R/ci_p_xxx.R
ci_p_wald_bs.RdThis function calculates the Wald Binomial Score confidence interval for a Binomial proportion. It is vectorized, allowing the evaluation of single values or vectors.
ci_p_wald_bs(x, n, conf.level = 0.95)A vector with the lower and upper limits of the confidence interval.
The Wald Binomial Score confidence interval is an adjusted version of the Wald interval, designed to improve accuracy in small samples and near the boundaries of the parameter space.
The estimated proportion is given by:
$$\hat{p} = \frac{x}{n}$$,
and its complement is:
$$\hat{q} = 1 - \hat{p}$$.
The Wald Binomial Score interval incorporates adjustments for the sample size and confidence level:
$$\text{wn\_z} = \sqrt{n - z^2 - \frac{2z}{\sqrt{n}} - \frac{1}{n}}$$,
where \(z\) is the quantile of the standard normal distribution corresponding to the confidence level. The final confidence interval is given by:
$$\text{Lower} = \hat{p} - \frac{z \sqrt{\hat{p}\hat{q} + \frac{1}{2n}}}{\text{wn\_z}},$$ $$\text{Upper} = \hat{p} + \frac{z \sqrt{\hat{p}\hat{q} + \frac{1}{2n}}}{\text{wn\_z}}.$$
This interval is particularly useful when \(n\) is small or when the proportion \(\hat{p}\) is close to 0 or 1.
Blyth, C.R. and Still, H.A. (1983). Binomial confidence intervals, Journal of the American Statistical Association, 78, 108–116.
ci_p.
ci_p_wald_bs(x = 15, n = 50, conf.level = 0.95)
#> [,1]
#> [1,] 0.1629798
#> [2,] 0.4370202