Difference between measuring rates as $E(X/Y)$ vs. $E(X)/E(Y)$?

Question

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?

Answer 1

Recall that the expectation is linear, so

$$ E[a + b X] = a + c E[X] $$

Let's say that $E[X] = 0$, in such a case if we have $Y = cX$, where $c$ is some constant, $E[Y] = E[cX] = c E[X] = c \times 0$, so $E[X]\,/\,E[Y]$ would be undefined regardless of what $c$ is. But we can shift $X$ and $Y$ by a constant $a$, so that $E[X + a] = 0 + a$, and $E[Y + a] = 0 + a$, then $E[X+a]\,/\,E[Y+a] = a \,/\, a = 1$, regardless of what $c$ is. This example shows that comparing expectations to learn $c$ easily breaks apart.

Another problem with comparing expectations is ecological fallacy and the related Simpson's paradox that show how the relations in aggregated vs raw data (or differently aggregated) may be completely different.

I am not saying that comparing expected values does not make sense. I am trying to say that it doesn't necessarily tell you how $X$ and $Y$ are related to each other, just how their averages are related.