Date:         Wed, 22 Nov 2006 13:54:40 -0800
Reply-To:     Dale McLerran <stringplayer_2@YAHOO.COM>
Sender:       "SAS(r) Discussion" <SAS-L@LISTSERV.UGA.EDU>
From:         Dale McLerran <stringplayer_2@YAHOO.COM>
Subject:      Re: poisson regression
In-Reply-To:  <6716d5d0611201008ye28d757tfa6a681c60d747d8@mail.gmail.com>
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--- Jeff <zhujp98@GMAIL.COM> wrote:

> I have poisson regression model.
> I express it as
> Log(P(Y=y))=b0+b1x1+b2x2+b3x3;
>
> My question is this the right expression for poisson regression?
>
> and how to interpret b1?
> Thanks.
>
> Jeff
>

Jeff,

Your regression function is not a standard model, and for good
reason.  I would note that if you have positive slope values,
then for sufficiently large X1 (or X2 or X3), the value of
Log(p(Y=y)) may exceed 0.  In that case, the value of p(Y=y)>1.
That is not a very satisfactory model, is it?

For Poisson regression, one usually assumes that

    log(mu) = b0 + b1*X1 + ... + bk*Xk

or

    mu = exp(b0 + b1*X1 + ... + bk*Xk)

where mu is the expected value of the response.  The log link
shown above is the canonical form of the link function for
a Poisson response.  Note that mu>0 when we use the log link
function.  Since the r.v. Y takes on nonnegative values, it is
fitting that our estimate of the mean be constrained to be
nonnegative, and this is accomplished when we employ the log link.

Fitting the Poisson regression involves modelling the mean such
that the estimated density model provides the best possible match
to the observed response probability structure.  Exactly how this
is done is beyond what I have time (or space) to write here.
You might get further understanding by reading the details section
for the GENMOD procedure.

Dale


---------------------------------------
Dale McLerran
Fred Hutchinson Cancer Research Center
mailto: dmclerra@NO_SPAMfhcrc.org
Ph:  (206) 667-2926
Fax: (206) 667-5977
---------------------------------------



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