Why One Fairness Number Is Not Enough
A model can have similar true-positive rates across groups and still flag different proportions of those groups. It can also have similar flag rates while missing more actual positive cases in one group. These findings are not necessarily contradictory: they answer different questions. A decision-maker needs to know what each measure means, which outcome and population it describes, and what trade-offs follow from using it.
This tutorial uses a hypothetical claim-review model. A positive prediction means that a claim is flagged for additional review; the observed positive outcome means that the claim is later confirmed to require that review. The numbers are illustrative, not data from an insurer. The same reasoning applies in other settings, but the meaning and consequences of a positive prediction must be defined for each decision.
Measures That Answer Different Questions
The following measures are often useful, but they are not interchangeable. Each rate is calculated within a group. For example, a true-positive rate for group \(g\) is the number of true positives in that group divided by the number of actual positives in that group.
Demographic parity compares selection rates across groups. It asks whether groups are flagged at similar rates, regardless of their observed outcome rates. Equal opportunity compares true-positive rates: among cases with the positive outcome, does the model flag similar proportions in each group? Equalized odds requires both true-positive rates and false-positive rates to be similar across groups. Predictive parity compares positive predictive values: among cases flagged, is the proportion with the positive outcome similar across groups?
These are statistical descriptions, not self-justifying ethical rules. For example, equal opportunity gives particular attention to missed positive cases, while demographic parity focuses on the overall frequency of a decision. Which harm matters most depends on the application and affected people. Even the choice of \(Y\) matters: a recorded outcome may be an imperfect measure of the underlying condition that a decision-maker cares about.
Worked Example: Reading Several Measures From Group Counts
A hypothetical review model is evaluated on 200 claims from each of two groups. In group A, 40 claims have the positive observed outcome and 160 do not. The model flags 32 of the 40 positive claims and 16 of the 160 negative claims. In group B, 20 claims have the positive outcome and 180 do not; the model flags 12 positive claims and 9 negative claims. Calculate each group’s true-positive rate, false-positive rate, selection rate, and positive predictive value.
Method. Calculate each measure from its own denominator. The TPR denominator is the number of actual positives; the FPR denominator is the number of actual negatives; selection rate and PPV use the number of all claims and the number flagged, respectively.
Group A. There are 32 true positives and \(40-32=8\) false negatives. There are 16 false positives and \(160-16=144\) true negatives. Therefore,
Group B. There are \(20-12=8\) false negatives and \(180-9=171\) true negatives. The model flags \(12+9=21\) claims. Thus,
The TPRs and FPRs differ, so these results do not show equalized odds. The selection rates and PPVs also differ. This does not, by itself, identify the reason for the differences or determine which policy is acceptable. The evaluation reports what happened on these cases under this threshold; interpretation requires context about the outcome, consequences, and intended use.
Common mistake avoided: Dividing group A’s 32 true positives by all 200 claims gives a rate of 0.16, but that is not its TPR. The TPR asks what fraction of actual positive cases were flagged, so its denominator is 40 and its value is 0.80.
Why Fairness Measures Can Conflict
The conflict between selection rate and error-rate measures can be shown directly. Let \(p_g=P(Y=1\mid G=g)\) be the prevalence of the observed positive outcome in group \(g\). Every case is either positive or negative, so the selection rate is the weighted sum of the group’s TPR and FPR:
This identity follows by dividing all members of group \(g\) into the disjoint cases \(Y=1\) and \(Y=0\), then adding the probability of a flag within each case, weighted by the case’s prevalence. It applies when the rates and prevalence refer to the same population and outcome definition. If TPR and FPR are equal across groups but prevalence differs, selection rates will generally differ. Therefore equalized odds does not automatically imply demographic parity.
Worked Example: Equal Error Rates, Different Flag Rates
Suppose two groups have different observed positive-outcome prevalence: \(p_A=0.20\) and \(p_B=0.10\). At a chosen threshold, the model has a TPR of 0.80 and an FPR of 0.10 in both groups. Calculate the selection rate and PPV in each group. State which measures are equal.
Selection rates. For group A, the positive cases contribute \(0.80(0.20)=0.16\) to the selection rate, while negative cases contribute \(0.10(0.80)=0.08\). The total is \(0.16+0.08=0.24\). For group B, the contributions are \(0.80(0.10)=0.08\) and \(0.10(0.90)=0.09\), for a total of 0.17.
Positive predictive values. Among selected cases, the positive-outcome share is the positive-case contribution divided by the total selection rate. Thus,
Both groups have equal TPRs and equal FPRs, so their error-rate measures match. Their selection rates and PPVs do not. The arithmetic does not show that either group should have a different threshold; it shows why a report should not claim that all fairness measures are satisfied merely because TPR and FPR match.
Common mistake avoided: Treating the shared TPR of 0.80 as the selection rate ignores the false-positive cases. The correct selection rate includes both positive and negative cases, as in \( \text{TPR}_g p_g+\text{FPR}_g(1-p_g) \).
Translate the Trade-Off Into Decision Terms
A decision-maker needs more than a ranking of metrics. The report should connect each metric to the decision and its consequences, then distinguish the observed quantities from choices that require judgment. A practical sequence is:
State what a positive prediction triggers, what \(Y=1\) means, and which population was evaluated. A review flag, denial, and payment are different decisions with different consequences.
Report whether a comparison concerns all cases, actual positive cases, actual negative cases, or flagged cases. A rate without its denominator is easy to misread.
For example, one policy may reduce missed positive cases in a group but generate more false alarms there. State both the measured change and its operational meaning.
Choosing which harms to prioritize, what differences are tolerable, and whether a proposed policy is appropriate are ethical and governance judgments, not outputs of the metric calculation.
