Negative values for AIC in General Mixed Model [duplicate]

Question

I'm trying to select the best model by the AIC in the General Mixed Model test. The best model is the model with the lowest AIC, but all my AIC's are negative!

• So is the biggest negative AIC the lowest value?
• Or is the smallest negative AIC the lowest value, because it's closer to 0?

For example is AIC -201,928 or AIC -237,847 the lowest value and thus the best model?

Answer 1

The AIC is defined as

$$\text{AIC} = 2k - 2\ln(L)$$

where $k$ denotes the number of parameters and $L$ denotes the maximized value of the likelihood function.

For model comparison, the model with the lowest AIC score is preferred. The absolute values of the AIC scores do not matter. These scores can be negative or positive.

In your example, the model with $\text{AIC} = -237.847$ is preferred over the model with $\text{AIC} = -201.928$.

You should not care for the absolute values and the sign of AIC scores when comparing models.

A good reference is Model Selection and Multi-model Inference: A Practical Information-theoretic Approach (Burnham and Anderson, 2004), particularly on page 62 (section 2.2):

In application, one computes AIC for each of the candidate models and selects the model with the smallest value of AIC.

as well as on page 63:

Usually, AIC is positive; however, it can be shifted by any additive constant, and some shifts can result in negative values of AIC. [...] It is not the absolute size of the AIC value, it is the relative values over the set of models considered, and particularly the differences between AIC values, that are important.