# Agresti-Coull Interval Method

The Agresti-Coull Interval Method calculates confidence intervals for proportions.  A proportion (p) is a type of ratio where the numerator is contained in the denominator.  Proportions can be expressed as percentages (per 100), per 1000, or per 100,000, etc.  The Agresti-Coull Interval Method was created by two statisticians, Alan Agresti and Brent Coull, who suggested adding 4 observations to the sample, two successes and two failures, and then using the Wald formula to construct a 95% confidence interval (CI).1 In other words, 2 counts are added to the numerator and 4 counts are added to the denominator.

Formula for Agresti-Coull 95% confidence interval:

lower limit =  max { (X + 2)/(n + 4) - 1.96 x [(X + 2)/(n + 4)2 x (1 - (X + 2)/(n + 4))]½ ; 0 }

upper limit = min { (X + 2)/(n + 4) + 1.96 x [(X + 2)/(n + 4)2 x (1 - (X + 2)/(n + 4))]½ ; 1 }

where X represents the number of observations that belong to a certain category of interest, n represents the denominator (total number of observations), and X/n is the proportion of interest (p).

The Agresti-Coull interval does not show the persistently chaotic coverage probabilities that characterize the standard Wald interval even when the denominator is large and p is not near it’s boundaries (0% or 100%).1,2,3  Other advantages of the Agresti-Coull interval are the following:  1) mean coverage probability is very close to 95%, 2) minimum coverage never dips below 92% for n >10, 3) the expected interval lengths are reasonable, 4) simple, straightforward method, and 5) well-regarded and recommended among statisticians.1,2,3

The Agresti-Coull interval can be used to assess whether an estimated proportion is statistically significantly different than a benchmark. When the upper limit of the interval is less than the benchmark value, it indicates that the proportion is significantly less than the benchmark value.  Likewise, a lower interval limit that is greater than the benchmark indicates that the proportion is significantly greater than the benchmark.  If the Agresti-Coull interval includes the benchmark value, then the proportion is not statistically significantly different from the benchmark.

REFERENCES

1. Agresti A., Coull BA.  Approximate is Better than “Exact” for Interval Estimation of Binomial Proportions.  The American Statistician 1998;52:119-126.

2. Brown LD, Cai TT, DasGupta, A. Interval Estimation for a Binomial Proportion.  Statistical Science 2001;16:101-133.

3. Pires AM, Amado C.  Interval Estimators for a Binomial Proportion: Comparison of Twenty Methods.  Statistical Journal 2008;6:165-197.