# Confidence interval for a relative risk when one proportion is 0

I am analysing data from a small study. There are two treatment groups and for each, one notes success and failures. For one of the groups, the number of successes is 0. I did an exact significance test, but reviewers of the paper want a confidence interval for the relative risk - in other words, a confidence interval for $p_1/p_2$, where $p_1=0$.

What should I do? The usual formula assumes that neither proportion is 0 or 1.

• Just to be clear, it seems you want a CI for $p_1/p_2$ where the estimate $\hat{p}_1$ is $0$. Because you did an exact significance test, you should be able to use essentially the same calculations to obtain an exact confidence interval. What formula, then, did you use? – whuber Sep 12 '13 at 20:57
• I used Fisher's exact test ... but I don't see how I would turn that into an expression in p1/p2. I am tending to the opinion that it makes no sense to put a CI around a relative risk where one of the risks is 0. – Placidia Sep 13 '13 at 14:05
• Why should a basic concept like a confidence interval fail to apply for specific values of the data? The answer is that it cannot: a CI makes perfect sense no matter what you observe. – whuber Sep 13 '13 at 14:21
• Good point ... but I am still trying to put a confidence interval around the true porportion p1/p2, whatever that is, when the estimate of p2 is 0. – Placidia Sep 13 '13 at 16:32
• That's right (which is why I have upvoted this question). You can compute a CI, and there are many ways to do it ranging from using a generalized linear model to a permutation test. I therefore anticipate seeing some interesting and useful answers appear. – whuber Sep 13 '13 at 16:55