Overview
In 1986 a study compared two kidney stone treatments. One won for small stones, won for large stones, and lost overall. Both conclusions came from the same table.
How to solve Simpson's Paradox
- Do not average the averages. Look at the group sizes behind each rate.
- Treatment A handled 263 large stones out of 350 patients; treatment B handled 270 small stones out of 350.
- Stone size decides both which treatment a patient received and how likely a cure was, so it confounds the pooled comparison.
The key insight
A pooled average silently answers a different question: 'how did patients who happened to get this treatment do?' rather than 'how well does this treatment work?'. When assignment correlates with severity, only the within-group rates answer the second question.
Variations & echoes
- The 1973 Berkeley admissions data is the other famous case: the university looked biased against women overall, yet most individual departments admitted women at a higher rate.
- Randomised assignment destroys the confounding, which is precisely why controlled trials randomise.
Frequently asked questions
Is the pooled number simply wrong?
It is arithmetically correct but causally misleading here. Which number to trust depends on the causal structure — and when a variable drives both treatment and outcome, you must split on it.