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 Three methods for computing the CI of a proportion

There are many methods to compute the confidence interval of a proportion. Your goal is to have a 95% confidence intervals, but in fact the actual confidence level (for all methods) depends on the precise values for the numerator and denominator.

Prism offers three methods to compute the confidence interval of a proportion:

The so called "exact method" of Clopper and Pearson (1). This is the only method Prism 6 (and earlier) used. No matter what data you enter, this method always ensures that the actual confidence level is greater than the level you requested (usually 95%). But often the actual confidence level is a lot higher. On average, therefore, these intervals have a greater confidence level than you requested so are wider than they need to be.

The method of Wilson (2). With some data the actual confidence level is greater than what you requested, and for some data the actual confidence level is less. On the average, the actual confidence level equals the confidence level you requested.  Wilson's method is great except when the probability is very close to 0 or 1. Note that while some programs offer a variation of this method with a continuity correction, but Prism does not.

The hybrid Wilson/Brown method (3). In most cases this method uses the Wilson method with no modifications.  The exception is when the proportion is very close to 0.00 or 1.00. In these cases, Brown prescribes using a Poisson approximation instead. Prism uses this approximation for the lower confidence limit when the numerator (N) equals 1 or 2, and when N=3 and the denominator (D) exceeds 50. It  uses this approximation for the upper limit when  N=D-1 or N=D-2 (or N=D-3, and N > 50). Note that the approximation is not needed when N=0. In that case the lower confidence limit is 0.0 and the upper limit is computed by Wilson's method. Similarly, when N=D, the upper confidence limit equals 1.00 and the lower limit is computed by Wilson's method. Brown and colleagues (3) call this hybrid method the modified Wilson method, but this name can be ambiguous because other modifications of Wilson's method have been proposed.

We recommend the third (Wilson/Brown hybrid) method. Use one of the first two methods only if you need to be sure that Prism's results match those of another program.

## References

1.Clopper, C.; Pearson, E. S. (1934). "The use of confidence or fiducial limits illustrated in the case of the binomial". Biometrika 26: 404–413. doi:10.1093/biomet/26.4.404

2.Wilson, E. B. (1927). "Probable inference, the law of succession, and statistical inference". Journal of the American Statistical Association 22: 209–212. JSTOR 2276774.

3.Brown, L., Cai, T., & DasGupta, A. (2001). Interval Estimation for a Binomial Proportion. Statist. Sci, 16(2), 101–133.