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What are the relative merits of each approach, and which circumstances call for one rather than the other?

To some extent I have a specific example in mind, which I've discussed here. In that example I am comparing various linear mixed models. However, I'm interested in general principles that can be applied across various research questions.

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    $\begingroup$ Anova and Likelihood ratio tests are for nested models, i.e. one of the model is a special case of the other one. AIC and BIC can be used for non-nested models. $\endgroup$ – user83346 Sep 14 '15 at 8:46
  • $\begingroup$ ANOVA can be used to determine partial probabilities of parameters, which can then be deleted and/or others added to obtain a best model. I compare the F-statistics between non-nested ANOVA tests. It is unnecessary to assume that residuals are normally distributed just to obtain an AIC result. More, AIC is not used to compare different regression types of even the same model, but ANOVA can be used in that way. ANOVA wins in my book. More here $\endgroup$ – Carl Jun 27 '16 at 13:29
  • $\begingroup$ @Carl ANOVA is also not used to compare different regression types... $\endgroup$ – SmallChess Feb 15 '17 at 9:07
  • $\begingroup$ @StudentT How many counter examples is enough? Try this ANOVA weighted regression, and if that is insufficient, I'll give you more. $\endgroup$ – Carl Feb 15 '17 at 21:58

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