[R] glm

Jim Lindsey jlindsey at alpha.luc.ac.be
Wed Feb 2 08:49:50 CET 2000

> On Tue, 1 Feb 2000, Jim Lindsey wrote:
> > > 4. I don't think AIC is stricly correct. My understanding of AIC is that
> > > it is the log likelihood maximised over all the parameters, including
> > > the dispersion parameter. Now the problem is that the mle of the
> > > dispersion parameter for the gamma family (and the exponential family in
> > > general) is really awkward - I think it has to be found by interative
> > > methods, using the derivative of the gamma function - I see from a print
> > > of Gamma that you are using dev/n, but I don't think this is the mle....
> > 
> > Yes you are right. I chose this only as a reasonable approximation for the
> > gamma and inverse Gaussian distributions. If you want the exact AIC,
> > you can get it from the gnlr function in my gnlm library.
> Another way to get the MLE of the dispersion parameter from the glm output
> is to use the function gamma.shape.glm (by Bill Venables) in the MASS
> package.  Its help says
> Details:
>      A glm fit for a Gamma family correctly calculates the maximum
>      likelihood estimate of the mean parameters but provides only a
>      crude estimate of the dispersion parameter.  This function takes
>      the results of the glm fit and solves the maximum likelihood
>      equation for the reciprocal of the dispersion parameter, which is
>      usually called the shape (or exponent) parameter.
> and yes it is optimized over an expression using the digamma and trigamma
> functions, but the code is about 30 lines only including all the warnings
> and housekeeping. (Such things show the power of S over GLIM: it would be
> `really awkward' in GLIM.)
> Note, though, that McCullagh and Nelder argue hard against the MLE of
> dispersion in the gamma family.

That argument can be made when a precision interval is required for
the dispersion parameter (conditional or REML estimate). I do not
believe that it can be made for substituting the estimate into the
likelihood or AIC for comparing different models which is the case in
question here. Jim

> Brian
> -- 
> Brian D. Ripley,                  ripley at stats.ox.ac.uk
> Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
> University of Oxford,             Tel:  +44 1865 272861 (self)
> 1 South Parks Road,                     +44 1865 272860 (secr)
> Oxford OX1 3TG, UK                Fax:  +44 1865 272595

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