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.
ci_p_wald_t(x, n, conf.level = 0.95)A vector with the lower and upper limits of the confidence interval.
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.
Pires, Ana M., and Conceiçao Amado. "Interval estimators for a binomial proportion: Comparison of twenty methods". REVSTAT-Statistical Journal 6.2 (2008): 165-197.
ci_p.
ci_p_wald_t(x = 0, n = 50, conf.level = 0.95)
#> [,1]
#> [1,] 0.0000000
#> [2,] 0.1090479
ci_p_wald_t(x = 15, n = 50, conf.level = 0.95)
#> [,1]
#> [1,] 0.1850598
#> [2,] 0.4445698
ci_p_wald_t(x = 50, n = 50, conf.level = 0.95)
#> [,1]
#> [1,] 0.8909521
#> [2,] 1.0000000