Model section when AIC and logLik differ To analyze my ordinal (Liker scale ratings) I used the clmm function for the ordinal package. Since I have fixed and random effect, I try do backward model selection (of nested model) using the anova function. When there is a significant difference given by Chisq, I used that model. However, my understand was that in the cases where the difference is not significant to select the model with the lowest AIC value, but I have also heard to select the model with the lowest logLik value. In my current model comparison those values don't go hand in had. Can someone tell me what selection criteria to follow in those cases and if selected the one with the lowest AIC is correct?
anova(ord.model1,
      ord.model2)


Likelihood ratio tests of cumulative link models:
 

           no.par    AIC  logLik LR.stat df Pr(>Chisq)
ord.model2     19 2247.5 -1104.8                      
ord.model1     20 2249.0 -1104.5  0.5187  1     0.4714



formula:                                                             link:      threshold:
ord.model2  rating ~ factor1 + (factor1 + factor2 | item) + (factor1 + factor2 | subject)    logit  flexible
ord.model1  rating ~ factor1 + factor2 + (factor1 + factor2 | item) + (factor1 + factor2 | subject) logit   flexible
```


If the models are nested then likelihood (and the logarithm) is always equal or higher for the model with more parameters, because the model with more parameters makes a better fit. So comparing likelihood, which one is higher or lower, says very little about which model is better.
The AIC compares the likelihood and the parameters.
$$AIC  = 2k - 2\text{LogLik} = \text{estimate of information loss}$$
Where $k$ is the number of model parameters and $\text{LogLik}$ is the likelihood.
The AIC is an estimate for the information loss (KL divergence between the model and the true data generating process). Using it for a model comparison makes sense when you wish to select a model that has the least information loss.
