# Re: st: Binomial confidence intervals

 From Constantine Daskalakis 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:
```
```At 11:47 AM 9/8/2004 +0100, Paul Seed wrote:
```
```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.
```
A few key quotes:

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."
The controversy can be summarized like this:

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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