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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.

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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

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  • $\begingroup$ Thank you very much for answering, We have one model with the best AIC but the others have the same AIC, it could indicate that models are wrong? Concretely these are our results Predictors in model________________________________loglik__AIC__Delta AIC None____________________________________________-281,79_569,58_-10,81 Development_stageN_applications+Altitude_____________-272,39_558,77__0 Type_treatmentDevelopment_stage+AltitudeN_applications_-271,03_560,07_1,29 Type_treatment+Development_stageAltitude*N_applications_271,03_560,07_1,29 ETC... Thanks in advance $\endgroup$ Jan 30 '19 at 10:26
  • $\begingroup$ I must say that the formatting of your output here makes it somewhat hard to read. However, If I understand the output correctly, we are still dealing with 2 coefficients in both of the bottom 2 models. These get the same loglikelihood. This can be because either both models perform equally bad, or because your estimation method somehow doesn't work as it should. What are the data types of Type_Treatment, Development_stage, Altitude and N_applications? The answer might be hidden in there, or otherwise you could revise your estimation methods $\endgroup$
    – J. Dekker
    Jan 30 '19 at 16:25
  • $\begingroup$ Sorry for the format. Comments don't allow to add a table like i want. Data is compound by 2 categorical variables, 2 numerical variables and our random factor: Type_Treatment (Yes/No), Development_stage (Yes/No), Altitude (number), N_applications (1,2 or 3) and Random Factor. Maybe we could disscus it by email. Thank you very much. $\endgroup$ Jan 30 '19 at 17:09
  • $\begingroup$ I think that either there is an error in your estimation method, or these models happen to be equally good. If it is neither, then I think the problem is beyond my knowledge/skill $\endgroup$
    – J. Dekker
    Jan 30 '19 at 17:53

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