Date:         Thu, 11 Mar 2004 06:27:53 -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: zero truncated poisson regression
Comments: To: Andrew Haywood <andrewHa@SF.NSW.GOV.AU>
Content-Type: text/plain; charset=us-ascii

Andrew,

I was mulling over my response from yesterday and realized that I
kind of got ahead of myself at one point.  I indicated that

> The probability function for the zero-truncated Poisson distribution
> is
>
>   p(y) = exp(-Xbeta)*(Xbeta**y)/(y!) /
>         (1 - exp(-Xbeta)*(Xbeta**0)*(0!))
>        = exp(-Xbeta)*(Xbeta**y)/(y!) / (1 - exp(-Xbeta))
>        = -Xbeta + y*log(Xbeta) - log(y!) - log(1 - exp(-Xbeta))

This last line is incorrect, of course.  I was anticipating the
log-likelihood function which the nlmixed procedure would solve.
Thus, I should have written:


    p(y) = exp(-Xbeta)*(Xbeta**y)/(y!) /
          (1 - exp(-Xbeta)*(Xbeta**0)*(0!))
         = exp(-Xbeta)*(Xbeta**y)/(y!) / (1 - exp(-Xbeta))

  and the log-likelihood function is

     ll(y) = -Xbeta + y*log(Xbeta) - log(y!) - log(1 - exp(-Xbeta))


The remainder of the post should be OK.

> As you suggest, nlmixed allows one to solve the likelihood function.
>
>   proc nlmixed data=mydata;
>     Xbeta = beta0 + x1*beta1 + x2*beta2 + ... + x<k>*beta<k>;
>     loglike = -Xbeta + y*log(Xbeta) - lgamma(y+1) -
>               log(1 - exp(-Xbeta));
>     model y ~ general(loglike);
>   run;

Just felt that I should set the record straight.

Dale


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

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