Agresti-Caffo confidence interval for Binomial proportion

This function calculates the Agresti-Caffo confidence interval for a Binomial proportion, which includes a Bayesian adjustment by adding 2 successes and 2 failures to stabilize the estimate. It is vectorized, allowing the evaluation of single values or vectors.

ci_p_agresti_caffo(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 Agresti-Caffo confidence interval incorporates a simple Bayesian adjustment by adding 2 successes and 2 failures to the data. The adjusted proportion is calculated as:

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

The standard error for the adjusted proportion is given by:

$$\text{se} = \sqrt{\frac{\hat{p} (1 - \hat{p})}{n + 4}}.$$

The confidence interval is then constructed using the $t$-distribution with $n - 1$ degrees of freedom:

$$\text{CI} = \hat{p} \pm t_{n-1, \alpha/2} \cdot \text{se},$$

where $t_{n-1, \alpha/2}$ is the critical value of the $t$-distribution at a two-tailed significance level of $\alpha = 1 - \text{conf.level}$.

References

Agresti, Alan, and Brian Caffo. "Simple and effective confidence intervals for proportions and differences of proportions result from adding two successes and two failures." The American Statistician 54.4 (2000): 280-288.

Author

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

Examples

# Example with a single value
ci_p_agresti_caffo(x = 15, n = 50, conf.level = 0.95)
#>           [,1]
#> [1,] 0.1878045
#> [2,] 0.4418251