Simpson’s paradox

Hospital B has a higher survival rate in both severe and mild cases — yet Hospital A’s overall survival rate looks better. How is that possible?

The answer: Hospital A treats mostly mild cases (which have high survival), while Hospital B takes on mostly severe cases. When you pool the groups, the case mix drowns out the per-group advantage.

Severe cases

30%

40%

A B B wins

Mild cases

70%

80%

A B B wins

Overall

62%

48%

A B A wins ← paradox!

Hospital A case mix: 20% severe, 80% mild

Hospital B case mix: 80% severe, 20% mild

Paradox active: B outperforms A in both subgroups, yet A looks better overall — because A treats proportionally more mild cases.

A: severe survival 30% A: mild survival 70% B: severe survival 40% B: mild survival 80% A: % severe cases 20% B: % severe cases 80%

What to notice

Default settings show the paradox. Hospital B is 10% better in both subgroups, yet Hospital A’s pooled rate is higher — because A’s patients are 80% mild, B’s are 80% severe.
Equalise the case mix. Drag both ”% severe cases” sliders to the same value and the paradox disappears. The per-group winner becomes the overall winner.
Flip the advantage. Make A better in both subgroups but give A the harder case mix. The paradox flips direction.

Why it matters

Simpson’s paradox is not just an academic curiosity. It has misled researchers in:

Medicine — UC Berkeley’s famous gender bias lawsuit (1973) showed the university appeared to favour men in admissions, but nearly every individual department favoured women. Men had applied to the less competitive departments.
Sports — A player can have a higher batting average than a rival in every individual season, yet a lower career average.
Epidemiology — Crude mortality rates can favour a treatment that harms every patient subgroup if the treatment group is younger.

The fix is always the same: identify the confounding variable (case severity, department selectivity, age) and control for it. Aggregate statistics without stratification are a trap.