How can I compute a one-sided confidence interval?

Typically, confidence intervals are expressed as a two-sided range. Given a set of assumptions, you can be 95% confident that this range includes the true value of a parameter such as mean, EC50, relative risk, difference. This  range is “two sided” because it is bounded by both a lower and an upper confidence limit.

In some circumstances, it can make more sense to express the confidence interval in only one direction – with only a lower confidence limit,  or only an upper confidence limit, but not both. This can best be illustrated by following an example.

A recent study was performed to evaluate the effectiveness of a new drug in the eradication of Heliobacter pylori infection, and to determine whether or not it was inferior to the standard drug. (This example was adapted from one presented in reference 1). The eradication rate for the new drug was 86.5% (109/126) compared with 85.3% (110/129) for patients treated with the standard therapy.

In this study, the difference between the eradication rates of the two treatments was 1.2%. The 95% confidence interval extends at the lower limit for the new drug from an eradication rate of 7.3% worse than standard drug, to the upper limit with an eradication rate of 9.7% better.

If we assume that the subjects of the study are representative of a larger population, this means there is a 95% chance that this range of values includes the true difference of the eradication rates of the two drugs. Splitting the remaining 5%, there is an additional 2.5% chance that the new treatment increases the eradication rate by more than 9.7%, and a 2.5% chance that the new treatment decreases the eradication rate by more than 7.3%.

In this case, our goal is to show that the new drug is not worse than the old one. So we can combine our 95% confidence level with the 2.5% upper limit, and say that there is a 97.5% chance that the eradication rate with the new drug is no more than 7.3% worse than the eradication rate with standard drug.

It is conventional, however, to state confidence intervals with 95%, not 97.5%, confidence. We can easily create a one-sided 95% confidence interval. To determine our lower limit, we lower our two-sided confidence interval from 95% to 90% -- extending, as above, both our upper and lower limits by an additional 2.5%.

The 90% CI for difference in eradication rate extends from -5.9% to 8.4%. We can restate this to say that the 95% confidence interval is greater than -5.9%. Thus, we are 95% sure that the new drug has an eradication rate not more than 5.9% worse than that of the standard drug. Note that by expressing the confidence limit in only one direction, we can set the lower limit to a higher value than when we use 95% confidence limits in both directions (5.9% vs 7.3%).

In this example of testing noninferiority, it makes sense to express a one-sided confidence interval as the lower limit only. In other situations, such as toxicology, it can make sense to express a one-sided confidence limit as an upper limit only.

GraphPad Prism does not compute one-sided confidence intervals directly. But, as the example shows, it is easy to create the one-sided intervals yourself. Simply ask Prism to create a 90% confidence interval for the value you care about. If you only care about the lower limit, say that you are 95% sure the true value is higher than that (90%) lower limit. If you only care about the upper limit, say that you are 95% sure the true value is lower than the (90%) upper limit.

1. S. J. Pocock, The pros and cons of noninferiority trials, Fundamental & Clinical Pharmacology, 17: 483-490 (2003).