RE: GLMs: show me the model!

From: Geoff Jones <G.Jones_at_massey.ac.nz>
Date: Fri, 20 Feb 2009 14:46:26 +1300

I've been waiting for someone to say something about quasilikelihood and
generalized estimating equations. Let me try.

Isn't a GEE a weighted sum of the additive "residuals", Y - Model? So this
way of looking at the model is actually quite useful. Geoff

At 10:49 a.m. 20/02/2009 +1000, Sama Low Choy wrote:
>Patrick,
>
>This debate has been very illuminating. Thank you Patrick for posing the
question.
>
>Bill's seemingly obvious comments are quite powerful - on writing down the
model so that the mean and variance are embedded within the key model
statement about the assumed distribution, including the (inverse) "link" to
covariates. Within a Bayesian context, it is perhaps even more important to
be aware of this way of "writing" down the model, as implied by Murray's
reference to Gelman & Hill's book.
>
>Thought I would just add a few of comments/queries that may be of interest
from a "pedagogical" standpoint (blame John Marriott for explaining this
word to us recently!)
>
>ERRORS AND LEAST SQUARES
>1. The way of writing down the Normal model with additive errors seems to
have been motivated by early least squares (including partial LS, weighted
LS) approaches to estimation. To do LS, you need the errors explictly e_i =
Y_i - X_i*beta, so that you can minimize the LS criterion on the errors min
sum_i e_i^2.
>
>When undertaking estimation based on the likelihood (eg maximum
likelihood), the distributional way of writing down the model becames more
useful. This includes Bayesian approaches where the likelihood is required
- either for a computational approach (eg in WinBUGS where GLMs can be
specified simply by specifying the likelihood, priors, and link to
covariates) or for an analytic approach where the marginal posterior
distributions of parameters are written down explicitly.
>
>ERRORS AND PARAMETERIZATION
>2. The parameterization of the model does not always lead to clear
separation of errors (ie to provide an additive or multiplicative
expression), let alone permit separation of the mean from the variance. I
encourage you to investigate alternative parameterisations of GLMs or other
non-Gaussian models where the covariates are related to the distribution of
the response (eg Beta regression).
>
>For example, a Beta regression may be parameterized so that the covariates
are linked to the mean or even perhaps the mode; the remaining parameter
(related to *both* the variance and the mean) can be expressed as the
variance, the "effective sample size" or as correlation.
>
>MODEL FORMULATION FOR ESTIMATION
>3. More generally, the "way of writing down the model" can be very closely
linked to the method of estimation; this seems to be true for a range of
statistical models beyond GLMs. For example, consider the suite of
clustering techniques (see nice exposition in Chambers, Tibshirani and
Friedman's Statistical Learning text). In most cases the "model" is written
down in a form that is suited to the usual method of estimation. See
Christian Robert's recent efforts to re-express the Nearest Neighbour
clustering technique to a particular form of Finite Mixture Models.
>
>Another example is classification trees. To explain, the Recursive
Partitioning Algorithm implemented in RPART (an R library by Atkinson &
Therneau 1997) the model is written down in a way that is linked to the
estimation method. However for a Bayesian estimation approach, the full
likelihood model is required - see O'Leary et al (2008; Journal of Applied
Statistics, 3(1)) with citation of seminal references Chipman, George &
McCulloch (1998) and Denison, Mallick & Smith (1998).
>
>ESTIMATION VS MODEL
>4. When working with ecologists, it has often been confusing to me that
sometimes the model is quoted via the estimation method. One example is the
terminology used when referring to alternatives to logistic regression when
modelling habitat. See Elith et al (2006; Ecography, 29(2): 129-151). Here
the "maximum entropy" approach is mentioned which seems to refer to
moment-matching on the mean and variance by applying the ubiquitous entropy
model from statistical physics.
>
>
>Regards,
>
>Dr Samantha Low Choy
>Research Fellow
>(1) ARC Discovery Grant: Bayesian Priors
>(2) Epidemiological study into the long-term respiratory health effects of
ultrafine air pollutants from traffic emissions on schoolchildren.
>
>School of Mathematical Sciences
>Queensland University of Technology
>Rm 0311, O Block, Gardens Point Campus
>2 George St, Brisbane Q 4001
>Ph: (07) 3138 8314
>Fax: (07) 3138 2310
>Email: s.lowchoy_at_qut.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: Friday, 20 February 2009 8:59 AM
>To: anzstat_at_lists.uq.edu.au
>Subject: GLMs: show me the model!
>
>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
>
>
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>FOR INFORMATION ABOUT "ANZSTAT", INCLUDING UNSUBSCRIBING, PLEASE VISIT
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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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Received on Fri Feb 20 2009 - 11:50:37 EST

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