Simpson’s Paradox

Simpson’s Paradox is one of the classic paradoxes, made famous by the UC Berkeley’s experience,

In 1973, the admission rate for men and women were noted to be very different for UC Berkeley. The admission rate for men was 44% and for women was 35%, The difference was statistically different.

Applicants Admitted
Men 8442 44%
Women 4321 35%

However, when investigated, it turned out that the admissions rate by department was actually higher for women in most cases. It was that women were applying to the competitive programs more than men as illustrated below.

Department Men Women
Applicants Admitted Applicants Admitted
A 825 62% 108 82%
B 560 63%  25 68%
C 325 37% 593 34%
D 417 33% 375 35%
E 191 28% 393 24%
F 373  6% 341 7%

Simpson’s paradox is very common and well-recognized. It illustrates the danger of drawing consluions on average, without knowing the overall distribution of the sample.

One real-life example comes from a study of kidney stones. A new treatment that used a percutaneous (by catherter) procedure to treat kidney stones was shown to be less effective vs. open surgery when considered for small and large stones separately. However, overall, it was better when the small and large stones were combined.

Treatment A Treatment B
Group 1
93% (81/87)
Group 2
87% (234/270)
Group 3
73% (192/263)
Group 4
69% (55/80)
Both 78% (273/350) 83% (289/350)

The reason for this is because there was a selection bias– the patients with large stones tended to be more likely to be treated with the percutaneous therapy.

Simpson’s Paradox is tangentially similar to Blotto Game.