ci_p_wald_t: Wald-t Confidence Interval for Binomial Proportion

In fhernanb/stests: Package with useful statistical tests

Wald-t Confidence Interval for Binomial Proportion

Description

This function calculates the Wald-t confidence interval for a Binomial proportion using the method XIX proposed by Pan (2002). It incorporates the use of a t-distribution with adjusted degrees of freedom \nu, as specified in equation (2.8). The function is vectorized, allowing for evaluation of single values or vectors.

Usage

ci_p_wald_t(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 Wald-t confidence interval is a modification of the classical Wald interval that uses the t-distribution instead of the normal distribution for greater accuracy in small samples. The estimated proportion \hat{p} is adjusted as:

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

which reduces bias in the interval estimation. The variance V(\hat{p}, n) is given by:

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

The degrees of freedom \nu are calculated using equation (2.8):

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

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

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

The confidence interval is then calculated as:

\text{Lower} = \max\left(0, \hat{p} - t \cdot \sqrt{V(\hat{p}, n)}\right),

\text{Upper} = \min\left(1, \hat{p} + t \cdot \sqrt{V(\hat{p}, n)}\right),

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

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.

Examples

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

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