R help - Jul 2008 - statistics question about a statement in julian faraway's "extending the linear model with R" text

In Julian Faraway's text on pgs 117-119, he gives a very nice, pretty 
simple description of how a glm can be thought of as linear model
with non constant variance. I just didn't understand one of his 
statements  on the top of 118. To quote :

"We can use a similar idea to fit a GLM. Roughly speaking, we want to 
regress g(y) on X with weights inversely proportional
to var(g(y). However, g(y) might not make sense in some cases - for 
example in the binomial GLM. So we linearize g(y)
as follows: Let eta = g(mu) and mu = E(Y). Now do a one step expanation 
, blah, blah, blah.

Could someone explain ( briefly is fine ) what he means by g(y) might 
not make sense in some cases - for example in the binomial
GLM ?

Thanks.

For the binomial the standard link function is the logit:

g(y) = log( y/(1-y) )

In the binomial glm model the observed y values are 0, or 1 which give g(0) =
-Inf and g(1) = Inf.  Switching to g(mu) with  0 < mu < 1 results in
finite values which are much easier for the computer to work with.

Hope this helps,

--
Gregory (Greg) L. Snow Ph.D.
Statistical Data Center
Intermountain Healthcare
greg.snow at imail.org
(801) 408-8111


> -----Original Message-----
> From: r-help-bounces at r-project.org
> [mailto:r-help-bounces at r-project.org] On Behalf Of
> markleeds at verizon.net
> Sent: Monday, July 14, 2008 2:48 PM
> To: r-help at stat.math.ethz.ch
> Subject: [R] statistics question about a statement in julian
> faraway's "extending the linear model with R" text
>
> In Julian Faraway's text on pgs 117-119, he gives a very
> nice, pretty simple description of how a glm can be thought
> of as linear model with non constant variance. I just didn't
> understand one of his statements  on the top of 118. To quote :
>
> "We can use a similar idea to fit a GLM. Roughly speaking, we
> want to regress g(y) on X with weights inversely proportional
> to var(g(y). However, g(y) might not make sense in some cases
> - for example in the binomial GLM. So we linearize g(y) as
> follows: Let eta = g(mu) and mu = E(Y). Now do a one step
> expanation , blah, blah, blah.
>
> Could someone explain ( briefly is fine ) what he means by
> g(y) might not make sense in some cases - for example in the
> binomial GLM ?
>
> Thanks.
>
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
> http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>

markleeds at verizon.net wrote:
> In Julian Faraway's text on pgs 117-119, he gives a very nice, pretty 
> simple description of how a glm can be thought of as linear model
> with non constant variance. I just didn't understand one of his 
> statements  on the top of 118. To quote :
>
> "We can use a similar idea to fit a GLM. Roughly speaking, we want to 
> regress g(y) on X with weights inversely proportional
> to var(g(y). However, g(y) might not make sense in some cases - for 
> example in the binomial GLM. So we linearize g(y)
> as follows: Let eta = g(mu) and mu = E(Y). Now do a one step expanation 
> , blah, blah, blah.
>
> Could someone explain ( briefly is fine ) what he means by g(y) might 
> not make sense in some cases - for example in the binomial
> GLM ?
>   
I don't know that text, but I'd guess he's talking about the fact
that
the expected value of a binomial must lie between 0 and N (or the 
expected value of X/N, where
X is binomial from N trials, must lie between 0 and 1).

Similarly, the expected value of a gamma or Poisson must be positive, etc.

Duncan Murdoch

markleeds at verizon.net wrote:
> In Julian Faraway's text on pgs 117-119, he gives a very nice, pretty 
> simple description of how a glm can be thought of as linear model
> with non constant variance. I just didn't understand one of his 
> statements  on the top of 118. To quote :
>
> "We can use a similar idea to fit a GLM. Roughly speaking, we want to 
> regress g(y) on X with weights inversely proportional
> to var(g(y). However, g(y) might not make sense in some cases - for 
> example in the binomial GLM. So we linearize g(y)
> as follows: Let eta = g(mu) and mu = E(Y). Now do a one step 
> expanation , blah, blah, blah.
>
> Could someone explain ( briefly is fine ) what he means by g(y) might 
> not make sense in some cases - for example in the binomial
> GLM ?
>
Note that he does say "roughly speaking". The intention is presumably 
that if y is a vector of proportions and g is the logit function, 
proportions can be zero or one, but then their logits would be minus or 
plus infinity. (However, that's not the only thing that goes wrong; the 
model for g(E(Y)) is linear, the expression for E(g(y)) in general is not.)

-- 
   O__  ---- Peter Dalgaard             ?ster Farimagsgade 5, Entr.B
  c/ /'_ --- Dept. of Biostatistics     PO Box 2099, 1014 Cph. K
 (*) \(*) -- University of Copenhagen   Denmark      Ph:  (+45) 35327918
~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk)              FAX: (+45) 35327907