# RE: Generalized linear models

From: Rissa Ota <Rissa.Ota001_at_msd.govt.nz>
Date: Thu, 19 Feb 2009 09:13:40 +1300

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 someone already done this (i.e., additive vs multiplicative errors) for various families and link functions?

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
P.S. You can view my selected research and sign up for notification of new 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 families and link functions in R. Any comments, links, or references appreciated. TIA.
--
-----
Patrick Cordue
Director
Innovative Solutions Ltd
www.isl-solutions.co.nz
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