Modified Wilson Confidence Interval Calculation in R

The modified Wilson confidence interval is an improved method for calculating confidence intervals for proportions, particularly when dealing with small sample sizes. This guide explains how to calculate it in R and provides an interactive calculator for quick results.

What is the Modified Wilson Confidence Interval?

The modified Wilson confidence interval is an adjustment to the traditional Wilson score interval that provides more accurate results, especially when the sample size is small or when the proportion is near 0 or 1. It's commonly used in statistics, survey analysis, and quality control applications.

Key characteristics of the modified Wilson interval include:

Better coverage probability than the standard Wilson interval
More accurate results for small sample sizes
Handles edge cases (proportions near 0 or 1) better
Provides symmetric confidence intervals

Note: The modified Wilson interval is particularly useful when dealing with binary outcomes in medical studies, A/B testing, and other applications where small sample sizes are common.

Formula and Calculation

The modified Wilson confidence interval for a proportion p is calculated using the following formula:

Lower bound = [p + z²/(2n) - z*sqrt((p*(1-p)+z²/(4n))/n)] / (1 + z²/n)

Upper bound = [p + z²/(2n) + z*sqrt((p*(1-p)+z²/(4n))/n)] / (1 + z²/n)

Where:

p = sample proportion (successes/trials)
n = sample size
z = z-score corresponding to the desired confidence level

The standard Wilson interval is similar but uses a different adjustment factor. The modified version provides better coverage properties, especially for small samples.

Comparison of Wilson and Modified Wilson Intervals
Characteristic	Standard Wilson	Modified Wilson
Coverage probability	Approximately correct	More accurate
Small sample performance	Good but not optimal	Better
Edge cases (p near 0 or 1)	Handled reasonably	Handled better
Computational complexity	Simple	Slightly more complex

Implementing in R

R provides several packages for calculating confidence intervals, including the binom package which includes the modified Wilson interval function. Here's how to implement it:

# Install package if needed
install.packages("binom")

# Load the package
library(binom)

# Calculate modified Wilson interval
modified.wilson.ci(successes, trials, conf.level = 0.95)

The function returns both the lower and upper bounds of the confidence interval. You can adjust the confidence level by changing the conf.level parameter.

For more control over the calculation, you can implement the formula directly in R:

modified_wilson <- function(successes, trials, conf.level = 0.95) {
p <- successes / trials
z <- qnorm(1 - (1 - conf.level)/2)
n <- trials
lower <- (p + z^2/(2*n) - z*sqrt((p*(1-p)+z^2/(4*n))/n)) / (1 + z^2/n)
upper <- (p + z^2/(2*n) + z*sqrt((p*(1-p)+z^2/(4*n))/n)) / (1 + z^2/n)
return(c(lower, upper))
}

Worked Example

Let's calculate the modified Wilson confidence interval for a scenario where 45 out of 100 patients responded positively to a treatment.

Given:
Successes = 45
Trials = 100
Confidence level = 95% (z ≈ 1.96)

Calculation:
p = 45/100 = 0.45
lower = [0.45 + 1.96²/(2*100) - 1.96*sqrt((0.45*0.55+1.96²/(4*100))/100)] / (1 + 1.96²/100)
≈ 0.343
upper = [0.45 + 1.96²/(2*100) + 1.96*sqrt((0.45*0.55+1.96²/(4*100))/100)] / (1 + 1.96²/100)
≈ 0.557

Interpretation: We can be 95% confident that the true proportion of patients who would respond positively to the treatment lies between approximately 34.3% and 55.7%.

FAQ

What is the difference between the standard Wilson and modified Wilson confidence intervals?

The modified Wilson interval provides better coverage probability, especially for small sample sizes and proportions near 0 or 1. The standard Wilson interval is simpler but may be less accurate in certain scenarios.

When should I use the modified Wilson interval instead of the standard Wilson?

Use the modified Wilson interval when you need more accurate results, especially with small sample sizes or proportions near the extremes (0 or 1). The standard Wilson is sufficient for most other cases.

How does the modified Wilson interval handle small sample sizes?

The modified Wilson interval performs better than the standard Wilson for small sample sizes, providing more accurate coverage probabilities and better handling of edge cases.