 Later we will show that the positive association in the subpopulations cannot vanish if the correlation of treatment with gender is broken (e.g., by balancing gender rates in both conditions). The weights in each line are then identical—\(p(\r{M}\mid \r{T}) = p(\r{M}\mid \neg \r{T})\)—and associations in subpopulations are preserved for the aggregate data  (Theorem 1 in Section 2.2).  In fact, the absence of such a correlation rules out Simpson’s Paradox.