Simpson’s paradox example: kidney stone treatments

What it shows

Simpson’s paradox in data from a real medical study: success rates of two kidney stone treatments (Charig and colleagues, 1986), split by stone size as in Julious and Mullee (1994). Treatment A is open surgery; treatment B is percutaneous nephrolithotomy.

Treatment A has the higher success rate for small stones and for large stones, yet the lower rate when the two groups are combined. The reversal arises because A was given mostly for large stones, which are harder to treat.

How to read it

1. 1

   Compare the bars within each stone size first: A is higher for small stones (93% against 87%) and for large stones (73% against 69%).

2. 2

   Then look at the combined bars: B is higher, 83% against 78%.

3. 3

   Check the counts under the bars: 263 of A’s 350 patients had large stones, against 80 of B’s.

4. 4

   Stone size is the confounder: it affected both the choice of treatment and the chance of success.

The data

• Small stones: A 81/87 = 93%, B 234/270 = 87%.
• Large stones: A 192/263 = 73%, B 55/80 = 69%.
• All patients: A 273/350 = 78%, B 289/350 = 83%.
• The totals are those Charig and colleagues (1986) report; the counts by stone size follow Julious and Mullee’s (1994) split and sum to the published totals.

Ways to use it in teaching

• Show only the combined bars first and ask which treatment students would choose, then reveal the split.

• Ask what random assignment would have done to the link between stone size and treatment.

• Set it beside the correlation vs causation diagram: stone size plays the confounder’s role.

Sources

• 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. https://doi.org/10.1136/bmj.292.6524.879
• Dunn, P. K. (2021). Scientific research and methodology. https://bookdown.org/pkaldunn/Book/
• Julious, S. A., & Mullee, M. A. (1994). Confounding and Simpson’s paradox. BMJ, 309(6967), 1480–1481. https://doi.org/10.1136/bmj.309.6967.1480
• Simpson's paradox. (n.d.). In Wikipedia. Retrieved October 18, 2026, from https://en.wikipedia.org/wiki/Simpson%27s_paradox