Simpson's Paradox
Can something be better for every group and worse overall?
The idea
A trend that holds in every subgroup can reverse when the groups are combined, if the groups differ in size or composition.
In the real world
A treatment better for both mild and severe cases appearing worse overall.
Going deeper
A trend can hold within every subgroup and reverse when the groups are combined, if the groups differ in size or composition. This is not a statistical curiosity; it happens routinely in real data.
The hospital case is the clearest: better survival on both mild and severe cases, worse overall, because one hospital treats far more severe cases. The aggregate is dominated by case mix, so it measures who arrived rather than what happened to them. Which number is correct depends on the question — and choosing whether to aggregate is a modelling decision that can invert the answer.
Where it stops applying
Splitting by every available variable creates its own problems, including small subgroups and multiple comparisons. The point is that aggregation is a choice, not that disaggregated is always right.
Why it matters
It shows that aggregation is a modelling choice which can invert the answer.
Try this today
Before trusting an overall rate, split it by the most obvious grouping.
Test yourself
A hospital has a worse overall survival rate than another, yet does better on both mild and severe cases. How?
Show the answer
It treats a far higher proportion of severe cases, which have lower survival regardless of care. The aggregate is dominated by case mix rather than quality, so combining the groups measures who walked through the door rather than what happened to them.
Learn this in the feed Answering from memory, then again days later, is what makes it stick.