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Fairness Metrics: A Comparative Analysis
Pratyush Garg, John Villasenor, Virginia Foggo
TL;DR
Assessing algorithmic bias requires choosing quantitative fairness measures, but their relationships and compatibility can be difficult to compare. This paper puts widely used measures into a common mathematical framework and derives their relationships, showing that compatibility can depend on base rates and threshold choice.
Problem
The growing set of fairness measures makes it difficult to determine how they relate and when they are compatible, complicating choices for system assessment.
Source: §1 · p. 1Method
The paper describes widely used fairness metrics in a common mathematical framework and derives relationships among them, including conditions for compatibility.
Source: §1 · p. 2, §3.1 · p. 5Results
Pairwise combinations of binary-prediction metrics can be satisfied only under limited conditions; with unequal base rates, statistical parity and predictive parity require inverse ratios of true positive rates and base rates.
Source: §4 · p. 9Takeaways & Limitations
Metric selection can account for practical observability: in loan approvals, positive predictive value may be easier to evaluate than statistics requiring outcomes for rejected applicants.
Source: §4 · p. 9In loan approvals, outcomes for rejected applicants are unobserved, making false negatives, true negatives, negative predictive value, and related rates impossible to observe.
Source: §4 · p. 9
Abstract
from arXiv · showhide
Algorithmic fairness is receiving significant attention in the academic and broader literature due to the increasing use of predictive algorithms, including those based on artificial intelligence. One benefit of this trend is that algorithm designers and users have a growing set of fairness measures to choose from. However, this choice comes with the challenge of identifying how the different fairness measures relate to one another, as well as the extent to which they are compatible or mutually exclusive. We describe some of the most widely used fairness metrics using a common mathematical framework and present new results on the relationships among them. The results presented herein can help place both specialists and non-specialists in a better position to identify the metric best suited for their application and goals.
1 Introduction
Algorithmic fairness requires quantitative measures of fairness, but the growing number of measures makes their relationships and compatibility difficult to assess. This paper compares widely used measures in a common framework and derives relationships among them.
1 Introduction
- Algorithm designers and users must choose which fairness measures to use when assessing systems, against the background of recognized concerns about algorithmic bias.The paper frames this as a need to connect normative conceptions of fairness with quantitative measurement.Source: §1 · p. 1
- Prior work showed that calibration within groups and balance for both classes cannot all be satisfied simultaneously except in highly constrained cases [2].Other cited work discusses conflicts between accuracy and fairness and disparate impact when recidivism prevalence differs across groups [3].Source: §1 · p. 1
- The paper presents widely used fairness measures in a common mathematical framework and derives relationships that clarify their compatibility and trade-offs.The framework also addresses variations in terminology and definitions.Source: §1 · p. 2
2 Fairness metrics
The paper defines fairness metrics for binary predictions and outcomes, and for continuous scores that may be thresholded into predictions. It then distinguishes the measures by what they compare and what conditions they impose.
2.1 Definitions
- The framework considers two groups, binary predictions and outcomes, and, for some metrics, a real-valued score that can be thresholded; its loan example uses small samples and assumes statistical significance.The positive class is y = 1, though a positive outcome need not be desirable; reliable real-world fairness assessments require much larger samples.Source: §2.1 · pp. 2–3
2.2 Equalized odds and equality of opportunity
- Equalized odds requires equal true positive and false positive rates across groups; equality of opportunity requires only equal true positive rates.Equalized odds implies equality of opportunity, but not vice versa.Source: §2.2 · p. 3
2.3 Statistical parity
- Statistical parity requires equal positive-prediction rates across groups without considering outcomes, so differing base rates rule out a perfect predictor satisfying it.In the loan example, the positive-prediction rates are 0.6 for the orange group and 0.3 for the blue group.Source: §2.3 · p. 3
2.4 Predictive parity
- Predictive parity requires equal positive predictive values (PPVs) across groups; some authors additionally require equal negative predictive values (NPVs).In the loan example, PPV is 7/9 = 0.77 for the orange group and 8/12 = 0.66 for the blue group.Source: §2.4 · p. 4
2.5 Calibration
- Calibration requires equal probabilities of a positive outcome across groups for each score; well-calibration additionally requires each score to equal that probability.Well-calibration implies calibration, but calibration does not imply well-calibration; a suitable transformation can convert calibrated scores to well-calibrated scores.Source: §2.5 · p. 4
2.6 Balance for positive/negative class
- Balance for the positive or negative class requires equal average scores across groups within that class, generalizing equalized odds to non-binary scores.When scores are limited to 0 and 1, the class-specific averages correspond to true and false positive rates.Source: §2.6 · p. 4
3 Comparisons and trade-offs between metrics
The section uses a common mathematical framework to examine when fairness metrics can coexist, focusing on groups with unequal base rates. It identifies both incompatibilities and conditions that permit simultaneous satisfaction, sometimes with practical costs.
