Functions to compute Akaike's information criterion (AIC), the second-order AIC (AICc), as well as their quasi-likelihood counterparts (QAIC, QAICc).
AICc(mod, return.K = FALSE, second.ord = TRUE, nobs = NULL, ...) # S3 method for aov
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for betareg
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for clm
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for clmm
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for coxme
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for coxph
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for fitdist
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for fitdistr
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for glm
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, c.hat = 1, ...)
# S3 method for glmmTMB
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, c.hat = 1, ...)
# S3 method for gls
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for gnls
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for hurdle
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for lavaan
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for lm
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for lme
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for lmekin
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for maxlikeFit
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, c.hat = 1, ...)
# S3 method for mer
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for merMod
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for lmerModLmerTest
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for multinom
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, c.hat = 1, ...)
# S3 method for negbin
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for nlme
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for nls
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for polr
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for rlm
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for survreg
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
# S3 method for unmarkedFit
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, c.hat = 1, ...)
# S3 method for vglm
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, c.hat = 1, ...)
# S3 method for zeroinfl
AICc(mod, return.K = FALSE, second.ord = TRUE,
nobs = NULL, ...)
AICc returns the AIC, AICc, QAIC, or QAICc, or the number of
estimated parameters, depending on the values of the arguments.
an object of class aov, betareg, clm,
clmm, clogit, coxme, coxph,
fitdist, fitdistr, glm, glmmTMB,
gls, gnls, hurdle, lavaan, lm,
lme, lmekin, maxlikeFit, mer,
merMod, lmerModLmerTest, multinom,
negbin, nlme, nls, polr, rlm,
survreg, vglm, zeroinfl, and various
unmarkedFit classes containing the output of a model.
logical. If FALSE, the function returns the information
criterion specified. If TRUE, the function returns K (number
of estimated parameters) for a given model.
logical. If TRUE, the function returns the second-order Akaike
information criterion (i.e., AICc).
this argument allows to specify a numeric value other than total
sample size to compute the AICc (i.e., nobs defaults to total number
of observations). This is relevant only for mixed models or various
models of unmarkedFit classes where sample size is not
straightforward. In such cases, one might use total number of
observations or number of independent clusters (e.g., sites) as the
value of nobs.
value of overdispersion parameter (i.e., variance inflation factor) such
as that obtained from c_hat. Note that values of c.hat different
from 1 are only appropriate for binomial GLM's with trials > 1 (i.e.,
success/trial or cbind(success, failure) syntax), with Poisson GLM's,
single-season occupancy models (MacKenzie et al. 2002), dynamic
occupancy models (MacKenzie et al. 2003), or N-mixture models
(Royle 2004, Dail and Madsen 2011). If c.hat > 1,
AICc will return the quasi-likelihood analogue of the
information criteria requested and multiply the variance-covariance
matrix of the estimates by this value (i.e., SE's are multiplied by
sqrt(c.hat)). This option is not supported for generalized
linear mixed models of the mer or merMod classes.
additional arguments passed to the function.
Marc J. Mazerolle
AICc computes one of the following four information criteria:
Akaike's information criterion (AIC, Akaike 1973), $$-2 * log-likelihood + 2 * K,$$ where the log-likelihood is the maximum log-likelihood of the model and K corresponds to the number of estimated parameters.
Second-order or small sample AIC (AICc, Sugiura 1978, Hurvich and Tsai 1989, 1991), $$-2 * log-likelihood + 2 * K * (n/(n - K - 1)),$$ where n is the sample size of the data set.
Quasi-likelihood AIC (QAIC, Burnham and Anderson 2002), $$QAIC =
\frac{-2 * log-likelihood}{c-hat} + 2 * K,$$ where c-hat is the
overdispersion parameter specified by the user with the argument
c.hat.
Quasi-likelihood AICc (QAICc, Burnham and Anderson 2002), $$QAIC = \frac{-2 * log-likelihood}{c-hat} + 2 * K * (n/(n - K - 1))$$.
Note that AIC and AICc values are meaningful to select among
gls or lme models fit by maximum likelihood. AIC and
AICc based on REML are valid to select among different models that
only differ in their random effects (Pinheiro and Bates 2000).
Akaike, H. (1973) Information theory as an extension of the maximum likelihood principle. In: Second International Symposium on Information Theory, pp. 267--281. Petrov, B.N., Csaki, F., Eds, Akademiai Kiado, Budapest.
Anderson, D. R. (2008) Model-based Inference in the Life Sciences: a primer on evidence. Springer: New York.
Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.
Burnham, K. P., Anderson, D. R. (2004) Multimodel inference: understanding AIC and BIC in model selection. Sociological Methods and Research 33, 261--304.
Dail, D., Madsen, L. (2011) Models for estimating abundance from repeated counts of an open population. Biometrics 67, 577--587.
Hurvich, C. M., Tsai, C.-L. (1989) Regression and time series model selection in small samples. Biometrika 76, 297--307.
Hurvich, C. M., Tsai, C.-L. (1991) Bias of the corrected AIC criterion for underfitted regression and time series models. Biometrika 78, 499--509.
MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A., Langtimm, C. A. (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology 83, 2248--2255.
MacKenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G., Franklin, A. B. (2003) Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84, 2200--2207.
Pinheiro, J. C., Bates, D. M. (2000) Mixed-effect models in S and S-PLUS. Springer Verlag: New York.
Royle, J. A. (2004) N-mixture models for estimating population size from spatially replicated counts. Biometrics 60, 108--115.
Sugiura, N. (1978) Further analysis of the data by Akaike's information criterion and the finite corrections. Communications in Statistics: Theory and Methods A7, 13--26.
AICcCustom, aictab, confset,
importance, evidence, c_hat,
modavg, modavgShrink,
modavgPred, useBIC,
##cement data from Burnham and Anderson (2002, p. 101)
data(cement)
##run multiple regression - the global model in Table 3.2
glob.mod <- lm(y ~ x1 + x2 + x3 + x4, data = cement)
##compute AICc with full likelihood
AICc(glob.mod, return.K = FALSE)
##compute AIC with full likelihood
AICc(glob.mod, return.K = FALSE, second.ord = FALSE)
##note that Burnham and Anderson (2002) did not use full likelihood
##in Table 3.2 and that the MLE estimate of the variance was
##rounded to 2 digits after decimal point
##compute AICc for mixed model on Orthodont data set in Pinheiro and
##Bates (2000)
if (FALSE) {
require(nlme)
m1 <- lme(distance ~ age, random = ~1 | Subject, data = Orthodont,
method= "ML")
AICc(m1, return.K = FALSE)
}
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