Akaike information criterion

Overview
Akaike's information criterion, developed by Hirotsugu Akaike under the name of "an information criterion" (AIC) in 1971 and proposed in Akaike (1974), is a measure of the goodness of fit of an estimated statistical model. It is grounded in the concept of entropy. The AIC is an operational way of trading off the complexity of an estimated model against how well the model fits the data.

Definition
In the general case, the AIC is


 * $$AIC = 2k - 2\ln(L)\,$$

where k is the number of parameters in the statistical model, and L is the likelihood function.

Over the remainder of this entry, it will be assumed that the model errors are normally and independently distributed. Let n be the number of observations and RSS be the residual sum of squares. Then AIC becomes


 * $$AIC=2k + n\ln(RSS/n)\,$$

Increasing the number of free parameters to be estimated improves the goodness of fit, regardless of the number of free parameters in the data generating process. Hence AIC not only rewards goodness of fit, but also includes a penalty that is an increasing function of the number of estimated parameters. This penalty discourages overfitting. The preferred model is the one with the lowest AIC value. The AIC methodology attempts to find the model that best explains the data with a minimum of free parameters. By contrast, more traditional approaches to modeling start from a null hypothesis. The AIC penalizes free parameters less strongly than does the Schwarz criterion.

AICc and AICu
AICc is AIC with a second order correction for small sample sizes, to start with:


 * $$AICc = AIC + \frac{2k(k + 1)}{n - k - 1}\,$$

Since AICc converges to AIC as n gets large, AICc should be employed regardless of sample size (Burnham and Anderson, 2004).

McQuarrie and Tsai (1998: 22) define AICc as:


 * $$AICc = \ln{\frac{RSS}{n}} + \frac{n+k}{n-k-2}\ ,$$

and propose (p. 32) the closely related measure:


 * $$AICu = \ln{\frac{RSS}{n-k}} + \frac{n+k}{n-k-2}\ .$$

McQuarrie and Tsai ground their high opinion of AICc and AICu on extensive simulation work.

QAIC
QAIC (the quasi-AIC) is defined as:


 * $$QAIC = 2k-\frac{1}{c}2\ln{L}\,$$

where c is a variance inflation factor. QAIC adjusts for over-dispersion or lack of fit. The small sample version of QAIC is:


 * $$QAICc = QAIC + \frac{2k(k + 1)}{n - k - 1}\,$$.