Computes the (generalized) Akaike An Information Criterion for a fitted parametric model.
extractAIC(fit, scale, k = 2, …)fitted model, usually the result of a fitter like
lm.
optional numeric specifying the scale parameter of the
model, see scale in step. Currently only used
in the "lm" method, where scale specifies the estimate
of the error variance, and scale = 0 indicates that it is to
be estimated by maximum likelihood.
numeric specifying the ‘weight’ of the
equivalent degrees of freedom (\(\equiv\) edf)
part in the AIC formula.
further arguments (currently unused in base R).
A numeric vector of length 2, with first and second elements giving
the ‘equivalent degrees of freedom’
for the fitted model fit.
the (generalized) Akaike Information Criterion for fit.
This is a generic function, with methods in base R for classes
"aov", "glm" and "lm" as well as for
"negbin" (package MASS) and "coxph" and
"survreg" (package survival).
The criterion used is
$$AIC = - 2\log L + k \times \mbox{edf},$$
where \(L\) is the likelihood and edf the equivalent degrees
of freedom (i.e., the number of free parameters for usual parametric
models) of fit.
For linear models with unknown scale (i.e., for lm and
aov), \(-2\log L\) is computed from the
deviance and uses a different additive constant to
logLik and hence AIC. If \(RSS\)
denotes the (weighted) residual sum of squares then extractAIC
uses for \(- 2\log L\) the formulae \(RSS/s - n\) (corresponding
to Mallows' \(C_p\)) in the case of known scale \(s\) and
\(n \log (RSS/n)\) for unknown scale.
AIC only handles unknown scale and uses the formula
\(n \log (RSS/n) + n + n \log 2\pi - \sum \log w\)
where \(w\) are the weights. Further AIC counts the scale
estimation as a parameter in the edf and extractAIC does not.
For glm fits the family's aic() function is used to
compute the AIC: see the note under logLik about the
assumptions this makes.
k = 2 corresponds to the traditional AIC, using k =
log(n) provides the BIC (Bayesian IC) instead.
Note that the methods for this function may differ in their
assumptions from those of methods for AIC (usually
via a method for logLik). We have already
mentioned the case of "lm" models with estimated scale, and
there are similar issues in the "glm" and "negbin"
methods where the dispersion parameter may or may not be taken as
‘free’. This is immaterial as extractAIC is only used
to compare models of the same class (where only differences in AIC
values are considered).
Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. New York: Springer (4th ed).