ci_p_add_4: Add-4 Wald-t Confidence Interval for Binomial Proportion

In fhernanb/stests: Package with useful statistical tests

Add-4 Wald-t Confidence Interval for Binomial Proportion

Description

This function calculates the confidence interval for a proportion. It is vectorized, allowing users to evaluate it using either single values or vectors.

Usage

ci_p_add_4(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.

Details

The Add-4 Wald-t confidence interval improves the performance of the Wald interval by adding 2 successes and 2 failures to the observed data, effectively modifying the estimated proportion:

\hat{p}=\frac{x + 2}{n + 4}.

The variance V(\hat{p}, n+4) is given by:

V(\hat{p}, n+4)=\frac{\hat{p}(1 - \hat{p})}{n + 4}.

The degrees of freedom \nu are calculated using equation (2.9):

\nu=\frac{2 V(\hat{p}, n+4)^2}{\Omega(\hat{p}, n+4)},

where \Omega(\hat{p}, n+4) is defined as:

\Omega(p, n)=\frac{p - p^2}{n^3} + \frac{p + (6n - 7)p^2 + 4(n - 1)(n - 3)p^3 - 2(n - 1)(2n - 3)p^4}{n^5} - \frac{2(p + (2n - 3)p^2 - 2(n - 1)p^3)}{n^4}.

The confidence interval is then calculated as:

\text{Lower}=\hat{p} - t \cdot \sqrt{\frac{\tilde{\pi}(1 - \hat{p})}{n+4}},

\text{Upper}=\hat{p} + t \cdot \sqrt{\frac{\tilde{\pi}(1 - \hat{p})}{n+4}},

where t is the critical value from the t-distribution with \nu degrees of freedom.

Value

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

Author(s)

David Esteban Cartagena Mejía, [e-mail]

References

Pan, W. (2002). Approximate confidence intervals for one proportion and difference of two proportions. Computational Statistics and Data Analysis, 40(1), 143–157

See Also

ci_p.

Examples

ci_p_add_4(x=15, n=50, conf.level=0.95)
ci_p_add_4(x=0,  n=50, conf.level=0.95)
ci_p_add_4(x=50, n=50, conf.level=0.95)

fhernanb/stests documentation built on March 29, 2025, 10:36 a.m.