Simpson's paradox

Simpson's paradox occurs when an association seen in aggregated data disappears or reverses after the data are divided into groups, often because group sizes differ.

Confounding: A distortion in the estimated relationship between variables caused by a third variable related to both. A confounder can shape group membership and outcomes, producing the reversal.
UC Berkeley gender bias admissions case: A 1973 analysis of graduate admissions at the University of California, Berkeley, examining sex differences across departments. Department-level admission patterns differed from the aggregate pattern that suggested bias against women.
Association (statistics): A statistical relationship in which the values of two variables vary together. The paradox concerns an association whose direction depends on how data are grouped.
Berkson's paradox: A spurious association between variables created by conditioning on a common effect or selection criterion. Both can involve conditioning, but Berkson's paradox centers on selection-induced association.
Aggregation bias: Error in inference caused by combining data across groups whose properties or relationships differ. Pooling unlike groups can conceal or reverse their individual associations.
Kidney stone treatment paradox: A comparison of two kidney-stone treatments whose success rates reverse when cases are separated by stone size. Stone size changes the treatment comparison and produces opposite aggregate and subgroup rankings.
Contingency table: A table that displays counts for combinations of categorical variables. Many demonstrations compare subgroup tables with the table formed by combining them.
Collider bias: Bias induced by conditioning on a variable caused by two other variables. It can create or alter associations after subgrouping, through a specific causal structure.
Stratification: Dividing observations into subgroups according to one or more variables before analysis. Comparing strata reveals whether the aggregate association holds within groups.
Treatment paradox: A reversal in the apparent effectiveness of a treatment when outcomes are examined across different patient groups. This broader class includes clinical examples of Simpson-like reversals.

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