[R] AIC for quasipoisson link

[Ben Bolker]

Marc Schwartz <marc_schwartz <at> comcast.net> writes:

> 
> on 10/31/2008 01:07 PM Antonio.Gasparrini <at> lshtm.ac.uk wrote:

> > I'm trying to extract the AIC statistic from a GLM model 
>  >with quasipoisson link.
> > The formula I'm referring to is 
> >  
> > AIC = -2(maximum loglik) + 2df * phi
> >  
> > with phi the overdispersion parameter, as reported in:
> >  
> > Peng et al., Model choice in time series studies os air pollution and
mortality. J R Stat Soc A, 2006; 162:
> pag 190.
> >  
> I was under the impression that there is no log likelihood for quasi*
> family models, thus no AIC, which is why they are not calculated/printed
> in the glm() summary outputs.
> 

  Yes, but ... this is a matter of some disagreement.  

Long answer: The purist
position (hi Prof. Ripley) is that quasi-likelihood estimation
does not produce a likelihood and should not return one.  
A common position in applied statistics (I think starting with 
a paper by Lebreton, but I can't find the ref right now:
see refs below) is that dividing the log-likelihood of a regular
likelihood fit by the estimated scale (overdispersion) parameter
of the quasi- variant gives a "quasilikelihood" that can be
used to compute a quasi-AIC that can then be used in model
selection.

 Short answer: I think that if you fit the non-quasi version
of the model (ie. Poisson family in your case) and extract
the likelihood from it, then divide by the overdispersion
parameter estimated from the "quasi" variant, that should
give you what you want.

  By the way, the formula quoted above looks funny.
Shouldn't it be

 QAIC = -2(maximum loglik)/phi + 2df

?  The formula quoted above (phi times my
version) should give the same ordering, but
model weights and interpretations of QAIC
differences will be wrong.

 cheers
  Ben Bolker


Anderson, D. R., K. P. Burnham, and G. C. White. 1994. AIC model selection in
overdispersed capture-recapture data. Ecology 75, no. 6: 1780-1793.

Richards, Shane A. 2008. Dealing with overdispersed count data in applied
ecology. Journal of Applied Ecology 45: 218-227.
doi:10.1111/j.1365-2664.2007.01377.x.