# Why do we get the same AIC for different models in a GLMM?

Our problem here described is to interprete the AIC from a GLMM negbin. Our data compose by 2 Categorical variables (Yes/Not), 2 Numerical variables and our random factor, all without any NA. We calculate AIC from all models possiblities, with synergy between variables, and we got this results:

Predictors in model_____loglik_____AIC Categorical1*Categorical2+Numerical1*Numerical2_____-271,03_____560,07 Categorical1+Categorical2*Numerical1*Numerical2_____-271,03_____560,07 Categorical1*Categorical2*Numerical1+Numerical2_____-271,03_____560,07 Categorical2*Numerical1*Numerical2_____-271,03_____560,07 ETC.........................................................

We obtain the same AIC for different combinations. What does it means? I just found that models could be equivalents, but any more information would be better to understand it.

Assuming that I understand your question correctly, you should first know how the AIC is computed: $$AIC = 2k - 2 \ln\left(\hat{L}\right)$$. Here, $$k$$ is the number of coefficients estimated, and $$L$$ is the likelihood. Your first 3 models have exactly 2 coefficients, hence the value of $$k$$ is the same across all 3 models. The log-likelihood is also the same for all 3 models, so necessarily the AIC has to be the same for all 3 models. As to why the log-likelihood is the same for all 3 models, that is a matter you should have a good look at I guess