# Confidence Interval: Adjusted Wald Method

#### erand

##### New Member
If I am conducting an experiment on 10 people and all 10 performed a task successfully, I want to know how to determine the confidence interval. I am using the Adjusted Wald Method, which apparently is the most accurate to for small binomial samples.

Using this method, I got a 95% 2-tailed confidence interval of a success rate of 64.03% - 107.40%.

What does one do to account for a success rate value that is over 100% (impossible). Do I take that percentage over 100% and add it to the other tail?

Thank you very much!

#### BioStatMatt

##### TS Contributor
OK, you said that these data represent a binomial experiment, that is, you have 10 observations and each responded either positive or negative. Correct?

If this is the case, it appears that you have used an approximate method to calculate the confidence interval.

There are methods to calcuate the exact confidence interval for the binomial proportion. These confidence intervals will always be between 0 and 1. The following url has a calculator for exact confidence intervals for the binomial proportion.

http://www.causascientia.org/math_stat/ProportionCI.html

~Matt

#### erand

##### New Member
Thank you, Matt. The calculator is very helpful. Yes, I am dealing with binomial data. Is calculating the confidence interval using the method that the calculator is using (do you know what the calculation is?) more accurate than the Adjusted Wald Method? I read a couple of papers by Jeff Sauer indicating that for a small sample, this is the most accurate method for calculating the confidence interval. There is another calculator for these methods (http://www.measuringusability.com/wald.htm). I just don't understand logically how to treat an interval where one side of the best point estimate is greater than 100%. The calculator that I sent you a link to doesn't go over, but why?

#### BioStatMatt

##### TS Contributor
One Wald interval is based on a standard normal approximation. The interval is

phat +- z_alpha/2 * sqrt(phat*(1-phat)/n)

Another Wald type interval may use the t distribution with a penalized degrees of freedom. That may be what you have seen. Since these approximations are based on distributions that are continuous from -infinity to infinity, the confidence interval can include values less than 0 and greater than 1. You are correct in thinking that it makes little sense to report a confidence interval for p that is less than 0 or greater than 1. These approximate methods were developed to make calculations easier. However, with advances in computer science, it is much easier to calculate exact confidence intervals.

The exact method always produces a more accurate confidence interval. One exact method (derived by Clopper and Pearson. 1934) is:

LBp = y/(y + (n-y+1) * F(alpha/2, 2*(n-y+1), 2*y))
UBp = (n-y)/(n - y + (y+1) * F(1-alpha/2, 2*(y+1), 2*(n-y)))

Where y in the number of successes and n is the total number of trials. F(a,b,c) is the upper 1-a percent point of an F-distribution with degrees of freedom b and c.

~Matt

#### erand

##### New Member
Thank you, Matt. I really appreciate your expertise.