Statistics Multivariate analysis Counterexample

Simpson's paradox

Builds on Contingency table Conditional probability

Counterexample

The claim refuted is this. If treatment $A$ has a higher success rate than treatment $B$ in every subgroup, then $A$ has the higher rate in the whole population.

Consider two subgroups, mild and severe cases, summarised in a contingency table for each subgroup.

Mild cases: $A$ succeeds in $9$ of $10$ patients ( $90\%$ ), $B$ in $80$ of $100$ patients ( $80\%$ ).
Severe cases: $A$ succeeds in $30$ of $100$ patients ( $30\%$ ), $B$ in $2$ of $10$ patients ( $20\%$ ).
All cases: $A$ succeeds in $39$ of $110$ patients (about $35\%$ ), $B$ in $82$ of $110$ patients (about $75\%$ ).

So $A$ is better in each subgroup, yet $B$ is better overall.

Remarks

The reason is that the overall rate is a weighted average of the subgroup rates, and the weights differ between the treatments. Most patients given $A$ are severe cases, and most patients given $B$ are mild cases. In terms of conditional probability, $P(S \mid A, G) > P(S \mid B, G)$ for each subgroup $G$ does not imply $P(S \mid A) > P(S \mid B)$ . The same reversal can happen for correlations: a trend in each group can be reversed when the groups are pooled. Which comparison is relevant depends on the causal structure, not on the numbers alone.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Tag	`T060JHSH05`
URI	`m2://T060JHSH05`
Kind	counterexample
Writing	W0 stub
MSC	62H17, 62H20
Uses	Contingency table, Conditional probability
Same as	Wikidata Q757290 Wikipedia Simpson's paradox

Simpson's paradox

Counterexample

Remarks

Reading path

Further reading

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