Binomial proportion confidence interval

In statistics, a binomial proportion confidence interval is a confidence interval for a proportion in a statistical population. It uses the proportion estimated in a statistical sample and allows for sampling error. There are several formulas for a binomial confidence interval, but all of them rely on the assumption of a binomial distribution. In general, a binomial distribution applies when an experiment is repeated a fixed number of times, each trial of the experiment has two possible outcomes (labeled arbitrarily success and failure), the probability of success is the same for each trial, and the trials are statistically independent.

A simple example of a binomial distribution is the set of various possible outcomes, and their probabilities, for the number of heads observed when a (not necessarily fair) coin is flipped ten times. The observed binomial proportion is the fraction of the flips which turn out to be heads. Given this observed proportion, the confidence interval for the true proportion innate in that coin is a range of possible proportions which may contain the true proportion. A 95% confidence interval for the proportion, for instance, will contain the true proportion 95% of the times that the procedure for constructing the confidence interval is employed. Note that this does not mean that a calculated 95% confidence interval will contain the true proportion with 95% probability. Instead, one should interpret it as follows: the process of drawing a random sample and calculating an accompanying 95% confidence interval will generate a confidence interval that contains the true proportion in 95% of all cases. The odds that any fairly drawn sample from all cases will be inside the confidence range is 95% likely, so there is a 5% risk that a fairly drawn sample will not be inside a 95% confidence interval.

There are several ways to compute a confidence interval for a binomial proportion. The normal approximation interval is the simplest formula, and the one introduced in most basic statistics classes and textbooks. This formula, however, is based on an approximation that does not always work well. Several competing formulas are available that perform better, especially for situations with a small sample size and a proportion very close to zero or one. The choice of interval will depend on how important it is to use a simple and easy-to-explain interval versus the desire for better accuracy.

Normal approximation interval

The most commonly used formula for a binomial confidence interval relies on approximating the distribution of error about a binomially-distributed observation, ${\hat {p}}$ , with a normal distribution. However, although this distribution is frequently confused with a binomial distribution, it should be noted that the error distribution itself is not binomial,^[1] and hence other methods (below) are preferred.

The approximation is usually justified by the central limit theorem. The formula is

{\hat {p}}\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {1}{n}}{\hat {p}}\left(1-{\hat {p}}\right)}}

where ${\hat {p}}$ is the proportion of successes in a Bernoulli trial process estimated from the statistical sample, $z$ is the $\scriptstyle 1 - \frac{1}{2}\alpha$ quantile of a standard normal distribution, $\alpha$ is the error quantile and n is the sample size. For example, for a 95% confidence level the error ( $\alpha$ ) is 5%, so $\scriptstyle 1 - \frac{1}{2}\alpha$ = 0.975 and $z$ = 1.96.

The central limit theorem applies poorly to this distribution with a sample size less than 30 or where the proportion is close to 0 or 1. The normal approximation fails totally when the sample proportion is exactly zero or exactly one. A frequently cited rule of thumb is that the normal approximation is a reasonable one as long as np > 5 and n(1 − p) > 5, however even this is unreliable in many cases; see Brown et al. 2001.^[2]

An important theoretical derivation of this confidence interval involves the inversion of a hypothesis test. Under this formulation, the confidence interval represents those values of the population parameter that would have large p-values if they were tested as a hypothesized population proportion. The collection of values, $\theta$ , for which the normal approximation is valid can be represented as

\left\{ \theta \bigg| y \le \frac{\hat p - \theta}{\sqrt{\frac{1}{n}\hat p \left(1 - \hat p\right)}} \le z \right\}

where $y$ is the $\scriptstyle \frac{1}{2}\alpha$ quantile of a standard normal distribution.

Since the test in the middle of the inequality is a Wald test, the normal approximation interval is sometimes called the Wald interval, but Pierre-Simon Laplace first described it in his 1812 book Théorie analytique des probabilités (page 283).

Wilson score interval

The Wilson interval is an improvement (the actual coverage probability is closer to the nominal value) over the normal approximation interval and was first developed by Edwin Bidwell Wilson (1927).^[3]

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

This interval has good properties even for a small number of trials and/or an extreme probability.

