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Re: st: Binomial confidence intervals

From   Constantine Daskalakis <>
Subject   Re: st: Binomial confidence intervals
Date   Tue, 07 Sep 2004 14:41:24 -0400

At 02:17 PM 9/7/2004, Roberto G. Gutierrez, StataCorp wrote:

The exact interval used by -ci, binomial- is the Clopper-Pearson interval,
but you must realize that "exact" is a bit of a misnomer.  It is exact in the
sense that it uses the binomial distribution as the basis of the calculation.
However, the binomial distribution is a discrete distribution and as such its
cumulative probabilities will have discrete jumps, and thus you'll be hard
pressed to get (say) exactly 95% coverage.
I do not think this is correct. For the CI, it is the parameter space, not the sample space, that matters (and the former is continuous). In other words, if we have k successes out of N trials, we are looking for limits {p_l, p_u}, such that

Pr [K <= k | p_l] = a/2


Pr [K >= k | p_u] = a/2

In general, there exist such limits that correspond to tail probabilities of (exactly) a/2. The fact that the sample space is highly discrete (when N is small) has nothing to do with it. The only exception is when the observed number of successes is either 0 or N; in that case, one limit is on the boundary of the parameter space (p_l=0 or p_u=1) and the corresponding tail probability on that side is exactly 0, not a/2 (as the manual correctly points out).

Or, am I missing something?

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Constantine Daskalakis, ScD
Assistant Professor,
Biostatistics Section, Thomas Jefferson University,
211 S. 9th St. #602, Philadelphia, PA 19107
Tel: 215-955-5695
Fax: 215-503-3804
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