Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 34 / 1 / f / Solution

As a rough rare-event calculation, divide the observed elective-group event count by the reported adjusted odds ratio:For a homogeneous illustrative risk, the exact odds conversion starts from $p_1=9/1515$ and givesThe corresponding baseline risk is about $0.22\%$, compared with $1.62\%$ in the observed reference group, roughly seven times lower. Thus taking the adjusted result literally implies exceptionally low background risk among the elective patients.

This is a diagnostic approximation, not an identified mean of potential outcomes obtainable from the aggregate table. A conditional odds ratio cannot generally be inverted at an aggregate mean: individual background risks are needed for outcome standardization. Interpreting the calculation causally also requires conditional exchangeability, the positivity assumption and a correctly specified outcome model. At very small risks the rare-event approximation makes the estimate of roughly three events useful for checking the plausibility of the adjustment.