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RE: st: Model for Poisson-shaped distribution but with non-count data


From   Cameron McIntosh <cnm100@hotmail.com>
To   STATA LIST <statalist@hsphsun2.harvard.edu>
Subject   RE: st: Model for Poisson-shaped distribution but with non-count data
Date   Tue, 6 Dec 2011 16:41:13 -0500

You might also consider a Bayesian multilevel model with a gamma prior: 

Griffin, J.E., & Brown, P.J. (2010). Inference with normal-gamma prior distributions in regression problems. Bayesian Analysis, 5(1), 171-188.http://ba.stat.cmu.edu/journal/2010/vol05/issue01/griffin2.pdf

Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models. Bayesian Analysis, 1(3), 515–533.http://www.stat.columbia.edu/~gelman/research/published/taumain.pdf

Lambert, P.C., Sutton, A.J., Burton, P.R., Abrams, K.R., & Jones, D.R. (2005). How vague is vague? A simulation study of the impact of theuse of vague prior distributions in MCMC using WinBUGS. Statistics in Medicine, 24, 2401–2428.http://www.yaroslavvb.com/papers/lambert-how.pdf

Browne W.J., & Draper, D. (2006). A comparison of Bayesian and likelihood-based methods for fitting multilevel models. Bayesian Analysis, 1(3), 473–514.http://ba.stat.cmu.edu/journal/2006/vol01/issue03/draper2.pdf

Cam

> Date: Tue, 6 Dec 2011 12:48:35 -0800
> Subject: Re: st: Model for Poisson-shaped distribution but with non-count data
> From: ogallupe@gmail.com
> To: statalist@hsphsun2.harvard.edu
> 
> Thank you for the input, Cam, Paul, Paul, Nick, David, and Bill.
> 
> You have given me some very good options to consider.
> 
> Regarding Cam's earlier question, the multimodality only surfaces when
> the DV is transformed using lnskew0. It is not an issue using the raw
> version.
> 
> All the best.
> 
> Owen
> 
> On Tue, Dec 6, 2011 at 9:34 AM, William Gould, StataCorp LP
> <wgould@stata.com> wrote:
> > David Hoaglin <dchoaglin@gmail.com>, in reference to the blog entry
> > "Use Poisson Rather Than Regress, Tell a Friend" at
> >
> >    http://blog.stata.com/2011/08/22/use-poisson-rather-than-regress-tell-a-friend/
> >
> >  wrote,
> >
> >> [..] I did not see any
> >> mention of the fact that the Poisson distribution is discrete.  In the
> >> limit (as the mean of the distribution becomes large), that matters
> >> less, but one would need to view the possible data values as discrete.
> >>
> >> Some of the equations in the blog are not quite correct.  For example,
> >> since Poisson regression is a form of generalized linear model, the
> >> linear predictor is fitted to log(E(y)), rather than to log(y).  The
> >> random component of the GLM is a Poisson distribution.
> >
> > I'm concerned that someone might interpret what David wrote to mean
> >
> >    1.  There may be practical problems using -poisson- to run
> >        log-linear regressions, depending on whether the LHS variable
> >        contains noninteger values.
> >
> >    2.  There may be theoretical problems using -poisson- to run
> >        log-linear regressions.
> >
> > Neither would be true.  My short-and-quick response is,
> >
> >    1.  -poisson- can handle non-discrete (non-integer) data values.
> >        Left-hand-side values do not have to be large to ammelorate any
> >        problem.
> >
> >    2.  The formulas in the blog are as intended and are correct.
> >
> > Let me explain.
> >
> >
> > Concerning #1, -poisson- does not round values when run on noninteger
> > data.  Instead, it gives the warning message "you are responsible for
> > interpreation of noncount dep. variable."
> >
> > An implication of that is that the objective function with non-integer
> > data may not be a true likelihood function.  Actually, I suspect that
> > it is, but that's irrelevant because we in the blog entry are doing M
> > estimation and I recommended you obtain standard errors using the
> > -vce(robust)- option.
> >
> > When -poisson- calculates the likelihood value associated with a
> > noninteger value, it does that using the standard formulas, but
> > substituting the Gamma function for factorial function.  That is
> > appropriate for M estimation.
> >
> > This generalization means that you can run -poisson- using a LHS
> > variable with noninteger values and there will be no problems.  All
> > the values, in fact, can even be less than 1!  Whether you run on y,
> > y/10, y/100, y/1000, ..., all that will change will be the intercept.
> >
> >
> > Concerning #2, the formulas written in the blog entry imply that
> > log(E(y)) = a + b*X.  It is true that I did write
> >
> >      y = exp(a + b*X + e)
> >
> > and that implies more than merely log(E(y)) = a + b*X.  I did that
> > because I was starting with the log-linear regression problem.
> > The purpose of the blog entry is to show that -poisson- could be
> > used as an alternative to linear regression on the ln(y).
> >
> > -- Bill
> > wgould@stata.com
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