# An exact confidence interval for a binomial proportion

## 2006-08-18

[StATS]: An exact confidence interval for a binomial proportion (created 2006-08-18).

A researcher came into my office this morning with some data that was strongly negative. Out of 15 patients, none showed a detectable improvement after the use of a controversial treatment. That sounds like a strong negative result to me, but a reviewer asked a legitimate question: How do you know that you are not having problems with a Type II error?

I would not have worded it quite that way, but the point is still good. Recall that a Type II error is accepting the null hypothesis when the null hypothesis is true. A common reason why a Type II error occurs is that the sample size was so small that there was insufficient power to reject the null hypothesis.

So how do we know that 15 patients is an adequate sample size? It gets a bit tricky in this particular problem because there was no control group and no formal research hypothesis. In some situations, you might want to make a hypothesis about a change score, but for this data set, the values both before and after were always below the limit of detection, so there is no formal way to quantify the amount of improvement or lack thereof.

The way around this problem is to note that the estimated proportion of improvements is zero and to place 95% confidence limits around these values. A simple approach that works is the rule of three, but a more precise answer is to use the Clopper-Pearson method.

• The Use of Confidence or Fiducial Limits Illustrated in the Case of the Binomial. C. J. Clopper, E. S. Pearson. Biometrika, Vol. 26, No. 4 (Dec., 1934), pp. 404-413.

As you can see, this approach has been around for quite some time. A review of Clopper Pearson approach and some competing formulas for confidence intervals appears in

• Approximate Is Better than “Exact” for Interval Estimation of Binomial Proportions. Alan Agresti; Brent A. Coull. The American Statistician, Vol. 52, No. 2. (May, 1998), pp. 119-126.