Confidence Interval Calculation for Binomial Proportions
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The most common method for calculating the confidence interval is sometimes called the Wald method, and is presented in nearly all statistics textbooks. It is so widely accepted and applied, that for many it is the only method they have used. For most others it is the technique of first choice. Careful study however reveals that it is flawed and inaccurate for a large range of n and p, to such a degree that it is ill-advised as a general method 1,2 . Because of this many statisticians have reverted to the exact Clopper-Pearson method, which is based on the exact binomial distribution, and not a large sample normal approximation (as is the Wald method). Studies have shown however that this confidence interval is very conservative, having coverage levels as high as 99% for a 95% CI, and requiring significantly larger sample sizes for the same level of precision 1,2,3 . An alternate method, called the Wilson Score method is often suggested as a compromise. It has been shown to be accurate for most parameter values and does not suffer from being over-conservative, having coverage levels closer to the nominal level of 95% for a 95% CI. In this discussion a brief review of the Wald, Wilson-Score, and exact Clopper Pearson methods of calculating confidence intervals for binomial proportions will be presented, focusing on differences between the Wald and Wilson Score methods. Sample size calculations for the Wald and Wilson Score methods will also be discussed. SAS programs for these formulas will be also presented and applied to a worked out example, which can be readily modified for other data. Finally the differences between the methods will be discussed in general.
[1] Jeff Sauro,et al. Estimating Completion Rates from Small Samples Using Binomial Confidence Intervals: Comparisons and Recommendations , 2005 .
[2] K. Krishnamoorthy,et al. Some Properties of the Exact and Score Methods for Binomial Proportion and Sample Size Calculation , 2007, Commun. Stat. Simul. Comput..
[3] S E Vollset,et al. Confidence intervals for a binomial proportion. , 1994, Statistics in medicine.
[4] A. Agresti,et al. Approximate is Better than “Exact” for Interval Estimation of Binomial Proportions , 1998 .