FAQ
Hi All:

I am really fascinated by the content and the depth of discussion of
this thread. This really exemplifies what I have come to love and
enjoy about the R user group - that it is not JUST an answering service
for getting help on programming issues, but also a forum for some
critical and deep thinking on fundamental statistical issues.

Kudos to the group!

Best,
Ravi.

----- Original Message -----
From: David Firth <d.firth@warwick.ac.uk>
Date: Monday, January 26, 2004 5:28 am
Subject: Re: [R] warning associated with Logistic Regression
On Sunday, Jan 25, 2004, at 18:06 Europe/London, (Ted Harding) wrote:
On 25-Jan-04 Guillem Chust wrote:
Hi All,

When I tried to do logistic regression (with high maximum
number of
iterations) I got the following warning message

Warning message:
fitted probabilities numerically 0 or 1 occurred in: (if
(is.empty.model(mt)) glm.fit.null else glm.fit)(x = X, y = Y,

As I checked from the Archive R-Help mails, it seems that this
happens>> when the dataset exhibits complete separation.
This is so. Indeed, there is a sense in which you are experiencing
unusually good fortune, since for values of your predictors in one
region you are perfectly predicting the 0s in your reponse, and for
values in another region your a perfectly predicting the 1s. What
better could you hope for?

However, you would respond that this is not realistic: your
variables> are not (in real life) such that P(Y=1|X=x) is ever
exactly 1 or
exactly 0, so this perfect prediction is not realistic.

In that case, you are somewhat stuck. The plain fact is that your
data (in particular the way the values of the X variables are
distributed)
are not adequate to tell you what is happening.

There may be manipulative tricks (like penalised regression) which
would inhibit the logistic regression from going all the way to a
perfect fit; but, then, how would you know how far to let it go
(because it will certainly go as far in that direction as you allow
it to).

The key parameter in this situation the dispersion parameter (sigma
in the usual notation). When you get perfect fit in a "completely
separated" situation, this corresponds to sigma=0. If you don't like
this, then there must be reasons why you want sigma>0 and this may
imply that you have reasons for wanting sigma to be at least s0
(say),> or, if you are prepared to be Bayesian about it, you may
be satisfied
that there is a prior distribution for sigma which would not allow
sigma=0, and would attach high probability to a range of sigma
values> which you condisder to be realistic.
Unless you have a fairly firm idea of what sort of values sigma is
likely to havem then you are indeed stuck because you have no reason
to prefer one positive value of sigma to a different positive value
of sigma. In that case you cannot really object if the logistic
regression tries to make it as small as possible!
This seems arguable. Accepting that we are talking about point
estimation (the desirability of which is of course open to
question!!),
then old-fashioned criteria like bias, variance and mean squared
error
can be used as a guide. For example, we might desire to use an
estimation method for which the MSE of the estimated logistic
regression coefficients (suitably standardized) is as small as
possible; or some other such thing.

The simplest case is estimation of log(pi/(1-pi)) given an
observation
r from binomial(n,pi). Suppose we find that r=n -- what then can
we
say about pi? Clearly not much if n is small, rather more if n is
large. Better in terms of MSE than the MLE (whose MSE is
infinite) is
to use log(p/(1-p)), with p = (r+0.5)/(n+1). See for example Cox
&
Snell's book on binary data. This corresponds to penalizing the
likelihood by the Jeffreys prior, a penalty function which has
good
frequentist properties also in the more general logistic
regression
context. References given in the brlr package give the theory and
some
empirical evidence. The logistf package, also on CRAN, is another
implementation.

I do not mean to imply that the Jeffreys-prior penalty will be the
right thing for all applications -- it will not. (eg if you
really do
have prior information, it would be better to use it.)

In general I agree wholeheartedly that it is best to get
more/better
data!
In the absence of such reasons,
(cut)

All good wishes,
David

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