Date: Tue, 14 Oct 2003 17:35:14 -0700
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: Negative Binomial Functions
In-Reply-To: <OF492C3EE9.36626CEC-ON88256DBF.007BDA57@epamail.epa.gov>
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David,
I don't think there is a logical way to connect the two
parameterizations. In one parameterizations, we have a
sequence of Bernoulli trials with constant probability of
success and we count the number of failures before the kth
success. Another parameterization proposes that there is
a Poisson response with subject specific mean following a
gamma distribution. For the Poisson distribution, we are
counting the number of objects over a fixed period of time.
Assuming that the Poisson response is a limiting form of
a binomial distribution, then we have a count of successes
in an infinite number of Bernoulli trials with subject
specific Bernoulli trial success probability.
These are just two of the ways in which the N.B. distribution
can arise. There are quite a few other ways. But the above
two illustrate situations in which we have two integer
valued parameters and one real valued parameter (failures,
successes, Bernoulli trial success probability) and also
a situation in which we have only one integer valued
parameter and two real valued parameters (object count,
mean for i-th subject, variance of the mean under a gamma
distribution).
I don't have time for any more right now. The density functions
can, of course, be written in mathematically similar ways which
is why they are both N.B.
Dale
--- "David L. Cassell" <cassell.david@EPAMAIL.EPA.GOV> wrote:
> Bruce Bradbury <BruceBrad@INAME.COM> wrote:
> > SAS has a function which returns the probability that a negative
> > binomial random variable X takes the value m (m integer >= 0).
> > This is given by pdf('negb',m,p,n) where 0<=p<=1 and n is an
> integer
> >0.
> >
> > In the literature (eg Winkelman, 1997, Econometric analysis of
> count
> > data) I have found an alternative description of the negative
> binomial
> > function. In this case P(X=m) = f(m,alpha,theta)
> > where alpha and theta are both real positive parameters.
> >
> > Is this the same function? The algebriac expression is quite
> > different, and the SAS formulation (which also appears often in the
> > literature) restricts one of the parameters to be an integer.
> >
> > For my purposes I would prefer to work with the continuous
> parameter
> > version. Is there a simple mapping between the two
> parameterisations?
> > Any pointers appreciated.
>
> This one caught my eye. I couldn't see a logical way to map the two
> parameterizations, since the negative binomial pdf is dealing with
> n successes in m independent trials, each with probability p. The
> formula
> for the pdf seems to have three independent variables.
>
> I checked. Winkelman 1997 (and Winkelman 1994, etc.) use a 'negative
> binomial regression function' which is *not* the same thing as the
> pdf
> of the negative binomial. They do have negative binomial
> distributions
> under the hood, though...
>
> So tell me (by writing back to SAS-L, please): how do you envision
> using
> the negative binomial with two continuous parameters, instead of 1
> continuous in [0,1] and two positive integers? If your goal is to
> use Winkelman's structure, then you can look at PROC NLMIXED and go
> from there.
>
> HTH,
> David
> --
> David Cassell, CSC
> Cassell.David@epa.gov
> Senior computing specialist
> mathematical statistician
=====
---------------------------------------
Dale McLerran
Fred Hutchinson Cancer Research Center
mailto: dmclerra@fhcrc.org
Ph: (206) 667-2926
Fax: (206) 667-5977
---------------------------------------
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