3 Comparisons and trade-offs between metrics
- The section frames metric compatibility as a question about when multiple fairness notions can be satisfied together, analyzing the case of unequal group base rates.It uses a common mathematical framework to examine trade-offs among statistical parity, equalized odds, and predictive parity.Source: §3 · pp. 4–5, §3.1 · p. 5
3.1.1 All three?
- With unequal base rates, statistical parity, predictive parity, and equalized odds cannot all hold, even with a perfect predictor.The proof shows that equalized odds and statistical parity force a positive-prediction-rate difference proportional to the base-rate difference.Source: §3.1.1 · p. 5
3.1.2 Statistical parity and predictive parity
- Statistical parity and predictive parity can coexist with unequal base rates, but a large base-rate ratio difference forces one group’s TPR very low; adding NPV parity generally makes the pair infeasible.The latter, more constrained definition is called “overall predictive parity”.Source: §3.1.2 · p. 6
3.1.3 Equalized odds and predictive parity
- Without perfect prediction, equalized odds and predictive parity require equal base rates; equal opportunity can coexist with predictive parity at unequal rates, but requires differing FPRs.Large base-rate differences may require substantial FPR variation, which practitioners should weigh against application needs.Source: §3.1.3 · p. 6
3.1.4 Equalized odds and statistical parity
- With unequal base rates, statistical parity and equalized odds can coexist only when TPR equals FPR, a condition the paper describes as not particularly useful.The paper’s rationale is that predictors typically aim for TPR substantially higher than FPR.Source: §3.1.4 · p. 7
3.2 Predictive parity and calibration
- At threshold 0.49, calibrated scores yield PPV 0.66 for orange and 0.583 for blue because thresholded groups can have different score distributions.The example shows that equal calibration across scores does not ensure equal PPV after thresholding.Source: §3.2 · pp. 7–8
- Standard statistical parity can coexist with calibration and well-calibration despite unequal base rates and imperfect prediction; threshold choice can alter binary parity.The section distinguishes fairness measures for continuous scores from those for thresholded binary predictions.Source: §3.2 · p. 9
4 Discussion and conclusions
With unequal base rates, fairness metrics can be incompatible or compatible only under restrictive conditions, and some compatible combinations may be problematic for policy. The discussion also highlights threshold choice and practical observability as considerations in selecting metrics.
4 Discussion and conclusions
- With unequal base rates, pairwise combinations of statistical parity, equalized odds, and predictive parity are often incompatible or possible only under limited conditions.Equalized odds and predictive parity are incompatible; statistical parity and equalized odds require TPR = FPR, while statistical parity and predictive parity constrain the ratio of group TPRs to the inverse base-rate ratio.Source: §4 · p. 9
- Calibrated scores, once thresholded, may still yield statistical parity, but whether they do can depend on the threshold.Source: §4 · p. 9
- In loan approvals, only positive predictive value may be readily observable because repayment outcomes are unavailable for applicants predicted not to repay.False negatives, true negatives, negative predictive value, TPR, and TNR cannot be observed in this setting.Source: §4 · p. 9