I'm working on a difference-in-differences (DiD) analysis for a healthcare study measuring # hospitalizations (X) per # eligible months (Y).
Simplifying the math a bit - my actuary colleagues like to measure the DiD in the aggregate as sum(X)/sum(Y) = E(X)/E(Y) vs. the member-level rate I prefer (the average treatment effect), average of (X/Y) = E(X/Y).
I know via Jensen's Inequality that E(X)/E(Y) <= E(X/Y) given 1/Y is convex and X and Y are independent (actually, a fair assumption in this case).
Question: Which method is preferred for estimating the DiD treatment effect? Is one more "correct" than the other, or are they simply addressing different questions?