Why Removing a Proxy Is Not the Same as Fixing the Data
A model can use a variable that stands in for a sensitive attribute even when that attribute is absent. A postal area, for example, may reflect patterns in access to services or past decisions. Deleting that variable may change predictions, but it does not necessarily remove the underlying pattern: other inputs may carry similar information, and the data may still reflect an uneven process of observation or outcome recording.
A different response is to examine how the data were generated. Were outcomes recorded consistently across groups? Did people have comparable opportunities to apply, receive a service, or have an event observed? Could a more direct, consistently collected measure replace a proxy? These questions concern the process producing the data, not just the list of model inputs.
A correlation between \(Z\) and \(G\) is a reason to investigate, not by itself a conclusion that \(Z\) is inappropriate. Its meaning depends on the intended decision, the way \(Z\) was produced, and what other variables remain. Conversely, a model that excludes an explicitly sensitive attribute can still reproduce group differences through proxies or through biased labels.
What Proxy Removal Can and Cannot Establish
Removing a feature can reduce some group disparities, but it can also remove predictive information. It does not guarantee that the remaining features are unrelated to group membership or that group error rates will become equal. Those outcomes must be measured on held-out data and interpreted in the context of the intended use. The earlier material on statistical parity, equalized odds, predictive parity, and calibration provides distinct ways to describe group results; none should be silently substituted for another.
One basic prediction result helps explain the possible accuracy tradeoff. Suppose the target is numeric, the evaluation uses squared-error loss, and the best possible predictor is available for each information set. Write \(m(X,Z)=E[Y\mid X,Z]\) and \(m(X)=E[Y\mid X]\). If \(X\) is contained in the information \((X,Z)\), then:
To see why, write \(Y-m(X)=[Y-m(X,Z)]+[m(X,Z)-m(X)]\), square, and take expectations. The cross-product has expectation zero because \(E[Y-m(X,Z)\mid X,Z]=0\), while \(m(X,Z)-m(X)\) is determined by \(X,Z\). Thus, the predictor using the extra information cannot have larger population squared-error risk when both conditional means are known. The second term is nonnegative and is the information about the conditional mean that \(Z\) adds beyond \(X\).
This result is limited. It concerns the best population predictors under squared error, not necessarily fitted models on finite samples, and it says nothing about whether using \(Z\) is appropriate, fair, lawful, or useful for the decision. Removing a harmful or unsuitable proxy may be justified even if predictive error rises. A carefully changed data process may instead provide more decision-relevant and consistently measured information. Neither choice should be judged by prediction accuracy alone.
Worked Example: Removing a Geographic Proxy
Worked Example: Removing a Geographic Proxy
A hypothetical insurer uses a model to select policies for a manual review intended to identify files needing additional verification. A postal-area variable is suspected of acting as a proxy for group membership. On a held-out sample, each group has 40 positive outcomes and 60 negative outcomes. Compare a model with the postal-area variable to one that omits it. The counts below show positive predictions as true positives (TP) and false positives (FP), and negative predictions as false negatives (FN) and true negatives (TN).
| Model and group | TP | FN | FP | TN |
|---|---|---|---|---|
| Postal area included, A | 32 | 8 | 12 | 48 |
| Postal area included, B | 24 | 16 | 8 | 52 |
| Postal area omitted, A | 28 | 12 | 10 | 50 |
| Postal area omitted, B | 26 | 14 | 10 | 50 |
Method. Calculate accuracy using all 100 policies in each group. Calculate the true positive rate (TPR) using the group’s positive outcomes as the denominator, and the false positive rate (FPR) using its negative outcomes. Calculate the selection rate as the number predicted positive divided by 100.
Postal area included. Group A accuracy is \((32+48)/100=0.80\); its TPR is \(32/40=0.80\), its FPR is \(12/60=0.20\), and its selection rate is \((32+12)/100=0.44\). Group B accuracy is \((24+52)/100=0.76\); its TPR is \(24/40=0.60\), its FPR is \(8/60=0.1333\), and its selection rate is \((24+8)/100=0.32\). Overall accuracy is \((80+76)/200=0.78\).
Postal area omitted. Group A accuracy is \((28+50)/100=0.78\); its TPR is \(28/40=0.70\), its FPR is \(10/60=0.1667\), and its selection rate is \((28+10)/100=0.38\). Group B accuracy is \((26+50)/100=0.76\); its TPR is \(26/40=0.65\), its FPR is \(10/60=0.1667\), and its selection rate is \((26+10)/100=0.36\). Overall accuracy is \((78+76)/200=0.77\).
The selection-rate difference falls from \(0.44-0.32=0.12\) to \(0.38-0.36=0.02\), while overall accuracy falls by \(0.78-0.77=0.01\). The TPR difference also narrows, from \(0.80-0.60=0.20\) to \(0.70-0.65=0.05\). This is evidence about these particular measures on this sample; it does not establish that all fairness concerns have been resolved.
