Why Proxy Variables Matter
A predictive model can use information about a sensitive attribute even when that attribute is not included as an input. A postal area, a purchasing pattern, or a record of service use may be associated with that attribute and may therefore carry some of the same information. The relevant question is not only whether a model contains a sensitive field, but also what its other inputs reveal and how those inputs affect predictions.
This matters in actuarial work because predictions can shape access, prices, service, or other consequential decisions. A feature that appears operationally neutral may reflect differences in geography, purchasing opportunities, or prior access to services. Identifying that possibility does not by itself establish that a model is unfair, unlawful, or intentionally discriminatory. It does establish a reason to investigate the feature’s meaning and consequences in context.
The distinctions and examples below are aligned with the SOA Exam PA (Predictive Analytics) syllabus. All data and scenarios are hypothetical.
Association, Encoding, and Model Influence
Suppose \(A\) denotes a sensitive attribute and \(X\) a proposed model input. A basic indication that \(X\) may encode information about \(A\) is that the distribution of \(X\) differs across values of \(A\). For a binary \(X\), this can be expressed by comparing \(P(X=1\mid A=1)\) with \(P(X=1\mid A=0)\). If those probabilities differ, learning \(X\) can provide information about \(A\). This comparison does not establish why the association exists.
A feature can also transmit information through a chain. For example, a location may be associated with access to a particular service, and a service-use measure may then reflect both underlying need and access. A model using the service-use measure can inherit information associated with location or group membership without containing either field. Such a chain is a reason to examine how data were generated, not a conclusion about the causal process.
A simple score makes the connection between a proxy and predictions concrete. If a model assigns \(S=\beta_0+\beta_1X\), then for any group \(A=a\), linearity of expectation gives
Thus, if the average value of \(X\) differs between groups and \(\beta_1\ne 0\), the groups’ average scores differ in this model. This calculation describes a model consequence under the stated score formula; it does not, on its own, establish a causal effect or determine whether a decision is appropriate.
Worked Example: A Geographic Feature That Carries Group Information
A hypothetical service model uses a binary location feature \(X\), equal to 1 for residents of North Zone and 0 for residents of South Zone. A separate, sensitive attribute \(A\) is available for a permitted internal review. In 60 fictional records, 24 have \(A=1\) and 36 have \(A=0\). Among the \(A=1\) records, 18 are in North Zone; among the \(A=0\) records, 12 are in North Zone. The model score is \(S=0.10+0.50X\). Calculate the North Zone rate in each group and the mean score in each group.
Method. Calculate each group’s North Zone proportion, then substitute that proportion for \(E[X\mid A]\) in the score formula. Since \(X\) is binary, its mean is the proportion with \(X=1\).
For \(A=1\), there are 18 North Zone records out of 24, so \(E[X\mid A=1]=18/24=0.75\). The mean score is \(0.10+0.50(0.75)=0.10+0.375=0.475\).
For \(A=0\), there are 12 North Zone records out of 36, so \(E[X\mid A=0]=12/36=1/3\). The mean score is \(0.10+0.50(1/3)=0.10+0.1667=0.2667\), rounded to four decimal places.
The North Zone feature is associated with the sensitive attribute in these fictional records, and the model assigns a higher mean score to the group with the larger North Zone proportion. Check: the two North Zone counts sum to 30, and \(30/60=0.50\), consistent with an overall North Zone proportion of one half. The group-specific rates, 0.75 and 0.3333, average to one half when weighted by group sizes. The common mistake is to treat the location label as inherently sensitive or inherently harmless; its meaning and model effect depend on context. The calculation shows association and score differences, not why they arose.
Three Common Routes to Proxy Information
Geography. Postal areas, regions, or distances to facilities may be associated with demographic characteristics, resources, local services, or historical patterns. A geographic feature can be predictive for legitimate operational reasons, but it may also encode group information. The relevant review asks what the location represents in this particular population and decision, rather than assuming that every use of geography has the same meaning.
