Akaike Information Criterion - How To Apply AIC in Practice

How To Apply AIC in Practice

To apply AIC in practice, we start with a set of candidate models, and then find the models' corresponding AIC values. There will almost always be information lost due to using one of the candidate models to represent the "true" model. We wish to select, from among R candidate models, the model that minimizes the information loss. We cannot choose with certainty, but we can minimize the estimated information loss.

Denote the AIC values of the candidate models by AIC₁, AIC₂, AIC₃, …, AIC_R. Let AIC_min be the minimum of those values. Then exp((AIC_min−AIC_i)/2) can be interpreted as the relative probability that the ith model minimizes the (estimated) information loss.

As an example, suppose that there were three models in the candidate set, with AIC values 100, 102, and 110. Then the second model is exp((100−102)/2) = 0.368 times as probable as the first model to minimize the information loss, and the third model is exp((100−110)/2) = 0.007 times as probable as the first model to minimize the information loss. In this case, we would omit the third model from further consideration. We could take a weighted average of the first two models, with weights 1 and 0.368, respectively, and then do statistical inference based on the weighted multimodel; alternatively, we could gather more data to distinguish between the first two models.

If all the models in the candidate set have the same number of parameters, then using AIC might at first appear to be very similar to using the likelihood-ratio test. There are, however, important distinctions. In particular, the likelihood-ratio test is valid only for nested models whereas AIC (and AICc) has no such restriction.

The quantity exp((AIC_min−AIC_i)/2) is the relative likelihood of model i.