Solution
Simpson's Paradox
Show the problem again
Treatment A has a higher success rate than Treatment B for small kidney stones AND a higher success rate for large kidney stones. Can Treatment B still have the higher overall success rate when the two groups are combined? Answer yes or no.
Worked solution
Yes, if treatment assignment is confounded with case severity. In the famous kidney stone data, A beat B on small stones (93% vs 87%) and large stones (73% vs 69%), yet B won overall (83% vs 78%), because A was disproportionately given the hard large-stone cases and B the easy small-stone ones. The aggregation weights differ across groups, reversing the comparison. Moral: a lurking variable can invert conclusions, so the subgroup analysis (here, favoring A) is usually the trustworthy one.
Source: Named for Edward Simpson (1951); the effect was described earlier by Udny Yule (1903). Statement written for AxiomIQ.