The drug that wins every subgroup but loses overall

Question

Two drugs were tested on the same illness, which comes in mild and severe forms.

Drug A cured 9 of 10 mild patients (90%) and 40 of 100 severe patients (40%).
Drug B cured 72 of 90 mild patients (80%) and 6 of 20 severe patients (30%).

Drug A has the higher cure rate among mild patients and among severe patients. Yet drug B cured a larger fraction of all its patients.

Show that drug B has the higher overall cure rate, and give that overall rate.

Accepted Answer

Compare the two drugs within each group first: mild, A wins 90\% to 80\%; severe, A wins 40\% to 30\%. So A is better for every individual patient. Now pool each drug's own patients. 	ext{Drug A overall} = \frac{9+40}{10+100} = \frac{49}{110} \approx 44.5\%. 	ext{Drug B overall} = \frac{72+6}{90+20} = \frac{78}{110} = \frac{39}{55} \approx 70.9\%. So drug B wins overall, 70.9\% against 44.5\%, even though it lost in both subgroups. The reversal comes from the case mix. Drug A was given mostly to severe patients (100 of its 110), where cure rates are low for everyone, dragging its average do…

The drug that wins every subgroup but loses overall

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