Simpson's paradox

What just happened?

Method A treats more effectively in both subgroups — small stones (93.1% vs 86.7%) and large stones (73.0% vs 68.8%). Yet after combining all patients, the higher overall success rate belongs to method B (82.6% vs 78.0%). The trend reversed just by changing the way of looking at the same data. This is Simpson’s paradox.

Comparing proportions — subgroups vs the whole:

that is, in percentages: $93.1\% > 86.7\%$ and $73.0\% > 68.8\%$, and yet $78.0\% < 82.6\%$. The inequality of fractions does not carry over to their “sum by numerators and denominators”: $\tfrac{a}{b}>\tfrac{c}{d}$ and $\tfrac{p}{q}>\tfrac{r}{s}$ does not guarantee that $\tfrac{a+p}{b+q}>\tfrac{c+r}{d+s}$.

Why does it work? (the invariant)

Aggregating data can reverse the trend when subgroups differ in size and in the distribution of a hidden confounding variable. Here the hidden variable is case difficulty: method A was applied mostly to difficult (large) stones — 263 of 350 patients (75%), and method B mostly to easy (small) ones — 270 of 350 (77%). Large stones are harder to cure regardless of method, so A’s overall result is “dragged down” precisely by those difficult cases. The group-composition panel on the right shows this disproportion.

History of the paradox

The phenomenon was described by Edward H. Simpson in 1951, though it had been noticed earlier by Karl Pearson (around 1899) and George Yule (1903) — hence also the name “the Yule–Simpson effect”. It has great practical importance: it warns against jumping to conclusions from aggregated statistics in medicine, sociology, economics and data analysis. The same set of numbers can “prove” opposite conclusions — depending on whether we look at the whole or at the subgroups.

Why you need this for the exam:

This is a lesson in critical thinking about data and proportions. Remember that inequalities of fractions must not be added “numerator to numerator, denominator to denominator”. With every aggregate statistic, ask: does it hide an important split into subgroups of different sizes?

Remember: an advantage visible in every subgroup can disappear or reverse after combining the data — the culprit is a hidden confounding variable and different group sizes. Always ask what an aggregated average conceals.