# AIC vs. Likelihood ratio test [duplicate]

Please, can someone explain the main differences of AIC and LRT? I understand, that both methods test, whether a new variable should be included in the model. Both methods integrated the Likelihood. The LRT is a hypothesis test, whether a complex model with more variables is better fitted on the data than a simpler nested model with less variables.

I hope you can help me! best regards!

• Welcome to Cross Validated! The more complex model cannot have a worse fit on the (in-sample) data, so your statement of the LRT isn’t quite right. – Dave Sep 10 '20 at 0:41

## 1 Answer

In brief, comparing two nested models with residuals which obey the same distribution, for example Gaussian, using a likelihood ratio test with a specific significance level can be considered as comparing two models with the AIC.

AIC scores are compared by differences so that the model is chosen which minimizes the AIC and thus, the model zero is accepted if $$AIC_0-AIC_1 < 0$$. This can be rewritten using the formula for AIC in the form $$2L_1 -2L_0 <2k_1-2k_0$$, which is $$\chi^2$$ distributed with $$k_1-k_0$$ degrees of freedom. That is actually a likelihood ratio test, substracting the log-likelihoods. The significance level can be determined by the critical value $$2k_1-2k_0$$ and a $$\chi^2$$ table.

This interpretation of the AIC is only possible when the models are nested. A nice overview about the similarities and differences between information criteria and hypothesis testing can be found for example in

Leontaritis, I. J., and S. A. Billings. "Model selection and validation methods for non-linear systems." International Journal of Control 45.1 (1987): 311-341.

• Thanks for the citation for the more general question. – Dave Harris Sep 10 '20 at 14:54