Common mistake and correction. Dividing Group B’s 8 false positives by all 100 Group B policies gives 8%, but that is not its FPR. The FPR denominator is the 60 policies with negative outcomes, so the correct rate is \(8/60=0.1333\). The final check is that each row sums to 100 and that the positive-outcome denominators are 40 in both groups.
The deletion changed both group results and overall accuracy, but it did not change how the target outcome was generated. Before choosing between these models, the decision maker would need to establish that the target is measured consistently and that a referral for manual review is justified by the intended purpose.
When the Data-Generation Process Is the Problem
Suppose a model is trained to predict whether a customer needs assistance, but the target is recorded only when a customer completes a follow-up process. If completion rates differ between groups, the recorded outcomes may not describe the same underlying population in the same way. Deleting a correlated input does not make unrecorded outcomes appear, nor does it make the recorded target comparable.
Changing the process might mean offering follow-up through consistent channels, recording the same outcome definition for everyone, or measuring a more direct feature with a standard procedure. Such a change can improve the information available for model development and evaluation. It also requires care: a new process may alter who responds or what is measured. Model evaluation should use a well-defined target and a suitable evaluation sample, rather than assuming the old and new labels mean the same thing.
Worked Example: Unequal Outcome Recording
A hypothetical support program wants to predict which customers would benefit from a check-in. In an initial dataset of 100 customers from Group A, the outcome is recorded for 60, of whom 24 are recorded as needing support. In Group B, the outcome is recorded for 40, of whom 20 are recorded as needing support. A later audit uses the same standardized follow-up protocol for all 100 customers in each group and records 30 needing support in each group. Compare what the two data practices show.
Initial recorded rates. Among customers with recorded outcomes, Group A’s rate is \(24/60=0.40\), and Group B’s is \(20/40=0.50\). The difference is \(0.50-0.40=0.10\), or 10 percentage points. But the outcome is missing for \(100-60=40\) customers in A and \(100-40=60\) in B. The rates therefore describe different subsets of the two groups.
Standardized follow-up rates. Under the later protocol, Group A’s rate is \(30/100=0.30\), and Group B’s is \(30/100=0.30\). The observed difference is now zero percentage points. The recorded positive counts in the original data do not contradict the audit: the original denominators excluded customers without recorded outcomes.
The calculation shows why changing the outcome-recording process can change the evidence used to train or evaluate a model. It does not prove that every customer’s underlying need was measured perfectly, or that the groups have identical needs in every population. A sound interpretation depends on the follow-up protocol being applied consistently and on the target matching the support decision.
Common mistake and correction. Comparing \(24/100=0.24\) with \(20/100=0.20\) treats customers with unrecorded outcomes as if they were known not to need support. That is not what the data say. The correct initial comparison among observed outcomes is \(24/60\) versus \(20/40\), accompanied by the warning that outcome observation differs. The later audit offers a more comparable denominator under its stated protocol.
Judge Changes Against the Intended Decision
The right comparison depends on what a prediction triggers. A small change in average error may matter greatly if errors cause a serious adverse consequence; a less accurate model may be preferable if it avoids an inappropriate use of a proxy. For an assistance or review decision, false negatives can mean missed support, while false positives can consume limited staff time or burden customers. State these consequences before treating one metric as decisive.
A data-practice change can also produce a more useful alternative to a suspected proxy. For example, instead of relying on a broad location category, a hypothetical property-review model might use a consistently collected inspection measure directly related to the review need. That substitute must itself be assessed: it may be costly, inaccurate, or unevenly available. The point is not that more data are always better, but that the data should correspond to the intended decision and be generated in a defensible way.
Worked Example: Compare Two Review Policies by Decision Cost
A hypothetical program uses a model to offer property-safety inspections. A positive outcome means an independent assessment found a condition warranting inspection. Each group has 30 positive and 70 negative outcomes in a common held-out sample. Policy R omits a suspected neighborhood proxy. Policy M uses a consistently collected, direct inspection-screening measure instead. For this illustration, each missed positive outcome has a stated cost of $300, and each false positive has a stated cost of $20. These are hypothetical decision costs, not estimates from an insurer or industry.
| Policy and group | TP | FN | FP | TN |
|---|---|---|---|---|
| R, A | 18 | 12 | 8 | 62 |
| R, B | 14 | 16 | 8 | 62 |
| M, A | 21 | 9 | 10 | 60 |
| M, B | 20 | 10 | 10 | 60 |
Method. For this stipulated cost calculation, total cost equals \(300\) dollars times the total number of false negatives plus \(20\) dollars times the total number of false positives. This simple calculation assumes the stated costs apply equally to both groups and that other costs are outside the comparison.
