% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/simplexreg_ic.R
\name{ic}
\alias{ic}
\title{Information Criteria for Simplex Regression}
\usage{
ic(...)
}
\arguments{
\item{...}{One or more objects of class \code{"simplexregression"} to be
evaluated.}
}
\value{
When a single model is provided, a named numeric vector with
components \code{AIC}, \code{AICc}, \code{BIC}, \code{BICc}, \code{HQIC},
and \code{HQICc}. When multiple models are provided, a data frame with one
row per model.
}
\description{
Extracts the Akaike information criterion (AIC), the Schwarz
(Bayesian) information criterion (BIC), and the Hannan-Quinn criterion
(HQIC), along with their finite-sample corrections AICc, BICc, and HQICc follow
Bayer and Cribari-Neto (2017).
}
\details{
Let \eqn{\ell(\widehat{\boldsymbol\beta}, \widehat{\boldsymbol\gamma})}
denote the maximized log-likelihood of the fitted model and \eqn{k} the
total number of estimated parameters. The AIC (Akaike, 1973) is
\deqn{AIC = -2\ell(\widehat{\boldsymbol\beta}, \widehat{\boldsymbol\gamma})
+ 2k.}
Sugiura (1978) proposed an unbiased estimator of the Kullback-Leibler
distance in linear regressions, later generalized by Hurvich and Tsai
(1989) to nonlinear regressions and autoregressive models. The resulting
finite-sample-corrected AIC is
\deqn{AICc = -2\ell(\widehat{\boldsymbol\beta}, \widehat{\boldsymbol\gamma})
+ \frac{2nk}{n-k-1}.}

The Schwarz (1978) information criterion, also known as the Bayesian
information criterion (BIC), is a consistent criterion given by
\deqn{BIC = -2\ell(\widehat{\boldsymbol\beta}, \widehat{\boldsymbol\gamma})
+ k\log(n).}
McQuarrie (1999) derived a finite-sample-corrected version, which remains
consistent:
\deqn{BICc = -2\ell(\widehat{\boldsymbol\beta}, \widehat{\boldsymbol\gamma})
+ \frac{nk\log(n)}{n-k-1}.}

The Hannan and Quinn (1979) criterion, proposed for autoregressive model
selection, is
\deqn{HQIC = -2\ell(\widehat{\boldsymbol\beta}, \widehat{\boldsymbol\gamma})
+ 2k\log(\log(n)).}
McQuarrie and Tsai (1998) proposed a finite-sample-corrected variant:
\deqn{HQICc = -2\ell(\widehat{\boldsymbol\beta}, \widehat{\boldsymbol\gamma})
+ \frac{2nk\log(\log(n))}{n-k-1}.}

Smaller values indicate better model fit.
}
\examples{
data(ReadingSkills, package = "SimplexRegression")
fit1 <- simplexreg(accuracy ~ dyslexia * iq | dyslexia + iq + I(iq^2),
                   data = ReadingSkills)

# Single model
ic(fit1)

# Comparing multiple models
fit2 <- simplexreg(accuracy ~ dyslexia * iq | dyslexia + iq + I(iq^2),
                   data = ReadingSkills, link.mu = "loglog")
ic(fit1, fit2)

}
\references{
Akaike, H. (1973). Information theory and an extension of the maximum
likelihood principle. In \emph{Proceedings of the 2nd International
Symposium on Information Theory}, 267--281.

Akaike, H. (1978). A Bayesian analysis of the minimum AIC procedure.
\emph{Annals of the Institute of Statistical Mathematics}, \bold{30}(1),
9--14. \doi{10.1007/BF02480194}

Bayer, F. M. and Cribari-Neto, F. (2017). Model selection criteria in beta
regression with varying dispersion. \emph{Communications in Statistics -
Simulation and Computation}, \bold{46}(1), 729--746.
\doi{10.1080/03610918.2014.977918}

Hannan, E. J. and Quinn, B. G. (1979). The determination of the order of
an autoregression. \emph{Journal of the Royal Statistical Society, Series
B}, \bold{41}(2), 190--195. \doi{10.1111/j.2517-6161.1979.tb01072.x}

Hurvich, C. M. and Tsai, C.-L. (1989). Regression and time series model
selection in small samples. \emph{Biometrika}, \bold{76}(2), 297--307.
\doi{10.1093/biomet/76.2.297}

McQuarrie, A. D. R. (1999). A small-sample correction for the Schwarz SIC
model selection criterion. \emph{Statistics & Probability Letters},
\bold{44}(1), 79--86. \doi{10.1016/S0167-7152(98)00294-6}

McQuarrie, A. D. R. and Tsai, C.-L. (1998). \emph{Regression and Time
Series Model Selection}. Singapore: World Scientific.

Schwarz, G. (1978). Estimating the dimension of a model. \emph{The Annals
of Statistics}, \bold{6}(2), 461--464. \doi{10.1214/aos/1176344136}

Sugiura, N. (1978). Further analysis of the data by Akaike's information
criterion and the finite corrections. \emph{Communications in Statistics -
Theory and Methods}, \bold{7}(1), 13--26.
\doi{10.1080/03610927808827599}
}
\seealso{
\code{\link{r2}}, \code{\link{press}}, \code{\link{ic.penalized}}
}
