Why Exposure Mappings Need Sensitivity Analysis
Under interference, one person’s outcome may depend on treatment assigned to other people. An exposure mapping summarizes that assignment—for example, by the number of treated neighbors or whether any neighbor is treated. This summary makes analysis possible, but it is also a substantive assumption: assignments given the same summary must be sufficiently alike for the outcome contrast of interest.
A mapping can omit relevant peers, group together meaningfully different neighborhood configurations, or assign the wrong exposure label. Comparing results across plausible mappings is a useful diagnostic, but the differences alone do not show which mapping is correct. Sensitivity bounds add a more explicit question: under stated restrictions on the omitted or misclassified information, how far could the population spillover contrast move?
It is important to distinguish two issues. A mapping can change the exposure conditions being compared, and thus change the estimand. Even when the target contrast is held fixed, a mapping can be an imperfect representation of the full assignment. Comparisons and bounds are interpretable only after stating which of these situations is being considered.
Set Up the Contrast Before Comparing Mappings
A spillover contrast compares outcomes under two specified exposure conditions while holding the unit’s own treatment fixed, if that is the intended estimand. More generally, it can compare two assignment policies. Write \(Q_{i1}\) and \(Q_{i0}\) for the distributions of full assignments under the two policies for unit \(i\). The population contrast is
This definition makes the comparison explicit: the two policies determine which full assignments are being compared. An exposure mapping may be used to summarize those assignments and estimate the contrast, but changing the policies or the exposure conditions can change the target itself. A sensitivity comparison should therefore either hold \(Q_{i1}\) and \(Q_{i0}\) fixed or clearly state that it is comparing different target contrasts.
For a mapping \(g\), let \(m_i^g(g_i(z))\) be the outcome value represented or predicted by that mapping for assignment \(z\). A mapping-based approximation to the policy contrast substitutes these represented values for the full potential outcomes. A useful sensitivity analysis specifies how far the representation could be from the truth. For example, suppose the discrepancy is bounded by \(b_i^g(z)\):
The bound is an assumption or a sensitivity parameter, not something established just by fitting a model. It should be motivated using the context, design information, or a range of values that readers can examine.
Deriving a General Mapping-Sensitivity Bound
Suppose \(\Delta_g\) is the population contrast obtained by replacing each full potential outcome in \(\Delta_Q\) with its mapping-based representation. The difference between the target and that approximation is bounded by the average discrepancy under each policy:
To see why, subtract the two contrasts. For each unit, the difference is the policy-1 expectation of \(Y_i(Z)-m_i^g(g_i(Z))\) minus the corresponding policy-0 expectation. The triangle inequality bounds the absolute value of that difference by the sum of the two expected absolute discrepancies. Averaging those unit-level inequalities gives the displayed result. Thus a bound on outcome representation error under both policies yields a bound on contrast error.
This derivation does not require the mapping to be exact. It does require that the discrepancy restriction apply to assignments that can occur under both policies being compared. If the mapping has no support for one of the exposure conditions, a sensitivity bound does not repair the lack of information; positivity and design support still matter.
Omitted Peers and Alternative Neighborhood Summaries
One interpretable restriction concerns omitted peers. Suppose a proposed mapping records treatment among a selected set of neighbors but omits others. For a paired comparison of two full assignment profiles, let \(d_i\) be the number of omitted peers whose treatment differs between the profiles. Assume that changing one omitted peer’s treatment can change unit \(i\)’s outcome by at most \(\gamma_i\) outcome units, with the bound applying successively as omitted peer assignments change. Then the omitted-peer contribution to the unit’s outcome contrast is at most \(\gamma_i d_i\), and the population contrast discrepancy is bounded by the average of these quantities.
This is a restriction on the outcome effect of the omitted assignments, not a claim that omitted peers have no effect. A larger plausible value of \(\gamma_i\), or more omitted-peer differences, produces a wider bound.
Worked Example: Bounding the Effect of Omitted Peers
Hypothetical calculation. A study compares two assignment profiles for the same population. Its proposed mapping records a focal person’s own treatment and the treatment of selected neighbors, but not all peers. Across the population, the average number of omitted peers whose treatment differs between the two profiles is 1.25. Investigators consider the sensitivity restriction that each such change contributes at most 0.8 points to the outcome contrast.
The bound on the difference between the full-profile population contrast and the mapped population contrast is
Under that restriction, the two population contrasts differ by no more than 1.00 outcome point. This does not imply that an estimated mapped contrast has a range of plus or minus 1.00 around it: sampling uncertainty in an estimate must be accounted for separately. The calculation also depends on the assumed per-peer limit; it is not inferred from the average number of omitted peers alone.
Alternative neighborhood summaries can also be assessed with an outcome-smoothness restriction. For example, one mapping might record whether any neighbor is treated, while another records the proportion of treated neighbors. If the outcome representation changes by at most \(L\) outcome units for a one-unit change in that proportion, discrepancies can be bounded using the distances between the actual proportions and the values represented by the mapping. The restriction should apply to the relevant assignments, not merely to average proportions if individual deviations could be larger.
Worked Example: Comparing a Proportion Summary With Representative Exposures
Hypothetical calculation. A proposed mapping represents a low-proportion policy by a treated-neighbor proportion of 0.25 and a high-proportion policy by 0.75. Under the actual assignment policies, the average absolute deviations from those representative proportions are 0.07 and 0.13, respectively. Suppose the outcome representation is assumed to differ by no more than 10 points for each one-unit change in the treated-neighbor proportion.
