```Date: Tue, 14 Oct 2003 17:35:14 -0700 Reply-To: Dale McLerran Sender: "SAS(r) Discussion" From: Dale McLerran Subject: Re: Negative Binomial Functions Comments: To: cassell.david@EPAMAIL.EPA.GOV In-Reply-To: Content-Type: text/plain; charset=us-ascii 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" wrote: > Bruce Bradbury 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 --------------------------------------- __________________________________ Do you Yahoo!? The New Yahoo! Shopping - with improved product search http://shopping.yahoo.com ```

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