These properties are obtained from its derivation from the binomial model. Consider a binomial population probability $P$ , whose distribution may be approximated by the normal distribution with standard deviation $\scriptstyle \sqrt{\frac{1}{n}P \left(1 - P \right)}$ . However, the distribution of true values about an observation is not binomial. Rather, an observation ${\hat {p}}$ will have an error interval with a lower bound equal to $P$ when ${\hat {p}}$ is at the equivalent normal interval upper bound (i.e. for the same $\alpha$ ) of $P$ , and vice versa.^[1]

The Wilson interval can also be derived from Pearson's chi-squared test with two categories. The resulting interval

\left\{\theta {\bigg |}y\leq {\frac {{\hat {p}}-\theta }{\sqrt {{\frac {1}{n}}\theta \left({1-\theta }\right)}}}\leq z\right\}

can then be solved for $\theta$ to produce the Wilson interval. The test in the middle of the inequality is a score test, so the Wilson interval is sometimes called the Wilson score interval.

The center of the Wilson interval

{\frac {{\hat {p}}+{\frac {1}{2n}}z^{2}}{1+{\frac {1}{n}}z^{2}}}

can be shown to be a weighted average of ${\hat {p}}=\scriptstyle {\frac {X}{n}}$ and $\scriptstyle {\frac {1}{2}}$ , with ${\hat {p}}$ receiving greater weight as the sample size increases. For the 95% interval, the Wilson interval is nearly identical to the normal approximation interval using $\tilde p \,=\, \scriptstyle \frac{X + 2}{n + 4}$ instead of ${\hat {p}}$ .

Wilson score interval with continuity correction

The Wilson interval may be modified by employing a continuity correction, in order to align the minimum coverage probability (rather than the average) with the nominal value.

Just as the Wilson interval mirrors Pearson's chi-squared test, the Wilson interval with continuity correction mirrors the equivalent Yates' chi-squared test.

The following formulae for the lower and upper bounds of the Wilson score interval with continuity correction $( w^- , w^+ )$ are derived from Newcombe (1998).^[4]

{\begin{aligned}w^{-}&=\operatorname {max} \left\{0,{\frac {2n{\hat {p}}+z^{2}-\left[z{\sqrt {z^{2}-{\frac {1}{n}}+4n{\hat {p}}(1-{\hat {p}})+(4{\hat {p}}-2)}}+1\right]}{2(n+z^{2})}}\right\}\\w^{+}&=\operatorname {min} \left\{1,{\frac {2n{\hat {p}}+z^{2}+\left[z{\sqrt {z^{2}-{\frac {1}{n}}+4n{\hat {p}}(1-{\hat {p}})-(4{\hat {p}}-2)}}+1\right]}{2(n+z^{2})}}\right\}\end{aligned}}

However, if p=0, $w^{-}$ must be taken as 0; if p=1, $w^{+}$ is then 1.

Jeffreys interval

The Jeffreys interval has a Bayesian derivation, but it has good frequentist properties. In particular, it has coverage properties that are similar to the Wilson interval, but it is one of the few intervals with the advantage of being equal-tailed (e.g., for a 95% confidence interval, the probabilities of the interval lying above or below the true value are both close to 2.5%). In contrast, the Wilson interval has a systematic bias such that it is centred too close to p = 0.5.^[5]

The Jeffreys interval is the Bayesian credible interval obtained when using the non-informative Jeffreys prior for the binomial proportion $p$ . The Jeffreys prior for this problem is a Beta distribution with parameters $(1/2, 1/2)$ . After observing $x$ successes in $n$ trials, the posterior distribution for $p$ is a Beta distribution with parameters $(x + 1/2, n - x + 1/2)$ .

When $x \neq0$ and $x \neq n$ , the Jeffreys interval is taken to be the $100(1 - α)%$ equal-tailed posterior probability interval, i.e., the $α / 2$ and $1 - α / 2$ quantiles of a Beta distribution with parameters $(x + 1/2, n - x + 1/2)$ . These quantiles need to be computed numerically, although this is reasonably simple with modern statistical software.

