Collider bias: SAT and GPA among admitted students

Question

At a selective college, SAT scores and high-school GPA are weakly positively correlated across the full applicant pool. Yet among admitted students the correlation is negative: admits with top SATs tend to have lower GPAs and vice versa. An admissions officer concludes that test-takers and hard workers are different types of student.

Explain why admission can flip the correlation from positive to negative, and why the officer's conclusion is unwarranted.

Accepted Answer

This is collider bias (Berkson's paradox). Admission is a common effect of both SAT and GPA, and selecting on it distorts, even reverses, their correlation. The mechanism. The college admits a student if their overall strength (roughly SAT plus GPA) clears a bar: a high SAT can compensate for a lower GPA, and a high GPA can compensate for a lower SAT. Among admitted students, then, a low GPA must be offset by a high SAT to have cleared the bar, and a low SAT by a high GPA. That compensation is a negative correlation within the admitted group. The bar is strong enough here to overwhelm the mil…

Collider bias: SAT and GPA among admitted students

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