Simpson’s Paradox

Simpson’s Paradox

Disclaimer: nothing to do with The Simpsons.

I've been delving into more statistical maths recently and it feels like my intuition is much less useful than with other branches of the subject; it is so paradox ridden. Simpson’s Paradox is infamous in various forms, but I'll present it in one of the classic presentations; the Berkley Applications Figures.

Here is a table of the number of male and female applicants to Berkley in Fall 1973 and they appear to show a gender bias against females:

Applicants Admitted
Male 8442 44%
Female 4321 35%

However, this isn't quite representative of the whole situation. A further study was commissioned and it concluded that there was a statistically significant bias for females. If you break the applications up by department there were more departments that gave an advantage to women rather than men, and the vast majority had no significant bias either way.

What had skewed the results was that in general females had applied for courses such as English and History where places were competitive and the overall acceptance figures were down around 35%, while courses that were male heavy like Chemistry and Engineering tended to have much higher acceptance rates for everyone applying; some around 80%.

Here is a table of the 6 largest departments so that you can see the trend:

Department Applicants Male Admitted Male Applicants Female Admitted Female
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%

Often it is hard to get to grips with this sort of logic, so let's look at a reduced example to see Simpson’s Paradox in its pure form. Alice and Bob like playing chess with strangers online. One week Alice plays a game and loses it, while Bob plays four and manages to win one, giving him a higher win ratio. The next week Alice wins three out of her four games, but Bob manages to get a higher win ratio again by winning his one and only game.

In both weeks Bob performed better, but overall Alice won three out of five, while Bob only won two out of five. This apparent reversal of the trend when you look at the data overall compared with if you look at it grouped is Simpson’s Paradox and it has major implications when doing large studies, particularly in medical trials. 

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