Re: st: Binomial confidence intervals

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

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

and

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
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   Email: c_daskalakis@mail.jci.tju.edu
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