubly iterative) a weighted normal mixed model with a working variate,[7] is implemented by various commercial and open source statistical programs.
Fitting a model
Fitting GLMMs via maximum likelihood (as via AIC) involves integrating over the random effects. In general, those integrals cannot be expressed in analytical form. Various approximate methods have been developed, but none has good properties for all possible models and data sets (e.g. ungrouped binary data are particularly problematic). For this reason, methods involving numerical quadrature or Markov chain Monte Carlo have increased in use, as increasing computing power and advances in methods have made them more practical.
The Akaike information criterion (AIC) is a common criterion for model selection. Estimates of AIC for GLMMs based on certain exponential family distributions have recently been obtained.[8]
Software
Several contributed packages in R provide GLMM functionality[9][10]
GLMM can be fitted using SAS and SPSS [11]
Matlab also provides a function called "fitglme" to fit GLMM models.
See also
Generalized estimating equation
Hierarchical generalized linear model
References
Breslow, N. E.; Clayton, D. G. (1993), "Approximate Inference in Generalized Linear Mixed Models", Journal of the American Statistical Association, 88 (421): 9–25, doi:10.2307/2290687, JSTOR 2290687
Stroup, W.W. (2012), Generalized Linear Mixed Models, CRC Press
Jiang, J. (2007), Linear and Generalized Linear Mixed Models and Their Applications, Springer
Fitzmaurice, G. M.; Laird, N. M.; Ware, J.. (2011), Applied Longitudinal Analysis (2nd ed.), John Wiley & Sons, ISBN 978-0-471-21487-8
Pawitan, Yudi. In All Likelihood: Statistical Modelling and Inference Using Likelihood (Paperbackition ed.). OUP Oxford. p. 459. ISBN 978-0199671229.
Breslow, N. E.; Clayton, D. G. (20 December 2012). "Approximate Inference in Generalized Linear Mixed Models". Journal of the American Statistical Association. 88 (421): 9–25. doi:10.1080/01621459.1993.10594284.
Wolfinger, Russ; O'connell, Michael (December 1993). "Generalized linear mixed models a pseudo-likelihood approach". Journal of Statistical Computation and Simulation. 48 (3–4): 233–243. doi:10.1080/00949659308811554.
Saefken, B.; Kneib, T.; van Waveren, C.-S.; Greven, S. (2014), "A unifying approach to the estimation of the conditional Akaike information in generalized linear mixed models" (PDF), Electronic Journal of Statistics, 8: 201–225, doi:10.1214/14-EJS881
Pinheiro, J. C.; Bates, D. M. (2000), Mixed-effects models in S and S-PLUS, Springer, New York
Berridge, D. M.; Crouchley, R. (2011), Multivariate Generalized Linear Mixed Models Using R, CRC Press
"IBM Knowledge Center". www.ibm.com. Retrieved 6 December 2017.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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