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Re: st: logistic tranformation, proportion variables


From   "Austin Nichols" <austinnichols@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: logistic tranformation, proportion variables
Date   Thu, 13 Dec 2007 22:12:43 -0500

Marcello--
Yes, I did mean to include the denominator in the first term--thanks
for the correction.  Just to reiterate the main point, if the
regression model
  logit(A/(A+B)) = Xb + e
seems to fit the data better than A/(A+B) = Xb + e, then
  ln(A) - ln(B) = Xb + e
is the same model, and has to fit just as well, as does the identical
  ln(A) = Xb + ln(B) + e
so it must be the case that
  ln(A) = Xb + aln(B) + e
must fit at least as well.  A similar argument applies to a
logit-transformed RHS var.  It may make sense to constrain the
elasticity of A w.r.t. B to be 1, i.e. constrain the regression
coefficient a to be 1, but not on the grounds that the residuals are
less heteroskedastic with a=1; it must be an appeal to theory, IMHO.

In the case where A can be zero (mentioned by Marck Butler as the
source of the problem), the nearly equivalent model
 A = exp[ Xb + aln(B) ]u
can be estimated by -poisson- (including the obs where A=0).  See also
the help file for -ivpois- (on SSC).

No help for the case where a RHS proportion is zero, though.

On Dec 13, 2007 3:32 PM, Marcello Pagano <pagano@biostat.harvard.edu> wrote:
> Surely you meant
>
> logit(a/(a+b))=ln(a/(a+b))-ln(b/(a+b)) = ln(a) - ln(b)
>
> m.p.
>
> On 12/13/2007 3:12 PM, Austin Nichols wrote:
> > Yes, thanks. Should have been:
> >
> > logit(a/(a+b)) = ln(a)-ln(1-a/(a+b)) = ln(a)-ln(b)
> >
> > Read "other component of denominator" for "denominator" e.g. "all
> > other bonds" in the example given.
> >
> > On Dec 13, 2007 2:51 PM, Steven Samuels <sjhsamuels@earthlink.net> wrote:
> >> Austin, Did you mean to say:
> >>
> >> logit(a/(a+b))=ln(a)-ln(b) ?
> >>
> >> -Steve
> >>
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