Policy R. Total false negatives are \(12+16=28\), and total false positives are \(8+8=16\). Its cost is \(28(300)+16(20)=8{,}400+320=\$8{,}720\). Its TPRs are \(18/30=0.60\) for A and \(14/30=0.4667\) for B.
Policy M. Total false negatives are \(9+10=19\), and total false positives are \(10+10=20\). Its cost is \(19(300)+20(20)=5{,}700+400=\$6{,}100\). Its TPRs are \(21/30=0.70\) for A and \(20/30=0.6667\) for B. The cost difference is \(\$8{,}720-\$6{,}100=\$2{,}620\) in favor of M under the stated assumptions.
Policy M has more false positives, but the lower number of false negatives gives it a lower stipulated total cost. Its TPR difference is about \(0.70-0.6667=0.0333\), compared with \(0.60-0.4667=0.1333\) for R. These calculations support considering M in this hypothetical setting, not adopting it automatically. The direct measure must be feasible, consistently collected, and relevant to the actual inspection decision. Costs should also be challenged with decision makers and affected stakeholders.
Common mistake and correction. Choosing the policy with fewer false positives alone would favor R, which has 16 rather than 20. That ignores the much larger stipulated cost of a missed positive outcome. Applying both costs gives the correct comparison: $8,720 for R and $6,100 for M. The check is that every group has 100 records and each has 30 positives and 70 negatives.
A Practical Comparison
When a proxy is suspected, frame the analysis as a comparison of possible changes rather than a single yes-or-no feature decision. Keep the model purpose, outcome definition, evaluation population, and decision rule clear. If those change at the same time as the predictor set, it becomes difficult to attribute a performance or group-outcome difference to any one cause.
Identify what a positive prediction causes, who is affected, and what false positives and false negatives mean in context.
Ask what \(Z\) measures, how it relates to \(G\), whether it reflects a relevant condition or past access, and whether other inputs carry similar information.
Compare definitions, recording procedures, missingness, and opportunities for observation across the groups being audited.
Evaluate proxy inclusion, proxy removal, and any feasible change to measurement or data collection on appropriate held-out evidence. Report predictive performance and relevant group results separately.
Explain which decision consequences improved or worsened, what assumptions matter, and what evidence remains uncertain. A better metric on one sample does not by itself establish an acceptable policy.
An essential trap is to describe proxy deletion as “making the model fair” without specifying what changed or what was measured. Removing \(Z\) can narrow one selection-rate difference while leaving unequal outcome observation untouched. Changing the outcome-recording process can improve comparability while leaving a proxy in the feature set. The analyses answer different questions, so a responsible recommendation says which problem is being addressed and what evidence supports the choice.
Check Your Understanding
For each problem, identify what the calculations establish and what they do not establish.
- A hypothetical model has 50 positive and 50 negative outcomes in each of two groups. With a suspected proxy included, Group A has TP 40, FN 10, FP 15, TN 35; Group B has TP 30, FN 20, FP 10, TN 40. After removing the proxy, A has TP 35, FN 15, FP 10, TN 40; B has TP 32, FN 18, FP 10, TN 40. Calculate the selection rates before and after deletion and the overall accuracies. (Answer: Selection rates change from 55% versus 40% to 45% versus 42%; overall accuracy changes from 72.5% to 73.5%. This does not establish that all fairness concerns are resolved.)
- A service records outcomes for 45 of 75 Group A customers, with 18 positive outcomes, and for 30 of 75 Group B customers, with 15 positive outcomes. Under a later standardized follow-up, 21 of 75 in each group have positive outcomes. Calculate the initial observed rates and the later rates. Why is dividing the initial positive counts by 75 misleading? (Answer: Initial observed rates are \(18/45=40\%\) and \(15/30=50\%\); later rates are \(21/75=28\%\) in both groups. Dividing by 75 treats unrecorded outcomes as known negatives.)
- A review policy has 12 false negatives and 6 false positives. A second policy has 8 false negatives and 11 false positives. If each false negative costs $250 and each false positive costs $30, calculate each total cost and identify the lower-cost policy under these assumptions. (Answer: First policy: \(12(250)+6(30)=\$3{,}180\). Second policy: \(8(250)+11(30)=\$2{,}330\). The second is lower by $850 under the stated costs.)
- A team removes a location variable and finds that the group selection-rate difference falls from 9 percentage points to 2 percentage points. It has not checked whether the target outcome was recorded consistently across groups. What can it conclude, and what should it investigate next? (Answer: It can report the measured selection-rate change on the evaluated sample, but cannot conclude that the data process or all fairness concerns are fixed. It should investigate outcome and input recording, relevant group errors, and the decision consequences.)
- A new direct measure is collected consistently and improves held-out predictive performance, but it is more expensive to obtain and changes the set of customers with available data. Name two considerations before recommending it. (Answer: Assess whether the measure is relevant and feasible for the intended decision, and whether its cost or availability changes coverage or group outcomes; evaluate on evidence appropriate to the new data process.)