# Comparison of quantities based on proportion

I would like to compare two experimental quantities in two conditions.

I have two replicated experiments A and B in control condition, where total number of replications is $N$. Number of successes in the experiment A is $m_{control}$, number of successes in experiment B is $k_{control}$. The frequency of successes for the first experiment A is $q_{control}=m_{control}/N$ and for the experiment B is $p_{control}=k_{control}/N.$

I have another set of experiments in different condition, where the number of replication is preserved:N. The only things that change is number of successes, therefore we have $q_{cond1}=m_{cond1}/N$ and $p_{cond1}=k_{cond1}/N$.

I would like to test if the following qualities statistically differ between control and "condition" group of experiments: $q_{control}/p_{control}$ vs $q_{cond1}/p_{cond1}$. Which test you would suggest in this case?

PS. The experimental observations are independent and $N>100$.

Given a flat prior, the posterior probability distribution for $q_{control}$ is $Beta(1+m_{control}, 1+(N-m_{control})$ (conjugate distribution, cf https://en.wikipedia.org/wiki/Conjugate_prior#Discrete_distributions ). Thus it is easy to draw a large number of candidates for $q_{control}$, $q_{cond1}$, $p_{control}$ and $p_{cond1}$ and simply count, how often one ratio is larger then the other. This is one of the examples where Bayesian flexibility makes things actually easy.