A popular model selection approach for generalized linear mixed-effects models is the Akaike information criterion, or AIC. Among others,  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 . We show that the similar strategy extends to Poisson regression with random effects, where conditional AIC can be obtained based on our observations. Simulation studies demonstrate the usage of the criterion.
"A note on conditional Akaike information for Poisson regression with random effects." Electron. J. Statist. 6 1 - 9, 2012. https://doi.org/10.1214/12-EJS665