Let's say we have k binomial variables, and a sample size of n observations per variable, where each variable occurs (positive case) at a given frequency/probability f. We would like to check if there is a correlation between these variables, so the null hypothesis assumes that they are all independent (no correlation).

Is there a formula to calculate the minimum sample size n required to identify a "significant" correlation (p <= alpha) between these variables?

That is, how are n, k, f and alpha linked to each other mathematically?

Are any other parameters involved in the formula, e.g., beta, $R^2$, etc.?

I have a feeling that this relates to power and/or sample size calculations for multiple logistic regression, but I'm really not sure. Could the Wald test or likelihood-ratio test be used here?

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    $\begingroup$ What exactly is involved in your "multiple regression," given you have assumed these variables are independent? Why do you need to identify a correlation when you have explicitly assumed there is none? $\endgroup$ – whuber Mar 22 '16 at 21:37
  • $\begingroup$ Sorry, I should clarify that the null hypothesis assumes there is no correlation, but that is what we are trying to test. (original post now edited). $\endgroup$ – Kelvin Mar 22 '16 at 21:38
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    $\begingroup$ Thank you. Since it's a little strange to be analyzing vectors of binomial variables using correlation, could you explain the purpose of this and perhaps describe what these variables represent? $\endgroup$ – whuber Mar 22 '16 at 21:42
  • $\begingroup$ Yes, for example, how many individual human genomes (n) do we need to study to see if there is a correlation between k genetic mutations, where each mutation occurs at the same frequency f? $\endgroup$ – Kelvin Mar 22 '16 at 21:45

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