I'm working with four populations consisting of true/false events. They each have a different mean and variance. I have samples from each, with different sample sizes. Call the percentage of observed true events to samples $p_{11}$, $p_{12}$, $p_{21}$, and $p_{22}$. What statistical test could I use to test the hypothesis that $\frac{p_{12}}{p_{11}} > \frac{p_{22}}{p_{21}}$?
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Assume they are four independent beta random variables $X_i$, with means $\mu_i$. Note that the density functions would depend on the observed samples. We could then test the hypothesis that $\mu_1\mu_2>\mu_3\mu_4$ by setting up a quadruple integral of the joint density function over the set $\{ (x_1,x_2,x_3,x_4)\mid x_1x_2>x_3x_4 \}$. If this integral is small, we reject the hypothesis $\mu_1\mu_2>\mu_3\mu_4$. 

