Recentered Wald Interval with Continuity Correction for Binomial Proportion

This function calculates the recentered Wald confidence interval with continuity correction for a Binomial proportion. It adjusts the classical Wald interval by introducing a recentering term and a continuity correction, improving accuracy for small sample sizes and boundary cases. The method is vectorized, allowing for evaluation of single values or vectors.

ci_p_wald_recentered_cc(x, n, conf.level = 0.95)

Arguments

x

A number or a vector with the number of successes.

n

A number or a vector with the number of trials.

conf.level

Confidence level for the returned confidence interval. By default, it is 0.95.

Value

A vector with the lower and upper limits of the confidence interval.

Details

The recentered Wald interval with continuity correction adjusts the classical Wald interval by incorporating a recentering term and a continuity correction to account for the discreteness of the binomial distribution.

The critical value \(z\) is obtained from the standard normal distribution for the specified confidence level:

$$z = \Phi^{-1}(1 - \alpha / 2),$$

where \(\alpha = 1 - \text{conf.level}\).

The confidence limits are calculated as:

$$\text{Lower} = \max\left(\frac{x + z^2 / 2}{n + z^2} - \left[z \sqrt{\frac{x}{n^2} \left(1 - \frac{x}{n}\right)} + \frac{1}{2n}\right], 0\right),$$ $$\text{Upper} = \min\left(\frac{x + z^2 / 2}{n + z^2} + \left[z \sqrt{\frac{x}{n^2} \left(1 - \frac{x}{n}\right)} + \frac{1}{2n}\right], 1\right).$$

Special cases are handled explicitly:

- If \(x = 0\), the lower limit is 0, and the upper limit is calculated as \((\alpha / 2)^{1/n}\).

- If \(x = n\), the upper limit is 1, and the lower limit is calculated as \(1 - (\alpha / 2)^{1/n}\).

These adjustments ensure that the confidence interval is valid and well-behaved, even at the boundaries of the parameter space.

References

Pires, Ana M., and Conceiçao Amado. "Interval estimators for a binomial proportion: Comparison of twenty methods". REVSTAT-Statistical Journal 6.2 (2008): 165-197.

See also

ci_p.

Author

David Esteban Cartagena Mejía, dcartagena@unal.edu.co

Examples

ci_p_wald_recentered_cc(x =  0, n = 50, conf.level = 0.95)
#>           [,1]
#> [1,] 0.0000000
#> [2,] 0.9288783
ci_p_wald_recentered_cc(x = 25, n = 50, conf.level = 0.95)
#>           [,1]
#> [1,] 0.3514096
#> [2,] 0.6485904
ci_p_wald_recentered_cc(x = 50, n = 50, conf.level = 0.95)
#>            [,1]
#> [1,] 0.07112174
#> [2,] 1.00000000