The problem of infra-marginality in outcome tests for discrimination
Open accessAbstract
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 suffer from the problem of infra-marginality: even absent discrimination, the repayment rates for minority and white loan recipients might differ if the two groups have different 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.
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Thresholds cannot be identified from observed data alone, so inferences depend on the Bayesian model's functional form and prior distributions.
“thresholds cannot be identified by the observed data alone 7 , and so inferences are dependent on the specific functional form of the underlying Bayesian model, including the prior distributions.”
From A large-scale analysis of racial disparities in police stops across the United States (2020) · cites this paper as [7]
Section 2.3, “Inferring search thresholds” (page unavailable), specifies a Bayesian model with priors, supporting the claim that inference uses modeling assumptions. But a passage reproduced under “Conclusion” says the model would be identified by priors rather than data *without* certain constraints; it does not say thresholds cannot be identified from observed data under the specified model. The paper also says its results are robust to the exact choice of priors, qualifying the claimed dependence on them. (LLM, conf=0.84)
Experimental — accuracy not yet measuredThe threshold test has been proposed as a more robust means for detecting discrimination than outcome tests.
“the threshold test has been proposed as a more robust means for detecting discrimination 7,22 .”
From A large-scale analysis of racial disparities in police stops across the United States (2020) · cites this paper as [7]
The protected attribute is frequently represented implicitly by some combination of proxy variables, so a classifier trained without it still learns discrimination reflected in training data.
“However, since the protected attribute is frequently represented implicitly by some combination of proxy variables, the classifier still learns the discrimination reflected in training data.”
From Interventional Fairness: Causal Database Repair for Algorithmic Fairness (2019) · cites this paper as [46]
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“∗We thank Cheryl Phillips and Vignesh Ramachandran of the Stanford Computational Journalism Lab for helping to compile the North Carolina traffic stop data, and the John S. and James L. Knight Foundation for partial support of this research.”
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Cai, W., Gaebler, J., Kaashoek, J., Pinals, L., Madden, S. R., & Goel, S. (2022) · PNAS Nexus · doi:10.1093/pnasnexus/pgac144 · Cited by 11 · see citing works
Self-citationOn the site of predictive justice
Lazar, S., & Stone, J. O. H. (2023) · Noûs · doi:10.1111/nous.12477 · Cited by 12 · see citing works
Demystifying Bayesian Inference Workloads
Wang, Y. E., Zhu, Y., Ko, G. G., Reagen, B., Wei, G.-Y., & Brooks, D. M. (2019) · IEEE International Symposium on Performance Analysis of Systems and Software (ISPASS) · doi:10.1109/ispass.2019.00031 · Cited by 11 · see citing works
Algorithmic fairness and resentment
Babic, B., & King, Z. J. (2023) · Philosophical Studies · doi:10.1007/s11098-023-02006-5 · Cited by 10 · see citing works
Fair Models for Impartial Policies: Controlling Algorithmic Bias in Transport Behavioural Modelling
Vega-Gonzalo, M., & Christidis, P. (2022) · Sustainability · doi:10.3390/su14148416 · Cited by 10 · see citing works
Racial Disparities in Motor Vehicle Searches Cannot Be Justified by Efficiency
Feigenberg, B., & Miller, C. (2020) · National Bureau of Economic Research · doi:10.3386/w27761 · Cited by 10 · see citing works
Removing biased data to improve fairness and accuracyPreprint
Verma, S., Ernst, M. D., & Just, R. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2102.03054 · Cited by 19 · see citing works
Detection and Mitigation of Algorithmic Bias via Predictive Parity
DiCiccio, C. J., Hsu, B., Yu, Y., Nandy, P., & Basu, K. (2023) · ACM Conference on Fairness, Accountability, and Transparency (FAccT) · doi:10.1145/3593013.3594117 · Cited by 10 · see citing works
High-frequency location data show that race affects citations and fines for speeding
Aggarwal, P., Brandon, A., Goldszmidt, A., Holz, J. E., List, J. A., Muir, I., Sun, G., & Yu, T. (2025) · Science · doi:10.1126/science.adp5357 · Cited by 9 · see citing works
TESTING ECONOMIC MODELS OF DISCRIMINATION IN CRIMINAL JUSTICEPreprint
Gelbach, J. B. (2021) · SSRN Electronic Journal · doi:10.2139/ssrn.3784953 · Cited by 9 · see citing works
Fairness On The Ground: Applying Algorithmic Fairness Approaches to Production SystemsPreprint
Bakalar, C., Barreto, R. D. C., Bergman, S., Bogen, M., Chern, B., Corbett‐Davies, S., Hall, M. A., Kloumann, I. M., Lam, M. S., Candela, J. Q., et al. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2103.06172 · Cited by 25 · see citing works
Self-citationThe Invisible Power of Fairness. How Machine Learning Shapes Democracy
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Perceptions of Police Conduct When Race and Gender Are Considered
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Probabilistic Machine Learning for Healthcare
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Comparing ‘fair’ machine learning models for detecting at-risk online gamblers
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Fairness Through the Lens of Proportional Equality
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Fairness without Demographics through Shared Latent Space-Based Debiasing
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Driving While Broke: The Role of Class Signals in Police Discretion
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The Many Facets of Data Equity
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Neighborhood Residence and Assessments of Racial Profiling Using Census Data
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Fairness Criteria through the Lens of Directed Acyclic Graphs
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Noise Induces Loss Discrepancy Across Groups for Linear Regression.Preprint
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- Authors
- Simoiu, C., Corbett-Davies, S., & Goel, S.
- Affiliations
- Stanford University (US)
- ORCID
- Goel, Sharad
- Venue
- The Annals of Applied Statistics
- Publisher
- Institute of Mathematical Statistics
- Published
- 2017-09-01
- Volume
- 11
- Issue
- 3
- Pages
- 1193-1216
- ISSN
- 1932-6157 (ISSN-L)
- 1941-7330
- OpenAlex
- W2950220130
- Cited by
- 154
- References
- 30
- Citations by year
- 2026: 6 · 2025: 10 · 2024: 16 · 2023: 21 · 2022: 17 · 2021: 22 · 2020: 28 · 2019: 17 · 2018: 15 · 2017: 2
- FWCI
- 38.87
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- The Annals of Applied Statistics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Stanford Bio-X
- Topics
- Social Sciences › Safety Research › Ethics and Social Impacts of AI
- Social Sciences › Sociology and Political Science › Names, Identity, and Discrimination Research
- Social Sciences › Law › Discrimination and Equality Law
- Keywords
- discrimination detection, threshold test, decision threshold, risk allocation, algorithmic fairness, police stops, racial disparities, loan repayment rates, hiring discrimination, credit discrimination
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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