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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.

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    $\begingroup$ 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? $\endgroup$ – whuber Sep 12 '13 at 20:57
  • $\begingroup$ 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. $\endgroup$ – Placidia Sep 13 '13 at 14:05
  • $\begingroup$ 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. $\endgroup$ – whuber Sep 13 '13 at 14:21
  • $\begingroup$ 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. $\endgroup$ – Placidia Sep 13 '13 at 16:32
  • $\begingroup$ 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. $\endgroup$ – whuber Sep 13 '13 at 16:55
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Thanks to @whuber's suggestions, I think I know what to do. The confidence interval (or region) contains those parameter values that would not be rejected were they the subject of a null hypothesis. I used a Fisher exact test and since the data represent the most extreme possibility (no failures in one arm), the test statistic for a given pair of parameters is simply the likelihood evaluated at those points. I can evaluate the likelihood over a grid of parameter pairs and select those with likelihood > 0.05 (i.e. not in the rejection region, were I doing a test). From these, I can calculate the ratio and the difference. The spread of values gives me the confidence intervals I need for the parameter functions of interest: relative risk and absolute difference.

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