[R-sig-eco] Offsets in Poisson or Neg. Bin regression

[Ivailo]

On Wed, Jun 26, 2013 at 12:42 PM, Scott Foster <scott.foster at csiro.au> wrote:
> Hi again Ivailo,
>
> Yes, the `offset' and the covariate are the same thing.  Including them both
> simply alters the functional form of the linear predictor in your model.
> No, they are not collinear in the typical sense as there is only one
> parameter (linear form) between them -- the offset term does not have a
> parameter that will be estimated associated with it.  For example, with log(
> effort) added as a linear covariate the log-link GLM is
>
> log( E(y)) = offset + beta * log( effort) + other_stuff = log( effort) +
> beta * log( effort) + other_stuff = beta_1 * log( effort) + other_stuff
> where beta_1=1+beta.
>
> If you test that beta==0 (which is not beta_1) then you are testing that the
> effect of effect is purely scaling (as per nomenclature before).  This is
> the same as McCullagh and Nelder's testing to see if beta_1==1.  Thanks for
> the pointer to McCullagh and Nelder -- I didn't know that they suggested
> that.

Thanks a lot for the brilliant explanation, Scott! Now things make
sense to me, and I'm interested what the modeling strategy would be if
beta_1 turns out to be significantly <> 1. Would the option you
mention below be viable alternative in that case?

> My depiction of the effect of effort as f( effort) is to allow for the
> possibility that the effect of effort may be non-linear on the link scale.
> A simple example is when f(effort) is a low-order polynomial.  Departures
> from effort being a purely scaling term may extend beyond linearity.  One
> may even want to consider regression splines or even more flexible GAMs.
> Having said all this though, it is my practice to be quite conservative with
> including effort as anything but a scaling variable (offset).  It seems to
> me that there needs to be good reason before jumping to strong conclusions
> that may have no basis in the phenomenon under study.

I imagine that the fishing-net example you mentioned earlier could be
a case of a non-linear effect of effort -- wouldn't this warrant
modeling the effort as being non-linear on the link scale?

Cheers,
Ivailo
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