Your firm hires candidates who score well on either mathematics or programming. Across all applicants the two scores are independent, yet among hires their correlation is -0.4. Explain the mechanism, and say what it does to a regression of on-desk performance on mathematics score fitted on hires only.

Your firm hires candidates who score well on either mathematics or programming. Across all applicants the two scores are independent, yet among hires their correlation is -0.4. Explain the mechanism, and say what it does to a regression of on-desk performance on mathematics score fitted on hires only.

Approach: Identify what the hiring rule conditions on relative to the two scores, then trace the induced dependence into the omitted variable formula for the hires-only regression.

Selection on a collider: hiring depends on both scores, so conditioning on being hired induces negative dependence between two variables that are independent in the applicant pool. If a hire needs a high total or a high maximum, then among hires a weak mathematics score can only appear alongside a strong programming score, which manufactures the -0.4. Nothing about the candidates changed; the sample was chosen on a common effect of both variables. A regression of on-desk performance on mathematics score fitted on hires alone inherits the distortion. If performance truly depends on both scores with positive coefficients, then omitting programming from a hires-only regression carries omitted variable bias equal to the programming coefficient times a negative slope, pushing the mathematics coefficient toward zero or below it. The estimate then appears to show that mathematics does not matter, on a sample built to say so. This is the same mechanism as range restriction in a validity study, and the fixes are to control for the selection variables, to model selection with a Heckman correction, or to keep a small random-acceptance stratum.

Follow-up: You keep a random-acceptance stratum of 20 hires a year. How would you use it to correct the coefficient estimated on the other 200?

Key concepts: collider bias, selection effect, omitted variable bias, range restriction.