Simpson's paradox: the safer hospital with worse overall survival

Question

Two hospitals report patient survival rates (survivors / admissions), split by how severe each patient's case was on arrival:

	Routine cases	Critical cases	Overall
Mercy	190/200 (95%)	560/800 (70%)	750/1000 (75%)
City	850/900 (94.4%)	65/100 (65%)	915/1000 (91.5%)

Mercy has higher survival on routine cases (95% vs 94.4%) and on critical cases (70% vs 65%), yet City's overall survival rate is far higher (91.5% vs 75%).

Which hospital is safer, and explain the paradox.

Accepted Answer

This is Simpson's paradox. The confounder is case severity, which drives survival enormously (about 95 percent for routine cases vs 65 to 70 percent for critical), and the two hospitals see opposite case mixes: - Mercy is a trauma center: 800 of its 1000 patients arrive critical. - City handles mostly routine cases: 900 of its 1000. City's higher overall survival reflects who walks through its doors, not better care. The overall rates are severity-weighted averages: 	ext{Mercy: } 	frac{200}{1000}(0.95) + 	frac{800}{1000}(0.70) = 0.75, \qquad 	ext{City: } 	frac{900}{1000}(0.944) + 	frac{…

Simpson's paradox: the safer hospital with worse overall survival

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