Where costs can reasonably be quantified, they can help describe consequences, but they do not settle every fairness question. Costs may omit burdens that are difficult to monetize, and different stakeholders may reasonably disagree about priorities. The numbers should therefore be presented as decision inputs, not as proof that a policy is fair.
Worked Example: Comparing Two Review Policies
A hypothetical organization compares two thresholds using 1,000 claims from each of two groups. Group A has 200 positive and 800 negative outcomes; group B has 100 positive and 900 negative outcomes. Policy X has TPR 0.80 in both groups, with FPR 0.20 in A and 0.10 in B. Policy Y has TPR 0.70 and FPR 0.10 in both groups. For illustration, assume each false negative has an estimated cost of $500 and each false positive creates $30 of unnecessary review cost. Calculate the selection rates and total error costs for both policies. Then explain the trade-off without declaring a fairness winner.
Policy X, group A. The number of true positives is \(0.80(200)=160\), so there are \(200-160=40\) false negatives. False positives number \(0.20(800)=160\). The selection rate is \((160+160)/1000=0.32\).
Policy X, group B. There are \(0.80(100)=80\) true positives and \(100-80=20\) false negatives. There are \(0.10(900)=90\) false positives. The selection rate is \((80+90)/1000=0.17\). Total false negatives are \(40+20=60\), and total false positives are \(160+90=250\). The illustrative total cost is
Policy Y, group A. There are \(0.70(200)=140\) true positives and \(200-140=60\) false negatives. False positives number \(0.10(800)=80\). The selection rate is \((140+80)/1000=0.22\).
Policy Y, group B. There are \(0.70(100)=70\) true positives and \(100-70=30\) false negatives. False positives number \(0.10(900)=90\). The selection rate is \((70+90)/1000=0.16\). Total false negatives are \(60+30=90\), and total false positives are \(80+90=170\). The illustrative total cost is
Policy X has equal TPRs but unequal FPRs, so it does not satisfy equalized odds. It has lower illustrative total cost under the stated assumptions, but its selection rates differ by 0.15 (0.32 versus 0.17). Policy Y has equal TPRs and FPRs, but more false negatives overall and a higher illustrative total cost. Its selection rates still differ, at 0.22 versus 0.16, because the outcome prevalence differs.
The calculation is checked by summing group-specific counts: for either policy there are 2,000 claims in total, with 300 positive outcomes and 1,700 negative outcomes. The rates and costs are based on those denominators. A decision-maker would still need to consider whether the assumed costs represent the relevant harms, whether the outcome is suitable, and what differences are acceptable. The cost total alone cannot answer those questions.
What to Report—and What Not to Claim
A clear report separates three kinds of statements:
- Quantitative finding: “At the evaluated threshold, the TPR was 0.80 in group A and 0.60 in group B.” This reports a defined comparison on evaluated data.
- Interpretation: “The model flagged a smaller proportion of observed positive cases in group B.” This describes what the rate difference means in context.
- Judgment or governance choice: “The difference is acceptable,” or “the organization should adopt group-specific thresholds.” These are decisions requiring reasons, authority, and consideration of consequences; they are not established by the arithmetic.
Avoid saying that a model is “fair” because one measure is close across groups. Say which measure was evaluated, how it was defined, and what it does and does not establish. If metrics conflict, present the conflict rather than hiding one measure. Explain which outcomes are prioritized and why, and identify decisions that remain unresolved. The aim is not to make every measure agree; it is to make the evidence and the choices understandable.
A useful summary, aligned with the SOA Exam PA (Predictive Analytics) syllabus, is to treat fairness measures as evidence about specified outcomes and decisions. They can quantify differences and help compare policies. They cannot, by themselves, determine the ethically correct outcome or substitute for explicit governance choices.
Check Your Understanding
For each problem, calculate the requested quantities before checking the final answer.
- In group A, 30 of 100 cases have a positive outcome; the model flags 24 of those and 14 of the 70 negative cases. In group B, 20 of 100 cases are positive; the model flags 16 of those and 8 of the 80 negative cases. Find each group’s TPR, FPR, and selection rate. Which of these measures match across groups? (Answer: Both TPRs are 0.80; FPRs are 0.20 and 0.10; selection rates are 0.38 and 0.24. Only the TPRs match.)
- Two groups have outcome prevalence 0.25 and 0.10. A model has TPR 0.72 and FPR 0.08 in both. Find the selection rate in each group. (Answer: 0.24 in the first group and 0.144 in the second.)
- A group has 50 positive outcomes and 450 negative outcomes. At a chosen threshold, TPR is 0.60 and FPR is 0.04. Find the true positives, false negatives, false positives, selection rate, and PPV. (Answer: 30 true positives, 20 false negatives, 18 false positives, selection rate 0.096, and PPV 0.625.)
- Two groups each contain 1,000 cases. Group A has 200 positive outcomes and 800 negative outcomes; group B has 100 positive outcomes and 900 negative outcomes. Policy P has TPR 0.80 in each group, with FPR 0.10 in A and 0.04 in B. Find the two selection rates and state whether equalized odds and demographic parity hold. (Answer: Selection rates are 0.24 in A and 0.116 in B. Equalized odds does not hold because FPRs differ; demographic parity does not hold because selection rates differ.)
- A proposed policy reduces false negatives but increases false positives for one group, while making the TPRs more similar. What additional information should a report present before a decision-maker chooses the policy? (Answer: Define the outcome, population, threshold, and denominators; quantify both changes and their operational consequences; state any relevant cost assumptions and limitations; and identify which harms to prioritize and what differences are acceptable as explicit judgment or governance choices.)