Simpson's paradox

Simpson's paradox (or the Yule-Simpson effect) is a statistical paradox described by E. H. Simpson in 1951 and G. U. Yule in 1903, in which the successes of several groups seem to be reversed when the groups are combined. This seemingly impossible result is encountered surprisingly often in social science and medical statistics.

A kidney stone treatment example

This is a real-life example from a medical study comparing the success rates of two treatments for kidney stone. Kidney Stone example

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The first table shows the overall success rates and numbers of treatments for both treatments.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

This seems to show treatment B is more effective. If we include data about kidney stone size, however, the same set of treatments reveals a different answer.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

small stones large stones

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Treatment A Treatment B Treatment A Treatment B

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

93% (81/87) 87% (234/270) 73% (192/263) 69% (55/80)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~