Confidence Intervals for the Binomial Proportion: A Comparison of Four Methods

This 
paper presents four methods of constructing the confidence interval for the 
proportion p of the binomial distribution. Evidence in the literature indicates the 
standard Wald confidence interval for the binomial proportion is inaccurate, 
especially for extreme values of p. 
Even for moderately large sample sizes, the coverage probabilities of the Wald 
confidence interval prove to be erratic for extreme values of p. Three alternative confidence 
intervals, namely, Wilson confidence interval, Clopper-Pearson interval, and 
likelihood interval, are 
compared to the Wald confidence interval on the basis of coverage probability 
and expected length by means of simulation.


Introduction
Estimation of a binomial proportion p is one of the most commonly encountered statistical problems, with important application in areas such as clinical medicine, business, politics and quality control. For instance, politicians are certainly interested in knowing what fraction of voters would favor them in the next election. The binomial data is obtained from a binomial experiment which consists of a fixed number n of independent Bernoulli trials, each of which can result in either a success or a failure. The success probability p is assumed fixed. The binomial probability distribution is used to model the total number x of success resulting from the Binomial experiment. Once data are available, then information about p can be summarized by the likelihood function and on the basis of this summary, a point estimate for the Binomial proportion, denoted by p is obtained by the method of maximum likelihood as x p n =  . A number of two-sided confidence intervals for p have been proposed by several authors. The Wald method is the most commonly used technique since it is based normal approximation to the binomial distribution. However, the approximation is inaccurate whenever the sample size is small (n < 30) or when the proportion p is close to zero or one; the Wald confidence interval may have low coverage probability even if p is not close to zero or one, and confidence limits outside the interval ( ) 0,1 . Matiri et al. [1] applied the Wald method to obtain interval estimates for the prevalence rate and encountered the problem of overshoot with negative lower confidence limits. Poor performance of the Wald confidence interval has been pointed out by many authors [ This paper considers an alternative method, called the likelihood method, for constructing the approximate confidence interval for the binomial proportion. The likelihood intervals are determined from the graph of the relative likelihood function or its logarithm for a fixed likelihood level [8]. They are fully conditioned on the shape of the likelihood function and hence are optimal. The likelihood method can be used to construct confidence interval for the proportion in situations where the traditional methods based on asymptotic normality are inaccurate.
In order to identify the best confidence interval for the binomial proportion p, the Wald, Wilson score, Pearson-Clopper and Likelihood methods of interval estimation are compared on the basis of coverage probability and interval width using simulated data. The four intervals are also applied to a real data example. The resulting confidence intervals for the binomial proportion are compared in terms interval width and plausibilities of the parameter values in them.
The paper is organized as follows: in Section 2, the four methods of interval estimation are described. In Section 3, the simulation results regarding coverage probability and expected length of the different intervals are presented and discussed. Section 4 applies the four intervals to a real-life data from a clinical study and compares them in terms of interval length and plausibilities of the parameter values inside them. Section 5 is devoted to concluding remarks. is at least 5 (or 10), otherwise it will produce unreliable interval estimates.

Clopper-Pearson Interval
Clopper-Pearson [9] proposed a method of constructing an exact two-sided confidence interval for the binomial proportion p using the equal-tail rule. The derivation of the two-sided

Proof
The density function of X is given by ( ) change of variable technique the density function of Z is obtained as which is the density function of a beta distribution with parameters β and α. Im- We use the above identity to obtain Hence it follows by Theorem 1 that

Theorem 3
If X has an F distribution with u and v degrees of freedom, then the random The above three theorems are now applied in the derivation of the closed

Likelihood Interval
Let The log-relative likelihood function of p, denoted by ( ) The likelihood intervals may be determined from a graph of ( ) , Bin n p using arbitrary value of p, the resulting population of level c likelihood intervals will contain this value of p with known frequency. They are therefore also confidence intervals and so are likelihood confidence intervals.

Wilson Interval
The Wilson score method for constructing confidence interval for binomial proportion p was developed by Edward B. Wilson [10] and is based on inverting the z-test for p. The endpoints of the . The score confidence interval is asymmetric and does not suffer from problems of overshoot and zero width confidence intervals associated with Wald confidence interval.

Simulations
In

Application to Real Example
The four methods of interval estimation are applied in a clinical study about the effectiveness of hyperdynamic therapy in treating cerebral vasospasm [11]. Each of these four confidence intervals is plotted on the graph of relative likelihood function as shown in Figure 5.      The likelihood interval looks optimal by evidence presented in Table 1. With these four confidence intervals we can conclude that hyperdynamic therapy is an effective method for treating ischaemic neurological symptoms due to vasospasm.

Conclusion
Clopper-Pearson interval is conservative for both small and large samples; however, it is always wider than it should. The Wald interval is well known and frequently used in statistical practice. Unfortunately, according to the above simulation study, its coverage probabilities are lower than the nominal level and are associated with problem of overshoot. Therefore, the inferential comparisons and judgements based on them might be misleading. On the other hand, Wilson and Likelihood intervals have coverage probabilities near the nominal level and shorter lengths. Wilson interval for the real data application is wider than the likelihood interval and includes implausible values of the parameter. In summary, the Wilson and Likelihood intervals are recommended to be used in practice. It is worth noting the Likelihood interval looks superior to Wilson interval in that it is shorter and includes plausible values of the parameter p. The likelihood method has one drawback in the sense that it does not produce an interval when the number of successes x is 0 or n.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.