The sum of the two policy-specific discrepancy bounds is
Thus the population contrast represented by the two fixed proportions can differ from the full-assignment population contrast by at most 2.00 points under the stated pointwise smoothness restriction. The calculation uses absolute deviations within each policy; it does not assume that an outcome at the average proportion equals the average outcome. If the smoothness bound is implausible, or if the mapping’s representative values do not describe the policies being compared, this bound is not justified.
Exposure Misclassification
A mapping can also assign the wrong exposure label. A simple bound is available when outcomes lie in a known interval and misclassification can be described as contamination within each comparison group. Suppose the outcome range is \([L,U]\), with width \(R=U-L\). For each of the two exposure groups, assume a fraction \(q_e\) of the mapped group consists of units that do not belong to the corresponding correctly classified group. If the mapped group mean is a mixture of the correct group mean and the contaminating mean, the discrepancy in that group mean is at most \(q_eR\): both component means lie in \([L,U]\), so their difference cannot exceed \(R\). The contrast discrepancy is therefore at most \(R(q_1+q_0)\).
Worked Example: Bounding a Contrast Under Exposure Misclassification
Hypothetical calculation. A health outcome is bounded between 0 and 20 points. In the mapped high-exposure group, 8% of units are assumed to be misclassified; in the mapped low-exposure group, the corresponding fraction is 12%. Suppose the mapped population contrast is 3.2 points and the contamination-mixture assumptions above hold.
The contrast discrepancy is bounded by
The correctly classified population contrast is therefore within 4.00 points of the mapped population contrast: from \(3.2-4.0=-0.8\) to \(3.2+4.0=7.2\) points. This range can include zero, so the sensitivity bound does not establish a positive population contrast. If 3.2 points were an estimate rather than the mapped population contrast, the range could not be centered on 3.2 without also accounting for sampling uncertainty.
Combine Mapping Sensitivity With Sampling Uncertainty
Sensitivity bounds address uncertainty about how well a mapping represents exposure. Confidence intervals address sampling uncertainty under a statistical model or design. Neither substitutes for the other. If a population contrast under mapping \(g\) is known to be within \(B\) points of the alternative-mapping population contrast, and a valid confidence interval for the first population contrast is available, a conservative interval for the alternative contrast can be formed by extending the confidence interval by \(B\) in each direction.
Worked Example: Do Not Center a Mapping Bound on an Estimate
Hypothetical calculation. The estimated population contrast under the proposed mapping is 6.0 points, with standard error 2.0 points. Under a suitable large-sample approximation, a 95% confidence interval for that mapped population contrast is
Suppose a separate sensitivity analysis justifies a population-level mapping discrepancy bound of 16 points. That bound says the alternative-mapping population contrast is within 16 points of the proposed-mapping population contrast. It does not say the alternative contrast is within 16 points of the estimate 6.0.
Extending the confidence interval by 16 points in both directions gives
Under the stated confidence-interval and sensitivity assumptions, this is a conservative 95% confidence set for the alternative population contrast. It is wide because it includes both sampling uncertainty and the possible mapping discrepancy. The mapping bound alone would give a range centered at 6.0 only if 6.0 were the population contrast, or if additional analysis justified substituting the estimate for that population value.
How to Read Comparative Analyses
A practical analysis can report estimates under several scientifically plausible mappings—for example, number treated among immediate neighbors, proportion treated in a wider neighborhood, and an indicator for any treated neighbor. The comparison is most informative when each mapping is applied to the same underlying target policies, outcome scale, and analysis population. If those features change, the estimates may differ because they answer different questions rather than because one mapping is misspecified.
A useful report separates three things: the contrast targeted, the estimate and its sampling uncertainty under each mapping, and the sensitivity assumptions used to bound departures from each mapping. A small difference across mappings is reassuring only about the particular alternatives examined; it does not prove that all relevant peer information has been captured. A large difference signals that conclusions depend on the exposure definition and deserves substantive interpretation.
Common Mistakes
- Centering a population bound on an estimate. A bound on the difference between two population contrasts does not automatically give a range centered on an estimated contrast. First account for sampling uncertainty, or explicitly assume the estimate equals the population value.
- Treating alternative summaries as the same estimand. “Any treated neighbor” and “proportion treated” may define different exposure conditions. Hold the assignment policies fixed when the goal is to assess representation error, or describe the result as a comparison of different target contrasts.
- Calling a sensitivity parameter an empirical fact. A per-peer effect limit, smoothness constant, or misclassification rate must be justified or varied over plausible values. The resulting bound is conditional on that choice.
- Interpreting robustness as proof. Similar results under a few mappings do not establish that interference is fully captured. The mappings may share the same omission or measurement error.
- Ignoring support. A bound cannot create information about exposure conditions that the design never assigns or observes. Check whether both comparison conditions are supported before interpreting estimates.
Check Your Understanding
Use the definitions and sensitivity calculations in this tutorial to answer the following.
- What assumption makes an exposure mapping exact for outcomes, and what does it mean for the mapping to be approximate?
- If changing each omitted peer’s treatment can alter an outcome by at most 0.5 points, and the average number of omitted-peer changes in a paired comparison is 2, what is the resulting population discrepancy bound?
- Why can changing from an “any treated neighbor” summary to a treated-neighbor proportion change the estimand as well as the approximation?
- For outcomes bounded between 5 and 25 points, with misclassification fractions of 0.10 and 0.05 in the two comparison groups, calculate the contamination bound on the contrast.
- A population-level mapping discrepancy bound is 3 points, but the mapped contrast is estimated with substantial sampling uncertainty. Why is it incorrect to report only the estimate plus or minus 3 points as a range for the alternative contrast?