# Negative values for AIC in General Mixed Model [duplicate]

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?

• There's nothing special about negative AIC. Smaller (i.e. more negative, for negative values) is better. – Glen_b Feb 1 '14 at 12:57
• Which place in the world is coldest today? The South Pole, at -40 degrees C, or Atlanta, GA, at -1 degrees C "because it's closer to 0"? This analogy is not facetious: like degrees Celsius, AIC is an additive scale with an arbitrary zero. – whuber Feb 1 '14 at 15:23

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.

• +1 to @Sven. Just one note: There is, I believe, some software which reports AIC just inverted from the above, so that higher is better. I remember this from a few years ago, and am not sure which software it was. However, SAS includes "lower is better" on some output re AIC, just because of this confusion. – Peter Flom Feb 1 '14 at 12:46
• @PeterFlom Thanks for pointing this out. One should check the manual of the software before comparing AIC values. However, the "classic" definition of AIC is the one above. – Sven Hohenstein Feb 1 '14 at 12:49
• It might help to realize that simply changing the units of the data can drastically change the AIC values, and even change the sign (positive or negative) of the AIC. But changing the units won't change the difference between the AIC of competing models. – Harvey Motulsky Feb 1 '14 at 15:25
• can anyone give some journal or citations about this sentence In your example, the model with AIC=−237.847 is preferred over the model with AIC=−201.928. Because in my study, i also got negative AIC? and i a bit confused ? – harithstats Jan 25 '15 at 6:24