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

[Scott Foster]

Hi Ivailo,

If the effort term is not just present in the model for the purpose of 
scaling the outcome random variable, then I think that it should just be 
treated as a regression-type problem.  All the questions your raised 
seem(?) to be standard in that setting too: Is the covariate acting 
linearly (on the link scale)?  Are any non-linearities (on the link 
scale) important enough to warrant using some curvi-linear or 
basis-expanded function of the effort variable?  And so on...

Yes, the fishing net example *may* be one where the (scaling) effort 
variable acts non-linearly.  I have not thought about this though. I 
typically use effort as a scaling factor only as I have a strong a 
priori belief that effort will be multiplicatively related to expected 
outcome (log offset with log-link).  I am sure that I will need to 
revise this belief sometime;-)

Scott

On 27/06/13 16:57, Ivailo wrote:
> 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
> --
> UBUNTU: a person is a person through other persons.
>
>

-- 
Scott Foster
CSIRO Mathematics, Informatics and Statistics
GPO Box 1538
Castray Esplanade
Hobart 7001
Tasmania
Australia

Phone:     (03) 6232 5178
Fax:       (03) 6232 5000
Email:     scott.foster at csiro.au