AIC
Akaike's An Information Criterion
Generic function calculating Akaike's ‘An Information Criterion’ for one or several fitted model objects for which a loglikelihood value can be obtained, according to the formula $2*loglikelihood + k*npar$, where $npar$ represents the number of parameters in the fitted model, and $k = 2$ for the usual AIC, or $k = log(n)$ ($n$ being the number of observations) for the socalled BIC or SBC (Schwarz's Bayesian criterion).
 Keywords
 models
Usage
AIC(object, ..., k = 2)
BIC(object, ...)
Arguments
 object
 a fitted model object for which there exists a
logLik
method to extract the corresponding loglikelihood, or an object inheriting from classlogLik
.  ...
 optionally more fitted model objects.
 k
 numeric, the penalty per parameter to be used; the
default
k = 2
is the classical AIC.
Details
When comparing models fitted by maximum likelihood to the same data, the smaller the AIC or BIC, the better the fit. The theory of AIC requires that the loglikelihood has been maximized: whereas AIC can be computed for models not fitted by maximum likelihood, their AIC values should not be compared.
Examples of models not ‘fitted to the same data’ are where the
response is transformed (acceleratedlife models are fitted to
logtimes) and where contingency tables have been used to summarize
data.
These are generic functions (with S4 generics defined in package
stats4): however methods should be defined for the
loglikelihood function logLik
rather than these
functions: the action of their default methods is to call logLik
on all the supplied objects and assemble the results. Note that in
several common cases logLik
does not return the value at
the MLE: see its help page.
The loglikelihood and hence the AIC/BIC is only defined up to an
additive constant. Different constants have conventionally been used
for different purposes and so extractAIC
and AIC
may give different values (and do for models of class "lm"
: see
the help for extractAIC
). Particular care is needed
when comparing fits of different classes (with, for example, a
comparison of a Poisson and gamma GLM being meaningless since one has
a discrete response, the other continuous).
BIC
is defined as
AIC(object, ..., k = log(nobs(object)))
.
This needs the number of observations to be known: the default method
looks first for a "nobs"
attribute on the return value from the
logLik
method, then tries the nobs
generic, and if neither succeed returns BIC as NA
.
Value

If just one object is provided, a numeric value with the corresponding
AIC (or BIC, or ..., depending on
k
).If multiple objects are provided, a data.frame
with rows
corresponding to the objects and columns representing the number of
parameters in the model (df
) and the AIC or BIC.
References
Sakamoto, Y., Ishiguro, M., and Kitagawa G. (1986). Akaike Information Criterion Statistics. D. Reidel Publishing Company.
See Also
extractAIC
, logLik
, nobs
.
Examples
library(stats)
lm1 < lm(Fertility ~ . , data = swiss)
AIC(lm1)
stopifnot(all.equal(AIC(lm1),
AIC(logLik(lm1))))
BIC(lm1)
lm2 < update(lm1, . ~ . Examination)
AIC(lm1, lm2)
BIC(lm1, lm2)