Track 13 · Resampling and Statistical Reasoning · Lesson 7 of 10
Simpson's paradox
Recognise when an aggregated comparison reverses within subgroups, explain the reversal through a confounder, and choose the comparison that answers the question.
A league table shows Hospital A with a 79% recovery rate against 86% at Hospital B. Yet for mild cases and for severe cases, A does better. Both statements are true at once. Which one to quote depends on the question being asked — and on how the third variable fits into the causal story.
Simpson's paradox occurs when a comparison between groups points one way in every subgroup but the opposite way when the subgroups are combined. Nothing is miscalculated. The reversal happens because the groups have different mixes of subgroups, and the subgroups have very different baseline rates.
The third variable that drives the reversal — here, how severely ill a patient is on admission — is a confounder: it is associated with the group (which hospital treats you) and with the outcome (whether you recover).
Included with a Formula Quest membership
It has 8 graded exercises with feedback, hints and worked solutions. After it you will be able to:
- Compute overall and subgroup rates and spot a Simpson's reversal
- Explain a reversal as confounding by a variable with different mixes across groups
- Compare groups fairly with direct standardisation
- Decide which variables to adjust for: pre-treatment confounders, not mediators or colliders
- Avoid the ecological fallacy: a pattern among group averages need not hold for individuals