StATS: Wikipedia entry on Binomial Confidence Interval (August 21, 2006). Category: Wiki pages

In reviewing resources about the Clopper Pearson confidence interval, I noticed a Wikipedia entry, Binomial Confidence Interval, that was tagged as needing attention from an expert on the subject.

{{expert}}

A '''Binomial Confidence Interval''' occurs in the [[Binomial]] model, in which an experiment with two outcomes, each occurring with fixed but unknown probability, (e.g. a coin flip) is repeated multiple times and the true probability of each event estimated from the proportion that occurs during the experiment. The [[confidence interval]] in this case is an interval that contains the probability of one of the two outcomes occurring (e.g. getting a heads) with a specified level of confidence. If the experiment is repeated "a large number of times" and the proportion estimated each time and a (for example) 95% confidence interval obtained each time, then approximately 95% of all the intervals obtained will contain the true proportion (with the approximation improving as the number of experiments increases). A confidence interval of this type is a product of the [[frequentist]] view of [[statistical inference]]. [[Frequentist|
Frequentism]] is one of several competing ways of explaining how probabilities should be interpreted.)

Confidence intervals have historically been estimated using a normal approximation, relying on the fact, resulting from the [[Central Limit Theorem]], that for a "large number" (50+ and often 30+) of trials, the distribution of a B(N; p) binomial experiment (with N events and a true proabability p of a given event occurring) behaves very similarly to a normal distributin N(np; np[1-p]) with mean np and variance np(1-p).

However, with increased computing power, it has become common to provide exact results based directly on the Binomial distribution. It is particularly advisable to do so when the value of np is small (less than 5 or 10), which can occur if there is a possibility of ending up with a small number of trials or a small true probability p (or both).

An early and very common method for calculating exact binomial confidence intervals is the method of Clopper and Pearson. This method calculates two half-intervals separately. For a $1-\alpha$ condidence interval, it requires each side to individually exceed $\alpha/2$. The lower and upper endpoints are the solutions to the equations.

*** Note: In progress, equations to be filled in

Clopper and Pearson confidence intervals have been criticized for a deficiency that arises from the discrete nature of the binomial. It is not possible to actually realize every possible proportion with a given sample size. (For example, with 5 coins, one can never get an even number of heads, so one can never get proportion p=0.5. One can get .4 and .6, but nothing in between). It is similarly generally not possible to get a confidence interval that exactly fits with a given sample. Clopper and pearson stretch their confidence intervals, in both directions separately, to get the next available discrete value. Because of the separate, individual adjustment to both sides, this can result in wider intervals than could be obtained with a single adjustment that considered both sides jointly. In addition, the resolving precision of these confidence intervals is not [[monotone function|monotone]]. Because of the way Clopper and Pearson confidence intervals adust for binomial discreteness, a slightly larger number of trials can often result in narrower confidence intervals than a smaller number of trials. Statisticians would like resolving precision to be monotone, that is, to increase (or at least stay the same) when sample size increases. Lack of monotonicity can make inference difficult.

In an effort to overcome these difficulties, other methods of calculating binomial confidence intervals have been devised. Two such mehods are the Wilson score method and the Wald method.

*** Note: In progress, to be continued

Some methods for calculating the [[confidence interval]]s for [[binomial proportion]]s are described by Alan Agresti and Brent A. Coull in their paper "Approximate is Better than 'Exact' for Interval Estimation of Binomial Proportions" published in The American Statistician (1998). The paper reports on [[coverage probability|coverage probabilities]] and interval widths for various methods.

The authors recommend the use of the '''Wilson score interval'''. The coverage probability is closest to the nominal level for all values of the population proportion. Furthermore, the interval widths are smaller than the Wald large sample intervals for proportions near .5 and smaller than the exact intervals for proportions near 0 and 1. The Wilson’s dcore interval is a compromise between the exact methods and the Wald large sample, normal approximation. Wilson uses an estimate of the standard error based on the [[null hypothesis]] rather than the [[statistical sample|sample]] estimate of the population proportion. The method effectively shrinks the sample estimate towards 0.5.

The formula for the Wilson score method of confidence intervals is

:$\frac{{\left( {\hat p + \frac{{z_{\frac{\alpha } {2}}^2 }} {{2n}} \pm z_{\frac{\alpha } {2}} \sqrt {\frac{{\left[ {\hat p\left( {1 - \hat p} \right) + \frac{{z_{\frac{\alpha } {2}}^2 }} {{4n}}} \right]}} {n}} } \right)}} {{\left( {1 + \frac{{z_{\frac{\alpha } {2}}^2 }} {n}} \right)}}$

==References==
* Agresti, A. ''Categorical Data Analysis: Second Edition''. John Wiley & Sons, 2002.
* Clopper, C. and Pearson, S. The use of confidence or fiducial limits illustrated in the case of the binomial. ''Biometrika'' 26:404-413, 1934.
* Fleiss, J., Levin, B., and Paik, M. ''Statistical Methods for Rates and Proportions: Third Edition.'' John Wiley & Sons, 2003.

[[Category:Statistical theory]]

The material enclosed inside double brackets (e.g., [[Binomial]]) will ink to the appropriate Wikipedia entry if it is available.

Note that the $section produces the following image: It uses a subset of TeX as described at For example, if you wanted to write the traditional confidence interval that uses a normal approximation to the binomial, it would look like this [itex]\hat p \pm z_{\alpha /2}$
$\sqrt{ \frac{\hat p \left ( 1- \hat p \right )}{n}}$

and would produce the following image

Actually, it produces two images side by side. The advantage of splitting an equation into two (or more) parts are that

• the equation is easier to proofread,
• if an error occurs, you can localize the error more quickly, and
• if you need a line break, the equation will split where you want it to.

You can experiment with editing in the Wikipedia sandbox

If I get some time, I might volunteer to provide the "expert" revision of this page.