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.
"A unifying approach to the estimation of the conditional Akaike information in generalized linear mixed models." Electron. J. Statist. 8 (1) 201 - 225, 2014. https://doi.org/10.1214/14-EJS881