Re: st: Binomial regression

[Constantine Daskalakis <C_Daskalakis@mail.jci.tju.edu>]

Hey Marcello.

Shouldn't you be on vacation? :)

Certainly, you can recover the absolute risks for each covariate pattern
(or observation), but what about the risk DIFFERENCE?

The point is this:

If the Xs are additive on the log-odds scale, we can fit a logistic
regression and report a single OR for each X as a summary measure.
Additivity on the log-odds scale implies non-additivity on the
probability scale, so there is no single RD for X (depends on the actual 
value of X).

But suppose the Xs are (more) additive on the original probability scale
rather than the log-odds scale. Then, it would be best to report a
single RD for each X as a summary measure of effect (rather than a
single OR, which is not really appropriate in this situation). Now, from
a logistic regression without interactions, we get a single OR for each
X, but an infinite number of RDs for the same X. That's not very good.

On the other hand, we can retrieve a single summary RD from a
main-effects-only binomial regression model. And, by the way, a logistic
regression would need a bunch of interactions to have comparable fit to
this binomial regression, and would still not provide us with a single
summary RD.

There's also causal inference stuff regarding causal interpretations for
RD, but not for RR or OR. So, that's additional motivation for focusing 
on the RD.

Bottom line: The goal here (get summary RD) CANNOT be achieved via
logistic regression.

Finally, there's nothing about a "uniform distribution" here. It's just
a generalized linear model with the identity link -- a different model
for how the risk changes as a function of covariates (and the same
binomial error structure that logistic regression uses). There's no
substantive reason to prefer one link over another. Why would the risk
follow the logistic function rather than any other curve (including a
line)? The correct/appropriate link function will depend on the data at
hand. Sometimes it will be a line, sometimes a logistic, sometimes some
other unknown beast.

I think that we are getting to the point where logistic has become so
ingrained that many people think of it (unconsciously?) as "have
screwdriver, will always use screwdriver (with binary outcome)." The
logit is the canonical link function for the Bernoulli/binomial and
implementation of regression with the logit link is easier than anything
else. But that's just ease of programming and custom.

For history buffs, this goes back to Cornfield (Bull Int Statist Inst, 
1961) who 'tricked' programs designed for discriminant analysis to do 
logistic regression and to Walker & Duncan (Biometrics, 1967) who looked 
at the maximum likelihood approach. At the time, lack of computing 
resources made such things impractical for other types of regressions 
for binary outcomes. But that's half a century ago.

CD


On 8/2/2007 10:45 PM, Marcello Pagano wrote:

> Sorry to disagree with your first sentence, Constantine.
> Logistic regression stipulates a linear relationship of covariates with 
> the log of the odds of an event (not odds ratios). From this it is 
> straightforward to recover the probability (or risk, if you prefer that 
> label) of the event.
> Don't understand your aversion to logistic regression to achieve what 
> you want to achieve.
> If you don't like the shape of the logistic, then any other cdf will 
> provide you with a transformation to obey the constraints inherent in 
> modeling a probability.  The uniform distribution that you wish to use 
> has to be curtailed, as others have pointed out.
> m.p.
>
>
> Constantine Daskalakis wrote:
>
> > No argument about logistic regression. But that gives you odds ratios. 
> > What if you want risk differences instead?
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Constantine Daskalakis, ScD
Assistant Professor,
Thomas Jefferson University, Division of Biostatistics
   1015 Chestnut St., Suite M100, Philadelphia, PA 19107
   Tel: 215-955-5695
   Fax: 215-503-3804
   Email: c_daskalakis@mail.jci.tju.edu
   Webpage: http://www.jefferson.edu/clinpharm/biostatistics/



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