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RE: st:Confidence interval of difference between two proportions and -csi-


From   "Dupont, William" <william.dupont@vanderbilt.edu>
To   <statalist@hsphsun2.harvard.edu>
Subject   RE: st:Confidence interval of difference between two proportions and -csi-
Date   Fri, 19 Mar 2004 09:08:39 -0600

Statalisters

I believe that there is some confusion about the meaning of exact
confidence intervals.  Confidence intervals are defined in two ways.
Let theta be a parameter and L  U be two statistics.  Then confidence
intervals are defined as follows:

Coverage definition:

(L, U) is a 95% confidence interval for theta if Pr[L < theta < U] =
0.95

Non-rejection definition:

A 95% confidence interval, (L, U), consists of all values of theta that
can not be rejected at the 5% significance level given the data.

These two definitions are equivalent for normally distributed data in
which the mean and variance are unrelated.  In epidemiology and other
disciplines we often work with statistics (e.g. odds ratios) in which
these definitions yield different intervals.  Exact confidence intervals
use the non-rejection definition.  When estimating odds ratios from 2x2
tables,  the total number of successes in both groups is close to being
an ancillary statistic in the sense that knowing this total tells us
nothing about the true odds ratio. The Conditionality Principle requires
that we condition our inferences on ancillary statistics.  It is for
this reason that we condition on the marginal totals of a 2x2 table when
making inferences about odds ratios.  

If you accept the conditionality argument then the usual exact
confidence interval is correctly derived from the hypergeometric
distribution.  It is an exact interval not because it uses the
hypergeometric distribution but because it complies with the
non-rejection definition given above.  It should be noted that when
these definitions disagree, the non-rejection confidence interval will
have a higher coverage probability than the analogous interval obtained
by the coverage definition.   In this sense, it is a more conservative
interval.

See Rothman and Greenland (1998) for further details.

Bill Dupont

-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Garry
Anderson
Sent: Friday, March 19, 2004 2:11 AM
To: statalist@hsphsun2.harvard.edu
Subject: Re: st:Confidence interval of difference between two
proportions and -csi-

...

>
>then the so-called "exact" confidence interval is generated. (Note,
>however, that this confidence interval is conservative, not exact. It
is 
>called "exact" because it uses the exact hypergeometric distribution to

>calculate conservative confidence limits.)
>
>I hope this helps.
>
>Roger
>
>
>--
>Roger Newson
>Lecturer in Medical Statistics
>Department of Public Health Sciences
>King's College London
>5th Floor, Capital House
>42 Weston Street
>London SE1 3QD
>United Kingdom
>
>Tel: 020 7848 6648 International +44 20 7848 6648
>Fax: 020 7848 6620 International +44 20 7848 6620
>   or 020 7848 6605 International +44 20 7848 6605
>Email: roger.newson@kcl.ac.uk
>Website: http://www.kcl-phs.org.uk/rogernewson
>
>Opinions expressed are those of the author, not the institution.
>
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