Why Bound an Effect Instead of Reporting One Estimate?
Matching can make treated and comparison groups similar on measured covariates, but it cannot establish that they are also comparable on every unmeasured factor related to treatment and outcome. A point estimate from a matched analysis may therefore depend on an assumption that residual confounding is exactly absent. Partial identification makes a different move: specify a plausible limit on the remaining confounding and report the full range of effects compatible with that limit and the observed outcome means.
The result is an identification region, not a single effect. Its width reflects how much the conclusion depends on the sensitivity restriction. The restriction must be stated in meaningful units and defended substantively; the observed data alone generally cannot determine whether it is correct.
The comparison mean \(m_0\) is intended to represent the untreated outcome mean for the treated target population after accounting for measured covariates through the matching or weighting design. The unknown \(q\) is that population’s actual mean outcome had its members not been treated. A nonzero \(d\) means the adjusted comparison mean still differs from this counterfactual mean. Neither matching nor balance diagnostics reveal the value of \(d\).
This setup uses consistency and a well-defined outcome and treatment. It does not assume that treatment is unconfounded given measured covariates. The sensitivity restriction introduced below is an additional assumption about the remaining discrepancy. For a matched analysis, its target is the treated population retained in the analysis; it should not be silently generalized to all treated individuals if matching or trimming changed the target.
Deriving the Bounds
Suppose a subject-matter argument supports a symmetric restriction \(|d|\leq\delta\), where \(\delta\geq0\) is expressed in outcome units. Since \(d=q-m_0\), this says \(m_0-\delta\leq q\leq m_0+\delta\). Substituting into \(\tau_{\mathrm{ATT}}=m_1-q\) reverses the endpoints:
These are the bounds implied by the bias restriction alone. But they need not all be feasible. If the outcome is known to lie in \([L,U]\), then its treated-group counterfactual mean must satisfy \(L\leq q\leq U\). The ATT must consequently satisfy \(m_1-U\leq\tau_{\mathrm{ATT}}\leq m_1-L\). The feasible bounds are the intersection of the two intervals:
The intersection is the appropriate range under these restrictions. If its lower endpoint exceeds its upper endpoint, the stated restrictions and observed means are incompatible; one must investigate the assumptions or inputs rather than report an empty-looking interval as a valid effect range. With no restriction on the discrepancy beyond the outcome range, the ATT itself still cannot exceed the feasible range \([m_1-U,m_1-L]\).
An asymmetric restriction can represent directional knowledge. If \(-\delta_-\leq d\leq\delta_+\), then the bias-based interval is \([m_1-m_0-\delta_+,\;m_1-m_0+\delta_-]\): a larger possible positive discrepancy lowers the smallest compatible effect. Intersect this interval with \([m_1-U,m_1-L]\) whenever the outcome is bounded.
Worked Example: Binary Risks and Outcome-Range Feasibility
Consider a hypothetical matched cohort with treated-group outcome risk \(m_1=0.95\) and weighted comparison risk \(m_0=0.05\). The outcome is binary, so \(L=0\) and \(U=1\). Suppose the sensitivity restriction is \(|d|\leq1\), which is deliberately broad.
The observed adjusted risk difference is \(m_1-m_0=0.95-0.05=0.90\). The bias-only interval is \(0.90-1\leq\tau_{\mathrm{ATT}}\leq0.90+1\), or \([-0.10,1.90]\). This interval is not fully feasible for a binary outcome. The treated group’s untreated risk \(q\) must be between 0 and 1, so the ATT must be between \(0.95-1=-0.05\) and \(0.95-0=0.95\).
Intersecting the intervals gives \(\max(-0.10,-0.05)=-0.05\) for the lower endpoint and \(\min(1.90,0.95)=0.95\) for the upper endpoint. Thus the feasible ATT region is \([-0.05,0.95]\), in risk-difference units. For example, the bias-only endpoint \(1.90\) would require \(q=0.95-1.90=-0.95\), which cannot be a risk. The observed means and the broad sensitivity restriction do not identify one effect.
How the Sensitivity Parameter Changes the Conclusion
A useful sensitivity analysis reports the bounds for several substantively interpretable values of \(\delta\), not just one convenient setting. Increasing \(\delta\) allows a greater difference between the treated group’s untreated mean and the adjusted comparison mean. The bias-only interval therefore widens or stays the same. After imposing outcome feasibility, however, the reported region cannot widen beyond the range allowed by \([L,U]\).
Worked Example: Varying the Bound for a Binary Outcome
In a hypothetical matched cohort, suppose the treated risk is \(m_1=0.30\), the weighted comparison risk is \(m_0=0.22\), and the outcome is binary. The adjusted risk difference is \(0.30-0.22=0.08\). The outcome-range constraint gives \([m_1-U,m_1-L]=[0.30-1,0.30-0]=[-0.70,0.30]\).
For \(\delta=0.08\), the bias-only bounds are \([0.08-0.08,\;0.08+0.08]=[0,0.16]\). Both endpoints are feasible, so the final region remains \([0,0.16]\). Under this restriction, the ATT is nonnegative, but its value is not point identified.
For \(\delta=0.15\), the bias-only bounds are \([0.08-0.15,\;0.08+0.15]=[-0.07,0.23]\). These also lie inside the outcome-feasible interval, so the final region is \([-0.07,0.23]\). The sign is now uncertain: this restriction permits both a small harmful effect and a beneficial effect.
