A note on conditional Akaike information for Poisson regression with random effects

A popular model selection approach for generalized linear mixed-effects models is the Akaike information criterion, or AIC. Among others, \cite{vaida05} pointed out the distinction between the marginal and conditional inference depending on the focus of research. The conditional AIC was derived for the linear mixed-effects model which was later generalized by \cite{liang08}. We show that the similar strategy extends to Poisson regression with random effects, where condition AIC can be obtained based on our observations. Simulation studies demonstrate the usage of the criterion.