From: Patrick Cordue <patrick.cordue_at_isl-solutions.co.nz>

Date: Fri, 20 Feb 2009 11:59:32 +1300

Date: Fri, 20 Feb 2009 11:59:32 +1300

I asked a question on GLMs a couple of days ago. In essence I was asking

"what is the model - please write it down - you, know, like for a linear

model: Y = a + bx + e, where e ~N(0,s^2) - can't we do that for a GLM?"

I come from a modelling background where the first step is to "write down

the model"; the second step is to look for tools which will provide

estimates of the unknown parameters; (I am assuming we already have a data

set). If my model is a GLM, then I can just use glm() in R. So, I wanted to

know the form of the GLM models for different families and link functions.

In particular, which implied simple additive errors (Y = mu + e) and which

implied simple multiplicative errors (Y = mu * e)?

(where mu = E(Y))

The answer provided by Murray Jorgensen is correct:

"In glms there is no simple characterisation of how the

systematic and random parts of the model combine to give you the data

(other than the definition of the glm, of course)."

Clearly for discrete distributions, it makes no sense to look for a

"building block" error e which can be added/multiplied to/by the expectation

to provide the response variable. My question was aimed at continuous

distributions.

Murray Smith (from NIWA) provided some useful comments (see below), which, I

think, get to the heart of my question.

However, I deduced the following results from first principles:

For the Gaussian family, Y = mu + e where e ~ N(0, s^2) (and E(Y) = mu =

m(eta) where eta is the linear combination of the explanatory/stimulus

variables, and m^-1 is the link function) is a GLM. I take this to imply

that when one fits a model using glm() with a Gaussian family and any link,

that the implied error structure is additive.

For the Gamma family, Y = mu * e where e ~ Gamma(k, 1/k) is a GLM. I take

this to imply that when one fits a model using glm() with a Gamma family and

any link, that the implied error structure is multiplicative.

For the inverse Gaussian family the implied model does not have a simple

additive or multiplicative error structure (someone might know how to write

down the model in this case - but not me).

Thanks to everyone who provided comments and references.

--------------------------------------

Murray H. Smith wrote:

"In most GLMs the error is neither multiplicative nor additive. Parameterize

the 1-parameter error family by the mean (fixing any dispersion or shape

parameters, which is what pure GLM is with the added constraint that the

error distribution belongs to a 1-parameter exponential family).

We can only write

y ~ mu + e or y ~ mu*e

for e not depending on mu, if mu is a location or scale parameter for the

error family. I.e.

y ~ f( y;mu) where f(y;mu) = f(y - mu; mu =0)

or

y ~ f( y;mu) where f(y;mu) =1/mu* f(y/mu; mu =1)

The variance function V(mu), the variance expressed as a function of the

mean, must be constant for an additive error and proportional to mu^2 for

multiplicative."

-- ----- Patrick Cordue Director Innovative Solutions Ltd www.isl-solutions.co.nz ---- FOR INFORMATION ABOUT "ANZSTAT", INCLUDING UNSUBSCRIBING, PLEASE VISIT http://www.maths.uq.edu.au/anzstat/Received on Fri Feb 20 2009 - 08:59:36 EST

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