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Simpson's paradox

Simpson's paradox

Q: If drug B has a higher success rate (%age of cures) than drug A when given to women, and also when given to men, does it have a higher success rate when given to people in general?
[2 Apr 1997]

A: Not necessarily, e.g.

 WomenMen
 SuccessFailure SuccessFailure
A853145
B3111

Then for women B (75%) is better than A (73%) and for men B (50%) is again better than A (44%), but for people in general A (71%) is better than B (67%).

Q: What's the smallest possible "paradoxical" situation (i.e. smallest total number of people)? There are two versions of the problem depending on whether we allow entries to be 0.
[2 Apr 1997]

A: When I first considered this puzzle I found the following two examples by hand, and wondered whether they were minimal:

Zeros not allowed (20 people):

  WomenMen
SuccessFailure SuccessFailure
A3415
B1114

Zeros allowed (9 people):

  WomenMen
SuccessFailure SuccessFailure
A2101
B1013

In 2011 I found that the first of these was quoted in Impossible? Surprising solutions to Counterintuitive Conundrums by Julian Havil (published 2008, ISBN 978-0-691-13131-3, available from Amazon for example). This inspired me to settle the question with a quick exhaustive computer search.

It turns out that for the zeros-not-allowed version the 20-person example above is not quite the best possible. The minimum possible total is 19, and there are two essentially different solutions:

  WomenMen
SuccessFailure SuccessFailure
A2125
B3213

and

  WomenMen
SuccessFailure SuccessFailure
A2135
B3212

For the zeros-allowed version the 9-person example above is minimal, and is essentially the only solution.
[20 Mar 2011]

Q: Here's another striking version. A greengrocer sells apples at a fixed price per fruit, and oranges similarly. Each day an apple costs more than an orange. I buy fruit on several days. On average, did my apples cost me more per fruit than my oranges?
[20 Mar 2011]

A: Not necessarily. For example:

On Monday apples cost 9p each and oranges cost 8p each, and I buy an apple and two oranges.

On Tuesday apples cost 3p each and oranges cost 2p each, and I buy two apples and an orange.

Overall I bought three apples which cost a total of 15p, and three oranges which cost a total of 18p. So on average my apples cost 5p each but my oranges cost 6p each.

Simpson's paradox can be interpreted as a sign that we're asking the wrong question. In the drugs trial we shouldn't be asking whether drug A is better than drug B, but rather why both drugs are so much more effective on women than on men. At the greengrocer's we shouldn't be asking whether apples cost more than oranges, but rather why the prices of both fruit changed so much on Tuesday.
[20 Mar 2011]

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