Fairness Metrics: Demographic Parity, Equalized Odds, and Trade-offs › Worked Example: Calculating Group Fairness Metrics
Language AI HandbookAll chapters Fairness Metrics: Demographic Parity, Equalized Odds, and Trade-offs › Worked Example: Calculating Group Fairness MetricsOne section · Sections in this chapter3 / 13 ↓
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Let's build intuition with a concrete example. Imagine a simplified hiring classifier that predicts whether a candidate will be hired () for two demographic groups. We will compute each fairness metric by hand to make the relationships concrete.
Suppose we have the following data, recorded as a confusion matrix split by group:
| Group | True label | Predicted | Count |
|---|---|---|---|
| A=0 | Y=1 | Ŷ=1 | 40 |
| A=0 | Y=1 | Ŷ=0 | 10 |
| A=0 | Y=0 | Ŷ=1 | 20 |
| A=0 | Y=0 | Ŷ=0 | 30 |
| A=1 | Y=1 | Ŷ=1 | 15 |
| A=1 | Y=1 | Ŷ=0 | 25 |
| A=1 | Y=0 | Ŷ=1 | 5 |
| A=1 | Y=0 | Ŷ=0 | 55 |
From this table we can compute all the rates:
- Group 0 (100 total, 50 truly positive): positive prediction rate = (40+20)/100 = 60%, TPR = 40/50 = 80%, FPR = 20/50 = 40%, Precision = 40/60 = 67%
- Group 1 (100 total, 40 truly positive): positive prediction rate = (15+5)/100 = 20%, TPR = 15/40 = 37.5%, FPR = 5/60 = 8.3%, Precision = 15/20 = 75%
Computing the fairness metrics:
- DPD = 0.20 - 0.60 = -0.40. Group 1 is selected at a rate 40 percentage points lower than group 0.
- DPR = 0.20 / 0.60 = 0.33. Well below the EEOC four-fifths threshold of 0.80; this would constitute adverse impact.
- Equal opportunity difference = |0.375 - 0.80| = 0.425. Among truly qualified candidates, group 1 is identified at a rate 42.5 percentage points lower. Qualified group 1 candidates are being passed over at a dramatically higher rate.
- Equalized odds difference = max(|0.083 - 0.40|, |0.375 - 0.80|) = max(0.317, 0.425) = 0.425. The larger disparity drives the equalized odds metric, coming from the TPR gap.
- Predictive parity difference = |0.75 - 0.67| = 0.08. Precision is close but not identical; a positive prediction for group 1 is slightly more reliable.
This example illustrates a pattern common in practice: the model simultaneously fails to find true positives in group 1 (low TPR) while producing too many false positives for group 0 (high FPR). These failures have different downstream harms. Group 1 candidates miss real opportunities they have earned. Group 0 candidates who do not deserve the positive outcome receive it, which may dilute program quality or create unfairness toward group 0 members who are displaced by less-qualified peers.
Notice that this model comes relatively close to predictive parity (0.08 difference in precision) even though it badly violates demographic parity and equalized odds. This is not a coincidence, and the mathematical reason for it will become clear when we examine the impossibility theorem.