In order to avoid the coverage probability tending to zero when $p \to 0$ or $1$ , when $x = 0$ the upper limit is calculated as before but the lower limit is set to 0, and when $x = n$ the lower limit is calculated as before but the upper limit is set to 1.^[2]

Clopper-Pearson interval

The Clopper-Pearson interval is an early and very common method for calculating binomial confidence intervals.^[6] This is often called an 'exact' method, but that is because it is based on the cumulative probabilities of the binomial distribution (i.e., exactly the correct distribution rather than an approximation), but the intervals are not exact in the way that one might assume: the discontinuous nature of the binomial distribution precludes any interval with exact coverage for all population proportions. The Clopper-Pearson interval can be written as

S_{\leq }\cap S_{\geq }

or equivalently,

(\inf S_{\geq }\,,\,\sup S_{\leq })

with

S_{\le} := \left\{ \theta \Big | P \left[ \mathrm{Bin}\left( n; \theta \right) \le X \right] > \frac{\alpha}{2} \right\} \mathrm{~~and~~} S_{\ge} := \left\{ \theta \Big | P \left[ \mathrm{Bin}\left( n; \theta \right) \ge X \right] > \frac{\alpha}{2} \right\},

where 0 ≤ X ≤ n is the number of successes observed in the sample and Bin(n; θ) is a binomial random variable with n trials and probability of success θ.

Because of a relationship between the cumulative binomial distribution and the beta distribution, the Clopper-Pearson interval is sometimes presented in an alternate format that uses quantiles from the beta distribution.

B\left(\frac{\alpha}{2}; x, n - x + 1\right) < \theta < B\left(1 - \frac{\alpha}{2}; x + 1, n - x\right)

where x is the number of successes, n is the number of trials, and B(p; v,w) is the pth quantile from a beta distribution with shape parameters v and w.

When $x$ is either $0$ or $n$ , closed-form expressions for the interval bounds are available: when $x=0$ the interval is $(0,1-({\frac {\alpha }{2}})^{\frac {1}{n}})$ and when $x=n$ it is $(({\frac {\alpha }{2}})^{\frac {1}{n}},1)$ .^[7] Some mathematical packages, e.g python scipy fail on these extreme cases and the closed-form expression should be used instead.

The beta distribution is, in turn, related to the F-distribution so a third formulation of the Clopper-Pearson interval can be written using F quantiles:

\left(1+{\frac {n-x}{\left[x+1\right]\,F\!\left[{\frac {\alpha }{2}};2(x+1),2(n-x)\right]}}\right)^{-1}<\theta <\left(1+{\frac {n-x+1}{x\,\,F\!\left[1-{\frac {1}{2}}\alpha ;2x,2(n-x+1)\right]}}\right)^{-1}

where x is the number of successes, n is the number of trials, and F(c; d1, d2) is the 1 - c quantile from an F-distribution with d1 and d2 degrees of freedom.^[8]

The Clopper-Pearson interval is an exact interval since it is based directly on the binomial distribution rather than any approximation to the binomial distribution. This interval never has less than the nominal coverage for any population proportion, but that means that it is usually conservative. For example, the true coverage rate of a 95% Clopper-Pearson interval may be well above 95%, depending on n and θ. Thus the interval may be wider than it needs to be to achieve 95% confidence. In contrast, it is worth noting that other confidence bounds may be narrower than their nominal confidence width, i.e., the Normal Approximation (or "Standard") Interval, Wilson Interval,^[3] Agresti-Coull Interval,^[8] etc., with a nominal coverage of 95% may in fact cover less than 95%.^[2]

Agresti-Coull Interval

The Agresti-Coull interval is also another approximate binomial confidence interval.^[8]

Given $X$ successes in $n$ trials, define

\tilde{n} = n + z^2

and

\tilde{p} = \frac{1}{\tilde{n}}\left(X + \frac{1}{2}z^2\right)

Then, a confidence interval for $p$ is given by

\tilde{p} \pm z \sqrt{\frac{1}{\tilde{n}}\tilde{p}\left(1 - \tilde{p} \right)}

where $z$ is the $1 - \frac{1}{2}\alpha$ quantile of a standard normal distribution, as before. For example, for a 95% confidence interval, let $\alpha = 0.05$ , so $z$ = 1.96 and $z^2$ = 3.84. If we use 2 instead of 1.96 for $z$ , this is the "add 2 successes and 2 failures" interval in ^[8]

Arcsine transformation

For more details on this topic, see Cohen's h.

