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

Your answer

More Probability questions