Purchasing patterns. A product category, order channel, or frequency of purchase may be associated with health, income, household circumstances, or other sensitive information. The association may arise through access, preferences, or a service pathway; a purchasing field is not automatically a direct measure of any one of these. A model may still use it as a signal, even if that signal was not deliberately designed to reveal sensitive information.
Utilization. Counts of visits, claims, calls, or other service interactions can reflect need, but can also reflect access, availability, administrative practices, or earlier service decisions. Interpreting a high count as high underlying need without examining those other possibilities can be misleading. If utilization is used as a predictor, the model may reproduce differences embedded in how the records were generated.
Worked Example: A Purchasing Feature and Expected Scores
In a hypothetical dataset of 100 households, a sensitive attribute \(A\) divides the records into two equally sized groups. For group \(A=1\), 30 households have \(C=1\), meaning they made a particular type of recurring home-delivery purchase, and 20 have \(C=0\). For group \(A=0\), 10 have \(C=1\) and 40 have \(C=0\). A model uses only \(C\), with score \(S=0.08+0.30C\). Calculate the proportion with \(C=1\) and the mean score in each group.
Method. First divide the number with \(C=1\) by the group size. Then use \(E[S\mid A=a]=0.08+0.30E[C\mid A=a]\).
Each group has 50 households. For \(A=1\), \(E[C\mid A=1]=30/50=0.60\). Its mean score is \(0.08+0.30(0.60)=0.08+0.18=0.26\).
For \(A=0\), \(E[C\mid A=0]=10/50=0.20\). Its mean score is \(0.08+0.30(0.20)=0.08+0.06=0.14\). The difference in mean scores is \(0.26-0.14=0.12\).
Check: the total with \(C=1\) is \(30+10=40\), out of 100 households, so the overall proportion is 0.40. The group-specific proportions average to that overall value because the groups are equal in size: \((0.60+0.20)/2=0.40\). In this hypothetical dataset, purchase status carries information about \(A\), and the model uses it to generate different group mean scores. The common mistake is to conclude that the purchase feature directly measures the sensitive attribute. The data show an association, not identity or causation.
Worked Example: Utilization Can Reflect More Than Need
A hypothetical model predicts a future service score from \(V\), the number of recorded visits in a year, using \(S=0.05+0.10V\). For two sensitive groups, E and F, the visit-count distributions are as follows: group E has \(V=0,1,2\) with probabilities \(0.40,0.30,0.30\); group F has \(V=0,1,3\) with probabilities \(0.20,0.30,0.50\). Calculate each group’s expected visit count and expected score.
Method. Find each expected visit count by multiplying each possible count by its probability and summing. Then apply linearity of expectation to the score.
For E, \(E[V\mid E]=(0)(0.40)+(1)(0.30)+(2)(0.30)=0+0.30+0.60=0.90\). Therefore, \(E[S\mid E]=0.05+0.10(0.90)=0.05+0.09=0.14\).
For F, \(E[V\mid F]=(0)(0.20)+(1)(0.30)+(3)(0.50)=0+0.30+1.50=1.80\). Therefore, \(E[S\mid F]=0.05+0.10(1.80)=0.05+0.18=0.23\).
The expected score for E is 0.14, and for F it is 0.23. Check: each set of probabilities sums to 1.00, and the F calculation uses the value 3, not 2, for its highest visit count. A common arithmetic mistake is to report \(E[V\mid F]=1.90\), which would incorrectly produce a score of 0.24. The correct weighted sum is 1.80, so the correct expected score is 0.23. Visit count may reflect underlying need, but it may also reflect access, availability, or past service processes.
Why Removing the Sensitive Field May Not Be Enough
Suppose a model is fitted without \(A\), but retains a feature \(X\) that is associated with \(A\). The model can still use the information in \(X\). Removing \(A\) changes the available inputs; it does not force the remaining inputs to be independent of \(A\), nor does it guarantee that predictions are unaffected by group-associated information.
A second issue is that proxy information can enter indirectly. A derived feature may combine several inputs, and a model may use interactions or transformations that preserve information even when a raw field has been removed. Likewise, a proxy can be introduced during data collection or feature construction rather than selected as an obvious model variable. A review should therefore consider the complete prediction process, including how inputs are defined and obtained.
