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Medicine A cures 30% of patients (95% CI: 17 to 45).

Medicine B cures 15% of patients (95% CI: 10 to 20).

So I can divide 30% by 15% and say that medicine A is twice as likely to cure the patients compared to medicine B, right?

My question is: how would I make this same calculation for the confidence intervals?

I want to say that medicine A is twice as likely (95% CI: X to Y) to cure the patient as medicine B. Conceptually, do I just divide the CIs or is there something else to do?

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You can use the Delta method to obtain an approximate distribution of your relative risk, as shown by that link. Then you can define a pivot and use this to obtain a CI.

I understand that there might be some confusion regarding the use of the Delta method, so here are a few simple steps that show how to construct an approximate CI for the relative risk.

  1. Estimate the RR from the data
  2. Find the natural log of RR: $\log(RR)$
  3. The confidence coefficient is from the standard normal distribution: 1.96 for a 95% confidence interval

Now you need the standard error. Using the Delta method for sample sizes $n$ and $m$ with probabilities $p$ and $q$ respectively, this is found to be

$$SE=\sqrt{\frac{1-p}{pn}+\frac{1-q}{qm}}$$

Of course you need to replace the unknown quantities with your estimates, let's denote them by $\widehat{p}$ and $\widehat{q}$. You might notice that this is the second approximation we are using.

Now that you have the formula, compute the standard error: $SE$

  1. Calculate the lower and upper limits on the log scale: $\log(RR) scale: \log(RR) ± 1.96 \times SE \log(RR)$

  2. Exponentiate!

You can find plenty such information throughout the internet and the above steps are taken from here. We all have Fisher to thank for these approximations!

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    $\begingroup$ There's also an R function for the delta method. $\endgroup$
    – Jeremy Miles
    Commented May 20, 2015 at 18:17
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    $\begingroup$ @JeremyMiles I am sure there is. They are in the middle of constructing one that makes coffee too :P $\endgroup$
    – JohnK
    Commented May 20, 2015 at 18:17
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    $\begingroup$ :) inside-r.org/packages/cran/car/docs/deltaMethod $\endgroup$
    – Jeremy Miles
    Commented May 20, 2015 at 18:18
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    $\begingroup$ @Alexander Just in case you come by the need to use the Delta method again-and this is not very unlikely- I recommend studying it now. It's quite easy to use as it is mainly based on a third order Taylor expansion. I will look for the R-function as well. Let me just say that if you were a Bayesian you would be done by now :) $\endgroup$
    – JohnK
    Commented May 20, 2015 at 18:21
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    $\begingroup$ @Alexander I have included some extra details in my answer that I believe illustrate how you can program R on your own. $\endgroup$
    – JohnK
    Commented May 20, 2015 at 18:34

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