R/ci_p_xxx.R
ci_p_clopper_pearson.RdThis function calculates the confidence interval for a proportion. It is vectorized, allowing users to evaluate it using either single values or vectors.
ci_p_clopper_pearson(x, n, conf.level = 0.95)A vector with the lower and upper limits.
The Clopper-Pearson interval is an exact confidence interval for the Binomial proportion \(p\). The limits of the interval are derived based on the Beta distribution. For the special cases where \(x=0\) or \(x=n\), the limits are calculated directly.
The mathematical definitions are as follows:
- If \(x=0\), the lower limit is \(0\), and the upper limit is \(1 - (\alpha / 2)^{1/n}\).
- If \(x=n\), the lower limit is \((\alpha / 2)^{1/n}\), and the upper limit is \(1\).
Otherwise, the limits are given by:
$$\text{Lower Limit}=B_{1-\alpha/2, x, n-x+1}$$
$$\text{Upper Limit}=B_{\alpha/2, x+1, n-x}$$
where \(B_{\omega, a, b}\) is the \(100\%(1-\omega)\) percentile of the Beta distribution with parameters \(a\) and \(b\).
Due to the relationship between Beta and F distributions, the limits can be written as:
$$\text{Lower Limit}=\frac{1}{1+\frac{n-x+1}{x}F_{\alpha/2, \, 2(n-x+1), \, 2x}}$$
$$\text{Upper Limit}=\frac{\frac{x+1}{n-x} F_{\alpha/2, \, 2(x+1), \, 2(n-x)}}{1+\frac{x+1}{n-x} F_{\alpha/2, \, 2(x+1), \, 2(n-x)}}$$
where \(F_{\omega, a, b}\) is the \(100\%(1-\omega)\) percentile of the F distribution with parameters \(a\) and \(b\).
Clopper, C. J., & Pearson, E. S. (1934). The use of confidence or fiducial limits illustrated in the case of the binomial. Biometrika, 26(4), 404-413.
ci_p.
ci_p_clopper_pearson(x=15, n=50, conf.level=0.95)
#> [,1]
#> [1,] 0.1786178
#> [2,] 0.4460823