Let X be the number of successes in n trials and let p = X/n. The variance of p is

\operatorname {var} (p)={\frac {p(1-p)}{n}}

Using the arc sine transform the variance of the arcsine of p is^[9]

\operatorname {var} \left(\arcsin({\sqrt {p}})\right)\approx {\frac {\operatorname {var} (p)}{4p(1-p)}}={\frac {p(1-p)}{4np(1-p)}}={\frac {1}{4n}}

So, confidence interval itself has the following form:

\sin^2\left(\arcsin(\sqrt{p}) - \frac{z}{2\sqrt{n}}\right) < \theta < \sin^2\left(\arcsin(\sqrt{p}) + \frac{z}{2\sqrt{n}}\right)

where $z$ is $\scriptstyle 1\,-\,{\frac {\alpha }{2}}$ quantile of a standard normal distribution

This method may be used to estimate the variance of p but its use is problematic when p is close to 0 or 1.

t_a transform

Let p be the proportion of successes. For 0 ≤ a ≤ 2

t_{ a } = \log\left( \frac{ p^{ a } }{ ( 1 - p )^{ 2 - a } } \right) = a \log( p ) - ( 2 - a )\log( 1 - p )

This family is a generalisation of the logit transform which is a special case with a = 1 and can be used to transform a proportional data distribution to an approximately normal distribution. The parameter a has to be estimated for the data set.

Special cases

In medicine, the rule of three is used to provide a simple way of stating an approximate 95% confidence interval for p, in the special case that no successes ( $\hat p = 0$ ) have been observed.^[10] The interval is (0,3/n).

By symmetry, one could expect for only successes ( ${\hat p}=1$ ), the interval is (1-3/n,1).

Comparison of different intervals

There are several research papers that compare these and other confidence intervals for the binomial proportion.^[1]^[4]^[11]^[12] Both Agresti and Coull (1998)^[8] and Ross (2003)^[13] point out that exact methods such as the Clopper-Pearson interval may not work as well as certain approximations.

Many of these intervals can be calculated in R using packages like proportion and binom.

References

1 2 3 Wallis, Sean A. (2013). "Binomial confidence intervals and contingency tests: mathematical fundamentals and the evaluation of alternative methods" (PDF). Journal of Quantitative Linguistics. 20 (3): 178–208. doi:10.1080/09296174.2013.799918.
1 2 3 Brown, Lawrence D.; Cai, T. Tony; DasGupta, Anirban (2001). "Interval Estimation for a Binomial Proportion". Statistical Science. 16 (2): 101–133. doi:10.1214/ss/1009213286. MR 1861069. Zbl 02068924.
1 2 Wilson, E. B. (1927). "Probable inference, the law of succession, and statistical inference". Journal of the American Statistical Association. 22: 209–212. doi:10.1080/01621459.1927.10502953. JSTOR 2276774.
1 2 Newcombe, R. G. (1998). "Two-sided confidence intervals for the single proportion: comparison of seven methods". Statistics in Medicine. 17 (8): 857–872. doi:10.1002/(SICI)1097-0258(19980430)17:8<857::AID-SIM777>3.0.CO;2-E. PMID 9595616.
↑ Cai, TT (2005). "One-sided confidence intervals in discrete distributions". Journal of Statistical Planning and Inference. 131: 63–88. doi:10.1016/j.jspi.2004.01.005.
↑ 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.
↑ Thulin, Måns (2014-01-01). "The cost of using exact confidence intervals for a binomial proportion". Electronic Journal of Statistics. 8 (1): 817–840. doi:10.1214/14-EJS909. ISSN 1935-7524.
1 2 3 4 5 Agresti, Alan; Coull, Brent A. (1998). "Approximate is better than 'exact' for interval estimation of binomial proportions". The American Statistician. 52: 119–126. doi:10.2307/2685469. JSTOR 2685469. MR 1628435.
↑ Shao J (1998) Mathematical statistics. Springer. New York, New York, USA
↑ Steve Simon (2010) "Confidence interval with zero events", The Children's Mercy Hospital, Kansas City, Mo. (website: "Ask Professor Mean at Stats topics or Medical Research Archived October 15, 2011, at the Wayback Machine.)
↑ Reiczigel, J (2003). "Confidence intervals for the binomial parameter: some new considerations" (PDF). Statistics in Medicine. 22: 611–621. doi:10.1002/sim.1320.
↑ Sauro J., Lewis J.R. (2005) "Comparison of Wald, Adj-Wald, Exact and Wilson intervals Calculator". Proceedings of the Human Factors and Ergonomics Society, 49th Annual Meeting (HFES 2005), Orlando, FL, p2100-2104
↑ Ross, T. D. (2003). "Accurate confidence intervals for binomial proportion and Poisson rate estimation". Computers in Biology and Medicine. 33: 509–531. doi:10.1016/S0010-4825(03)00019-2.

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