Simpson's paradox

Sometimes a trend that appears in every group reverses when the groups are combined. This is Simpson's paradox, and it is not a trick: it happens in real data.

The Berkeley case

In 1973, the University of California, Berkeley's graduate admissions looked biased against women: about 44% of male applicants were admitted, against about 35% of female applicants.

But when Bickel and colleagues looked department by department, most departments admitted women at a similar or slightly higher rate than men.

How can both be true?

Women had applied more often to departments that were very competitive and rejected most applicants of either sex. Men applied more often to departments with high admission rates. The overall gap came from where people applied, not from how departments treated them.

The hidden factor (department) is a confounder: it is linked to both the group (sex) and the outcome (admission).

Why it matters for you

• An ML model can look better overall, yet worse on every subgroup, if the test sets have different mixes of easy and hard cases.
• A treatment can look harmful overall because sicker patients were more likely to get it.

What to do

• Look at results by subgroup as well as overall.
• Ask what could differ between groups besides the thing you're studying.
• Use causal reasoning, not just the numbers, to decide which view answers your question.

RememberWhen groups are combined, trends can flip. Always ask what hidden factor might differ between the groups.