Which hospital is safer?

Question

Two hospitals report patient survival, split by how sick patients were on arrival.

Hospital A: 90 of 100 mild patients survive (90%), and 200 of 400 severe patients survive (50%).
Hospital B: 800 of 900 mild patients survive (88.9%), and 40 of 100 severe patients survive (40%).

Hospital A has the higher survival rate for mild patients and for severe patients. Yet hospital B has the higher overall survival rate.

Show that hospital B has the higher overall survival rate, and give that rate. Which hospital would you choose as a patient?

Accepted Answer

Within each severity group, A wins: mild 90\% to 88.9\%, severe 50\% to 40\%. So for any individual patient, A gives a better chance. Now pool each hospital's patients. 	ext{Hospital A overall} = \frac{90+200}{100+400} = \frac{290}{500} = 58\%. 	ext{Hospital B overall} = \frac{800+40}{900+100} = \frac{840}{1000} = 84\%. Hospital B looks safer overall, 84\% to 58\%, yet it is worse for both mild and severe patients. The explanation is patient mix. Hospital A is a specialist center treating mostly severe cases (400 of 500), whose survival is low no matter where they go, so A's overall rate is…

Which hospital is safer?

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