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When I have a two mixed models (lme function) with different df then ANOVA summary shows the p-value of likelihood ratio test as following :

         Model df   AIC      BIC      logLik   Test  L.Ratio  p-value
fsModel3     1 14 -1429.098 -1367.059 728.5489                        
fsModel1     2 15 -1428.438 -1361.968 729.2192 1 vs 2 1.340498  0.2469

However, when I compared two or more models with the same df ( same random effect ) only different fixed effects ( same number of fixed effects ). My ANOVA table look like this :

         Model df    AIC       BIC     logLik
fsModel2     1 14 -1428.469 -1366.431 728.2346
fsModel3     2 14 -1429.098 -1367.059 728.5489
fsModel4     3 14 -1428.886 -1366.847 728.4429

My intuition tell me that then I am selecting the model with the lowest AIC/BIC. Am I right ? Is there a reason why the L.Ratio can't be calculated on models with same df ?

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    $\begingroup$ You can calculate likelihood ratio on models with same df (non-nested models), and the intuition behind likelihood ratios should still be valid, but the usual mathematics is not: You cannot any more expect a chisquare distribution, not even asymptotically, as the ratio can vary on both sides of one! $\endgroup$ – kjetil b halvorsen Jan 8 '19 at 23:07
  • $\begingroup$ Thank you for your comment. Then is there a way how to compare two (non-nested models) and obtain meaningful p-value ( different test ) ? As you probably know, in research papers everybody want to see a p-value ( don't ask me why :) ). For me the AIC/BIC is sufficient, however I am affraid that for my collegues not. $\endgroup$ – user210804 Jan 9 '19 at 8:36
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Likelihood ratio tests are only appropriate on nested models.

If your models have the same df, they must not be nested: indeed, you state that they have different fixed effects.

You can still compare AIC/BIC, but your models are all sufficiently similar in AIC and BIC that under normal conventions there isn't much meaningful difference between them.

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