Hi all,

 

In preparing some pedagogy, my coauthor and I came across the following
in Stata's stats FAQ:

 

"Generating the confidence intervals for the index [i.e., for XB] and
then converting them to probabilities to get confidence intervals for
the predicted probabilities is better than estimating the standard error
of the predicted probabilities and then generating the confidence
intervals directly from that standard error. The distribution of the
predicted index is closer to normality than the predicted probability."

Title                Prediction confidence intervals after logistic
regression 

Author            Mark Inlow, StataCorp 

Date                April 1999; minor revisions May 2005 

http://www.stata.com/support/faqs/stat/prep.html

 

This struck us as problematic; given Jensen's inequality, it would seem,
if we are correct, that this would be accurate, if at all, only in near
p=0,.5,1, where the function p=F(XB) is/becomes
approximately/increasingly linear. An alternative, the delta method:

V(p) approx= [dF/dB]'V(B)[dF/dB] for levels,

V(p) approx= [dF(X1,*)/dB- dF(X0,*)/dB]'V(B)[ dF(X1,*)/dB- dF(X0,*)/dB]
for differences,

V(dp/dx)= [d(dF/dx)/dB]'V(B)[ d(dF/dx)/dB] for marginals,

seems more appropriate to us (and apparently to Greene also: pp. 674-76
of 5th ed.). But, then again, it, too, relies on a (Taylor series)
linear approximation, so maybe it's no more or less appropriate than the
easier tack suggested by Inlow of calculating V(XB) and c.i.'s for that,
and transforming those results to p, delta(p), or d(p). (The
Taylor-series linear approximation aspect of the delta method, for its
part, would seem to us least problematic, if it's problematic at all,
for the last and first of these.)

 

Anyone have some thoughts or definitive references on the matter? Are we
right to doubt the quality of answers derived from the simpler
V(XB)-first-then-plug-and-chug method? (Yes, we know you can just
simulate any of these instead; we want some valid analytics also
though.)

 

Thanks!

Rob

 

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Robert (Rob) J. Franzese, Jr.                  US Mail:   (ISR Room
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The University of Michigan                       Ann Arbor, MI
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Research Assoc. Prof.                        TeleComm:
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http://www-personal.umich.edu/~franzese

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