Science Puzzle
The Treatment That Wins and Loses
A hospital study compared two treatments for kidney stones: traditional open surgery (A) and a newer keyhole technique (B). For patients with small stones, A cured a higher percentage. For patients with large stones, A cured a higher percentage again. But when all the patients were added together, B came out ahead.
Which treatment actually worked better?
The Answer
Treatment A. It did better for small stones and better for large stones. B only looks better overall because it was mostly used on the easy cases.
Look at who got which treatment. Large stones are harder to cure; doctors tended to use open surgery on them: 263 of A's 350 patients had large stones. Keyhole surgery was mostly used on small stones: 270 of B's 350 patients had small ones. So B's overall score is mostly made of easy cases, and A's is mostly made of hard ones. Adding them together compares an easy test with a hard test, and the easy test wins.
This reversal, where a pattern that holds in every group flips when the groups are combined, is called Simpson's paradox. The figures come from a real study published in 1986, and it's now a classic example taught to doctors and statisticians. Nothing in the arithmetic is wrong: the combined numbers are correct, they just answer a misleading question.
The lesson is to compare like with like. Whenever groups differ in something that affects the outcome, here the size of the stones, the combined figures can point the wrong way. Fair trials avoid this by giving treatments at random, so that each treatment gets a similar mix of easy and hard cases.
The principle: Simpson's paradox. A trend that appears in every group can reverse when the groups are combined, if the groups are mixed in different proportions, so always compare like with like.