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I want to perform a test for differences between 2 binomial populations. The probabilities are small (usually less than 10 %). I can define the successes as "rare events".

I think that the z-test for comparing means is not suitable since the binomial distribution is a lot different than the normal distribution in case of extremely low probabilities. What is the best test to use?

gung - Reinstate Monica
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TalG
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  • How many observations did you use for computing each proportion ? i.e. what are the denominators ? –  Aug 19 '15 at 08:11
  • It depends. Usually the number of observation is big (minimum 1000). The number of successes is relatively small and this is the reason the proportion is small. Thanks (By the way this is Tal G) – Tudmotu Aug 19 '15 at 08:27
  • Well if the number is large enough, I think that you can work with the normal approximation for the Binomial distribtion, for small Binomials $X_1$ and $X_2$, you can compute the exact distribution of $X_1 - X_2$ (or even another function than '-') and then use this distribution to do the hypothesis test (assuming independence is a reasonable assumption) –  Aug 19 '15 at 09:28
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    Please register & merge your accounts (you can find out how in the **My Account** section of our [help]), then you will be able to comment on & edit your own question. – gung - Reinstate Monica Aug 19 '15 at 09:34
  • Did you happen to check [link](http://stats.stackexchange.com/q/51494/67822) I think it will answer the question (?) – Antoni Parellada Aug 19 '15 at 14:26
  • Hi Antoni, Yes :), just now (thanks a lot) The thing is that I'm still not sure what is the answer for my questions, there is 1 recommendation for exact tests like fisher, and 1 comment that says that the Z test is not that bad because the mean is approximately equal to 50 Did I mess something? Thanks :) – TalG Aug 19 '15 at 14:42
  • @whuber: I can't add comments to a deleted answer and as you seem to know the topic well I want to ask whether this may work: if $H_0$ is true then $p_1=p_2$ so I can pool and use the pooled $p$. With the $p_1=p_2$ the pooled variable is $bin(p;n_1+n_2)$ and I can use the exact probabilities of this to do hypothesis tests? –  Aug 21 '15 at 05:10

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