# RE: Generalized linear models

From: Geoff Jones <G.Jones_at_massey.ac.nz>
Date: Thu, 19 Feb 2009 11:05:07 +1300

So it's all really about how you define your residuals. You can define them
as additive or multiplicative, on the observational scale or the linear
predictor scale. Any or all of these may or may not be useful. Look at
survival analysis, where there are about six different kinds, all for
different purposes.

In some special situations it may be useful to consider the residuals as
multiplicative. You can even have both - see Rocke amd Lorenzato (1995),
Technometrics. Geoff

At 08:41 a.m. 19/02/2009 +1100, Ken Russell wrote:
>... and doesn't even seem plausible for discrete distributions. Take a
Binomial distribution, with n = 5 and p (modelled by the logistic link) =
0.48. Then E(Y) = 2.4, but the possible values of Y are 0, 1, ... , 5. If
you insist on Y = E(Y) + error, then the errors can be -2.4, -1.4, ... ,
2.6. Yes, you can use Binomial probabilities to say how likely each error
is, but the errors don't have a Binomial distribution.
>
>
>Regards,
>Ken Russell
>
>A/Prof Ken Russell Ph.D., A.Stat.
>School of Mathematics & Applied Statistics
>University of Wollongong NSW 2522
>AUSTRALIA
>Phone: + 61 2 4221 3815
>Fax: + 61 2 4221 4845
>Email: kgr_at_uow.edu.au
>________________________________________
>From: owner-anzstat_at_lists.uq.edu.au [owner-anzstat_at_lists.uq.edu.au] On
Behalf Of Patrick Cordue [patrick.cordue_at_isl-solutions.co.nz]
>Sent: Thursday, February 19, 2009 8:22 AM
>To: Rissa Ota
>Cc: anzstat_at_lists.uq.edu.au
>Subject: RE: Generalized linear models
>
>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
>
>--
>-----
>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
>
>--
>-----
>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
>
>
>----
>
>http://www.maths.uq.edu.au/anzstat/
>
>--
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>----
>
http://www.maths.uq.edu.au/anzstat/
>----
>
http://www.maths.uq.edu.au/anzstat/
>
Geoff Jones
Senior Lecturer in Statistics
Institute of Fundamental Sciences (PN322)
Te Kura Putaiao o Hangarau-a-Mohiotanga
AgHort A, College of Sciences, Massey University
Private Bag 11222, Palmerston North, New Zealand

Phone: +64-6-350 5799 x2468 FAX: +64-6-350 2261
E-mail: G.Jones_at_massey.ac.nz

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