I have found the following formulae (from http://www.itl.nist.gov/div898/handbook/prc/section2/prc242.htm) to estimate the sample size required to test the binomial proportion in a single sample.

I would like to know if this is the best formula, and if $1/delta$ is really the best continuity correction, when the expected proportion is extremely low ($p_0$ < 10^-12). Is there a better, more robust formula and/or correction factor which I should use in this case?

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    $\begingroup$ Sorry but I don't see the point to this question. You seem to be asking someone else to provide you with risk-free confidence and security in the solution you've referenced. Like everything else in statistics, there are controversies, differing points of view handed down by differing traditions, training and education which can lead to differing interpretations of what is "best." In other words, there is no fixed, immutable standard or ground truth for an "optimal" anything. This leaves you with an uncertainty that can only be dealt with in terms of building on the plausible and defensible. $\endgroup$ – Mike Hunter Mar 30 '16 at 13:20
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    $\begingroup$ DJohnson, I guess I am asking whether this formula I found is fit for purpose (to test ultra low proportions), and whether there is any consensus around what might be a better formula. If you don't know, then don't answer, but I think it's a perfectly reasonable question to ask given that's exactly what I want to know after doing my best to get to the answer myself. Thanks. $\endgroup$ – Kelvin Mar 30 '16 at 13:25
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    $\begingroup$ Being reasonably familiar with the sampling literature, I'm not aware of any "formulas" for sampling rare events. In other words, the solution you've cited conforms to standard "best" practices. In general, this issue is treated on the applied side where the general Rx is to oversample those rare events to compensate for the sparsity of the information. $\endgroup$ – Mike Hunter Mar 30 '16 at 13:32
  • $\begingroup$ Then, in the post-hoc analysis, reweighting the information back to the appropriate proportions in the population. $\endgroup$ – Mike Hunter Mar 30 '16 at 13:48
  • $\begingroup$ Then if others agree (and unless somebody can suggest a clear improvement) - perfect answer, basically what I wanted to know, thanks. $\endgroup$ – Kelvin Mar 30 '16 at 14:08

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