There is an important limit to what a proxy analysis can establish. An association between \(X\) and \(A\) does not prove that someone intended \(X\) to stand in for \(A\), and it does not identify the cause of the association. Conversely, lack of an obvious direct relationship does not establish that a feature is harmless. Interpretation requires context about the population, the decision, and how the data were generated.
A Practical Review of Potential Proxies
Specify what prediction or action is affected and which sensitive attributes are relevant in that context. The same feature can have different implications in different uses.
For a location, purchase, or utilization feature, identify how it was recorded and what processes may influence it. Do not rely only on the feature’s short name in a data dictionary.
Where collection and use are permitted, compare feature distributions across relevant groups and consider how the feature might carry information about them. An observed association is a signal to interpret, not a causal finding.
Determine whether the feature is used directly or through a transformation, and whether its contribution changes the model output in the intended decision context. Document important assumptions and limitations.
Depending on the purpose and constraints, a response might include revising a feature, restricting its use, collecting better context, or seeking further review. Removing a feature is not automatically the right remedy: it can change predictive performance while leaving other proxies in place.
Information about sensitive attributes should be handled with appropriate authorization, safeguards, and data-minimization practices. A team may need suitable information to understand potential associations, but that does not imply that every model user should have access to it. The decision about whether a feature is acceptable also depends on the applicable legal, regulatory, and organizational requirements.
Common Mistakes
- Assuming that omission means no influence. A model without a sensitive field may still use correlated features. Consider the information in the remaining predictors, not just the input list.
- Treating correlation as proof of cause or intent. An association shows that a feature carries information in the observed data; it does not explain why or how the association arose.
- Assuming a neutral-sounding feature is harmless. Geography, purchases, and utilization can have consequential meanings that are not apparent from their labels.
- Assuming that every proxy must be removed. A proxy concern calls for contextual review. Any change should consider the decision, available alternatives, model consequences, and relevant requirements.
- Interpreting utilization as need alone. Recorded service use can reflect access and prior processes as well as underlying need.
Check Your Understanding
For each problem, identify the relevant proxy issue and show the calculation where requested.
- In a fictional sample, 15 of 50 residents in District H have a sensitive attribute \(A=1\), while 25 of 50 residents in District L have \(A=1\). Calculate \(P(A=1\mid H)\) and \(P(A=1\mid L)\). What does the difference indicate about district as a possible proxy? (Answer: The rates are \(15/50=0.30\) and \(25/50=0.50\); district is associated with \(A\) in these data, but the difference alone does not establish why.)
- A purchase indicator \(C\) has \(P(C=1\mid A=1)=0.70\) and \(P(C=1\mid A=0)=0.20\). A model uses \(S=0.04+0.20C\). Calculate the expected score for each group and the difference. (Answer: \(E[S\mid A=1]=0.04+0.20(0.70)=0.18\); \(E[S\mid A=0]=0.04+0.20(0.20)=0.08\); difference \(=0.10\).)
- For group G, a hypothetical visit count \(V\) is 0, 1, or 2 with probabilities 0.50, 0.25, and 0.25. For group H, it is 0, 2, or 4 with probabilities 0.20, 0.30, and 0.50. If \(S=0.02+0.05V\), calculate each expected score. (Answer: \(E[V\mid G]=0.75\), so \(E[S\mid G]=0.0575\); \(E[V\mid H]=2.60\), so \(E[S\mid H]=0.1500\).)
- A model’s input list excludes a sensitive attribute but includes a postal-area code that is associated with it. Explain why the omission alone cannot establish that the model’s predictions are independent of the sensitive attribute. (Answer: The postal-area code may carry information about the attribute, so the model can still use associated information through that input.)
- A team finds that a service-use feature differs between groups and concludes that the difference proves deliberate use of a sensitive attribute. Is that conclusion justified? State a more careful interpretation. (Answer: No. The difference is evidence of an association in the observed data, not proof of intent or cause; access, need, and prior processes are possible explanations to investigate.)