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From |
Constantine Daskalakis <C_Daskalakis@mail.jci.tju.edu> |

To |
statalist@hsphsun2.harvard.edu |

Subject |
Re: st: Binomial confidence intervals |

Date |
Wed, 08 Sep 2004 12:35:56 -0400 |

At 08:03 AM 9/8/2004, Richard Williams wrote:

The controversy can be summarized like this:At 11:47 AM 9/8/2004 +0100, Paul Seed wrote:A few key quotes:Dear all, As I don't have access to a decent Stats library here, I tried to obtain the recommended paper (Brown, Cai, & DasGupta. Interval Estimation for a Binomial Proportion. Statistical Science, 2001, 16, pp. 101-133.) over the internet; but it is currently behind a "rolling firewall", until 2005.

p. 115 - "Based on this analysis, we recommend the Wilson or the equal-tailed Jeffreys prior interval for small n (n is less than or equal to 40). These two intervals are comparable in both absolute error and length for n is less than or equal to 40, and we believe that either could be used, depending on taste."

p. 115 - "For larger n, the Wilson, the Jeffreys and the AgrestiCoull intervals are all comparable, and the AgrestiCoull interval is the simplest to present....we recommend the AgrestiCoull interval for practical use when n is greater than or equal to 40. Even for small sample sizes, the easy-to-present AgrestiCoull interval is much preferable to the standard one."

p. 113 - "The ClopperPearson interval is wastefully conservative and is not a good choice for practical use."

A. Suppose I have a true probability of success P that I am trying to estimate with a sample of size N. If I draw 1 million samples and compute my interval, I want at least 95% of those to cover the true P. I want this to happen, irrespective of the true value of P and the sample size N (ie, 100% of the time). So, if you have a different situation with probability P* and sample size N*, you should also have coverage of 95% or better. In other words, this means that I want an interval with coverage AT LEAST 95% ALL THE TIME (for some situations, it will have more).

B. In contrast, I may want the 95% coverage to be "on average," across different situations. In other words, in some situation of P and N, I am willing to accept coverage less than 95%, while for some other situation P* and N*, I will have coverage better than 95%. All I want is my interval to give at least 95% coverage ON AVERAGE.

(A) leads us to conservative intervals such as Clopper-Pearson. They have to cover the truth 95% of the time no matter what the situation. Even a single situation where they fall to 94.9% is not acceptable. However, they need not be as conservative as Clopper-Pearson. The Blyth-Still-Casella is much better (ie, it stays at 95% or better no matter what, but does not become as conservative as Clopper-Pearson) and is very competitive with those advocated in Agresti & Coull and Brown et al.

(B) leads us to generally shorter intervals, but in some situations they fall below 95% (sometimes way below that). Brown et al. propose certain "corrections" to remove the most severe dips in coverage probability. Still, none of those intervals is guaranteed to have 95% coverage or better in

a particular situation. And the problem is we cannot be sure how much below 95% they can fall. All we know is that they will have 95% coverage on average.

I go with A. I think that B is a bit ad hoc and a dip below 95% may be acceptable to me but unacceptable to you. But either approach seems quite defensible.

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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

Email: c_daskalakis@mail.jci.tju.edu

Webpage: http://www.jefferson.edu/medicine/pharmacology/bio/

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**Follow-Ups**:**Re: st: Binomial confidence intervals***From:*Marcello Pagano <pagano@hsph.harvard.edu>

**References**:**Re: st: Binomial confidence intervals***From:*Richard Williams <Richard.A.Williams.5@nd.edu>

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