The Problem of Infra-marginality in Outcome Tests for Discrimination⇤

Outcome tests are a popular method for detecting bias in lending, hiring, and policing decisions. These tests operate by comparing the success rate of decisions across groups. For example, if loans made to minority applicants are observed to be repaid more often than loans made to whites, it suggests that only exceptionally qualified minorities are granted loans, indicating discrimination. Outcome tests, however, are known to su↵er from the problem of infra-marginality : even absent discrimination, the repayment rates for minority and white loan recipients might di↵er if the two groups have di↵erent risk distributions. Thus, at least in theory, outcome tests can fail to accurately detect discrimination. We develop a new statistical test of discrimination—the threshold test—that mitigates the problem of infra-marginality by jointly estimating decision thresholds and risk distributions. Applying our test to a dataset of 4.5 million police stops in North Carolina, we find that the problem of infra-marginality is more than a theoretical possibility, and can cause the outcome test to yield misleading results in practice. ⇤We thank Cheryl Phillips and Vignesh Ramachandran of the Stanford Computational Journalism Lab for helping to compile the North Carolina tra c stop data, and the John S. and James L. Knight Foundation for partial support of this research. We also thank Stefano Ermon, Avi Feller, Seth Flaxman, Andrew Gelman, Lester Mackey, Jan Overgoor, and Emma Pierson for helpful comments. Our dataset of North Carolina tra c stops is available at https://purl.stanford.edu/nv728wy0570, and our analysis code is available at https://github.com/5harad/ threshold-test. This paper is forthcoming in the Annals of Applied Statistics.