For \(\delta=0.50\), the bias-only interval is \([-0.42,0.58]\), but the feasible region is only \([-0.70,0.30]\). The intersection is \([-0.42,0.30]\). The upper endpoint is clipped by outcome feasibility, because it would otherwise imply an untreated risk below zero. All endpoints are in absolute risk-difference units; for example, \(0.16\) is a 16-percentage-point ATT.
Sensitivity parameters need not be symmetric. A directional restriction may be appropriate when substantive knowledge supports the belief that, even after matching, treated individuals would have had worse outcomes under no treatment than the weighted comparison group. The direction must be justified for the particular outcome: the same selection pattern could imply different directions for outcomes where higher values are better versus worse.
Worked Example: A Directional Restriction for a Continuous Outcome
Suppose a hypothetical matched analysis measures a continuous outcome in points, with known range 0 to 100. Let \(m_1=68\) points and \(m_0=60\) points, giving an adjusted difference of \(8\) points. Assume the defensible sensitivity restriction is \(0\leq d\leq10\) points: the treated group’s mean untreated outcome is no lower than the weighted comparison mean, and is at most 10 points higher.
The ATT equals \(8-d\). As \(d\) ranges from 0 to 10, the bias-based bounds are \([8-10,\;8-0]=[-2,8]\) points. The outcome range implies feasible ATT values from \(68-100=-32\) to \(68-0=68\) points, so intersecting leaves \([-2,8]\) points. Under this directional restriction, both a two-point decrease and an eight-point increase remain compatible with the means; the effect is not known to be positive. The conclusion depends on accepting the directional restriction, not on the observed difference alone.
Interpretation, Diagnostics, and Alternatives
A sensitivity-bound interval is a set of parameter values compatible with the observed means, the stated sensitivity restriction, and any outcome-range constraints. It is not automatically a 95% confidence interval, and it is not a probability distribution over the ATT. Sampling uncertainty in \(m_1\) and \(m_0\) is a separate issue: an applied analysis should account for it when making inferential claims, rather than treating plug-in bounds as if the means were known exactly.
Diagnostics cannot prove a chosen \(\delta\), but they can show whether the analysis that supplies \(m_0\) is fragile. Check the target population and weights, overlap, retained observations, and balance on measured covariates. Examine whether plausible residual differences in important prognostic factors could shift the outcome mean by the proposed amount. Negative control checks and Rosenbaum sensitivity analysis address related concerns in different ways; they do not mechanically establish the mean-discrepancy restriction used here.
The following distinction is important:
| Approach | What it provides | Key dependence |
|---|---|---|
| Point estimate under no residual confounding | A single effect estimate | Assumes the adjusted comparison mean equals the treated group’s counterfactual mean |
| Partial identification bounds | A range of effects compatible with explicit restrictions | Depends on the chosen bias limit and feasibility constraints |
| Sampling confidence interval | Quantifies sampling uncertainty for an estimator or bound procedure | Depends on the sampling model and inferential procedure |
A common mistake is to calculate the bias-only interval and report it as the complete feasible range. In the first example, that would include an ATT of \(1.90\), even though it implies a negative untreated risk. The correction is to constrain the unknown counterfactual mean \(q\) to the outcome’s possible range and intersect the resulting ATT range with the bias-based interval. Another mistake is to describe the range as evidence that every value is equally likely; bounds establish compatibility, not likelihood.
Partial identification is useful when an exact no-unmeasured-confounding claim is not credible but a quantitative restriction is defensible. Its limitation is equally important: if the bounds span effects with substantively different signs or magnitudes, the available information and restrictions do not support a more decisive causal conclusion. Reporting a narrow interval from an implausibly restrictive \(\delta\) does not solve that problem.
Specify the treated population, the weighted comparison mean \(m_0\), the treated mean \(m_1\), and the outcome units.
Define \(d=q-m_0\) and justify either \(|d|\leq\delta\) or a directional interval for \(d\), in outcome units.
Obtain the bias-based ATT bounds, then intersect them with \([m_1-U,m_1-L]\) when \(Y\in[L,U]\).
Report how the feasible range and conclusions change as the sensitivity parameter changes; distinguish this range from sampling uncertainty.
Check Your Understanding
For each question, calculate or explain the relevant bound. Solutions follow each prompt so the role of feasibility is explicit.
- A binary-outcome analysis has \(m_1=0.40\), \(m_0=0.30\), and \(\delta=0.05\). Calculate the feasible ATT bounds. Solution: The bias-based bounds are \([0.10-0.05,\;0.10+0.05]=[0.05,0.15]\). The outcome-feasible interval is \([0.40-1,\;0.40-0]=[-0.60,0.40]\). Their intersection is \([0.05,0.15]\), in risk-difference units.
- For a binary outcome, \(m_1=0.90\), \(m_0=0.10\), and \(\delta=1\). Why should the bias-based interval not be reported without adjustment? Solution: It is \([-0.20,1.80]\). The feasible interval is \([0.90-1,\;0.90-0]=[-0.10,0.90]\), so the intersection is \([-0.10,0.90]\). Values above \(0.90\) would require \(q<0\), an impossible risk.
- For a continuous outcome, \(m_1=50\), \(m_0=46\), and \(d\in[0,3]\). Find the bias-based ATT bounds. Solution: Here \(m_1-m_0=4\), and \(\tau_{\mathrm{ATT}}=4-d\), so the bounds are \([4-3,\;4-0]=[1,4]\) outcome units. If the outcome is bounded, intersect this result with its corresponding outcome-feasible interval.
- Does a narrower bound under a smaller \(\delta\) show that the data have ruled out larger confounding? Solution: No. A smaller \(\delta\) is a stronger sensitivity assumption. The narrower region follows from that restriction, not from the observed means alone.