Meaning
Simpson's paradox occurs when data split into subgroups and the same data pooled together point to opposite conclusions. The British statistician Edward H. Simpson described it in 1951, though Karl Pearson and Udny Yule had noted similar effects earlier.
Classic examples
- Kidney stone treatment (Charig et al., 1986): open surgery (A) had higher success rates than percutaneous nephrolithotomy (B) for both small and large stones, yet B looked better overall (A 78%, B 83%) because the harder large-stone cases mostly received A.
- Berkeley graduate admissions (Bickel et al., 1975): overall, women appeared to be admitted at a lower rate, but within most departments there was no gap or a small one favoring women. Women had applied more often to highly competitive departments.
Why it happens
A third variable (a confounder) affects the outcome and is linked to group assignment. When group sizes differ sharply, the pooled rate is a weighted average that can flip direction.
How to interpret it
Whether the pooled or the stratified answer is right cannot be decided by the numbers alone; it depends on the causal structure. Judea Pearl argues that causal diagrams tell us which variables to adjust for. Randomization, stratified analysis and regression adjustment reduce confounding, which matters when reading studies for evidence-based practice.
References
- Simpson, E. H. (1951). The interpretation of interaction in contingency tables. Journal of the Royal Statistical Society, Series B, 13(2), 238-241.
- Bickel, P. J., Hammel, E. A., & O'Connell, J. W. (1975). Sex bias in graduate admissions: Data from Berkeley. Science, 187(4175), 398-404.
- Charig, C. R., Webb, D. R., Payne, S. R., & Wickham, J. E. (1986). Comparison of treatment of renal calculi by open surgery, percutaneous nephrolithotomy, and extracorporeal shockwave lithotripsy. BMJ, 292(6524), 879-882.
- Pearl, J. (2014). Comment: Understanding Simpson's paradox. The American Statistician, 68(1), 8-13.