# RE: Generalized linear models

From: Patrick Cordue <patrick.cordue_at_isl-solutions.co.nz>
Date: Thu, 19 Feb 2009 10:22:34 +1300

Hi Rissa,

What you say is perhaps often assumed but it is not true for all families
and link functions. The formulation you gave requires E(e) = 0, which will
work for the Gaussian family but will fail for Gamma or Inv. Gaussian.

Regards
Patrick

```--
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Patrick Cordue
Director
Innovative Solutions Ltd
www.isl-solutions.co.nz
-----Original Message-----
From: Rissa Ota [mailto:Rissa.Ota001_at_msd.govt.nz]
Sent: Thursday, February 19, 2009 9:14 AM
To: 'Patrick Cordue'; Chris Lloyd
Cc: anzstat_at_lists.uq.edu.au
Subject: RE: Generalized linear models
Hi Patrick,
From what I understand, the model is
g(E(Y)) = a+bx, where g is the link function
so Y = g^(-1)(a+bx)+e, where g^(-1) is the inverse of link function
I imagine this is true for all link functions.
Cheers,
Rissa
-----Original Message-----
From: owner-anzstat_at_lists.uq.edu.au [mailto:owner-anzstat_at_lists.uq.edu.au]
On Behalf Of Patrick Cordue
Sent: Wednesday, 18 February 2009 7:28 p.m.
To: Chris Lloyd
Cc: anzstat_at_lists.uq.edu.au
Subject: RE: Generalized linear models
Hi Chris,
I was thinking that the family and the link were available, e.g., for a
particular dataset the "best model" is found to be Gaussian with a log link,
say log(E(Y)) = a + bx. My question is what more can be said about how the
error combines with "a +bx", i.e., is the glm compatible with a model: Y =
exp(a + bx) + e where e ~ N(0,s^2), and is it compatible with Y = exp(a + b)
* e where e ~ N(1, s^2). Now, I think I know the answer to that and I can
check other specific examples such as gamma with a log link etc. But, has
Regards
Patrick
--
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Patrick Cordue
Director
Innovative Solutions Ltd
www.isl-solutions.co.nz
-----Original Message-----
From: Chris Lloyd [mailto:C.Lloyd_at_mbs.edu]
Sent: Wednesday, February 18, 2009 5:41 PM
To: Patrick Cordue
Subject: RE: Generalized linear models
It is supposed to be linear on the link scale. If they do not specify the
link then it is not true that "The model is clearly described in terms of
variables and structure for the mean response."
Regards
Professor Chris J. Lloyd
Associate Dean of Research
Ph: 613 -9349-8228
work at: http://works.bepress.com/chris_lloyd/
Statistics Blog: http://blogs.mbs.edu/fishing-in-the-bay/
Personal Homepage: http://www.mbs.edu/home/lloyd/homepage/
-----Original Message-----
From: owner-anzstat_at_lists.uq.edu.au
[mailto:owner-anzstat_at_lists.uq.edu.au] On Behalf Of Patrick Cordue
Sent: Wednesday, 18 February 2009 3:23 PM
To: anzstat_at_lists.uq.edu.au
Subject: Generalized linear models
Hi All,
For any given model it is usually straightforward to check if it satisfies
the assumption of a GLM (and if it does, one can use glm() in R, for
example, with an appropriate distribution family and link function, to
obtain estimates of the coefficients, etc).
However, given a dataset, which is analyzed using glm(), one may arrive at a
"best model" which uses a particular family and link function (and a set of
explanatory variables). The model is clearly described in terms of variables
and structure for the mean response, and the distribution of the response
variables is explicit, but the "exact" form of the error structure is not
explicit (e.g., are the errors additive or multiplicative?, e.g., Y = a + bx
+ e, or Y = (a + bx) * e, both can have E(Y) = a + bx). I assume that
+ some
general results are available - can anyone point me to, for example, an
online list of implicit error structure given the currently implemented
appreciated. TIA.
--
-----
Patrick Cordue
Director
Innovative Solutions Ltd
www.isl-solutions.co.nz
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