Conditional estimates for OR

When two independent samples from separate population are collected, a product binomial likelihood can be used. However, conditioning on margins, a different formulation is achieved, as follows (shai being population OR):

Comparing Risks

To evaluate whether an exposure is at all contributing to risk of disease, the simplest form of comparison is as follows:
x1 ~ Binomial(n1, pi1)
x2 ~ Binomial(n2, pi2)