# [R] aic calculus for glm models

Pierre-Andre Cornillon pac at Uhb.Fr
Wed Jun 26 16:50:06 CEST 2002

I am trying to know exactly the formulas used to calculate aic for glm
models.  In  glm.fit, the calculus of  aic is:
aic.model  <- aic(y, n,mu, weights, dev) + 2 * fit$rank where 2 * fit$rank is (may be am i wrong?) twice the  numbers of
parameters p and aic(y,  n, mu, weights, dev)
refers to  the function defined in  the family function  (which is for
Gamma family, for instance)

aic <- function(y, n, mu, wt, dev) {
n <- sum(wt)
disp <- dev/n
-2 * sum(dgamma(y, 1/disp, scale = mu * disp, log = TRUE) *
wt) + 2
}
Its seems to me that 1/disp = \phi is the ML estimator of \phi,
-2 * sum(dgamma(y,  1/disp, scale = mu  * disp, log = TRUE)  * wt) is
twice the log  likelihood of the observations at  the ML estimators mu
and disp. But what is the + 2 ?

The same thing occurs in the gaussian family: aic = function(y, n, mu,
wt, dev) sum(wt) * (log(dev/sum(wt) *  2 * pi) +  1) + 2
with dev/sum(wt) is the ML estimator of sigma^2, thus aic is
twice the likelihood function of the observations evaluated at ML
parameters + 2

I have probably not understood something but ??

Pierre-Andre Cornillon

PS may be some indications in  the help page of summary.glm to explain
that  the  dispersion  parameter  \phi  displayed  by  glm.summary  is
estimated as advocated  by Mc Cullagh & Nelder  (1989, p.296) could be
useful for some users
PS2 : I am using R Version 1.4.1

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