I have collected data on offenders’ number of offences (0,1,2,3,4,5) (DV) in the previous year in different correction centres (level 2 as site) and thought about using multilevel Poisson regression (e.g., GLMER in R) but a preliminary result showed that the data were overdispersed. I tried re-grouping the DV data into a binary variable (0 vs. 1+) and using binary logistic regression model, and another option could be using negative binomial regression. I run both multilevel logistic and negative binomial regression models, and found that the results in terms of significant predictors were similar. However, both AIC and BIC from the logistic regression model were much smaller than those from Poisson and negative binomial regression models.

My question is: in this case, can I use logistic regression model and use the argument that both AIC and BIC are much smaller from logistic regression model?

Any suggestions, comments or references are much appreciated. Thank you.