Sensitivity analysis and partial identification

Consider binary treatment and potential outcomes constrained to the interval zero through one. Assume consistency and no interference, with both observed treatment groups having positive probability, but do not assume exchangeability. Let p be the treatment prevalence and μ₁,μ₀ the observed arm means. The treated potential-outcome mean averages its known treated-group component and its unknown untreated-group component. Bound that unknown component between zero and one. Do the corresponding calculation for the untreated potential-outcome mean, whose treated-group component is missing.

Subtract the largest possible untreated mean from the smallest possible treated mean for the lower ATE bound; reverse those choices for the upper bound. The resulting interval has width one, lies within the logical effect range, and contains zero. It is sharp under these minimal restrictions because the unobserved potential outcomes can be chosen to attain the endpoints without changing the observed distribution. Its width represents ignorance about the counterfactual law, not imprecision from a small observed sample.

The bounds are often called worst-case or no-assumptions bounds, but the latter phrase is shorthand: intervention definitions, consistency, outcome support, and the observed population are still specified. Mean independence of potential outcomes from treatment collapses the bounds to the observed mean difference. Monotone treatment response, requiring every unit’s treated outcome to be at least its untreated outcome, instead rules out negative effects while leaving room for partial identification. It is different from monotone selection, which compares baseline prospects across treatment groups.

An identified set presumes knowledge of the observed distribution. With sampled data, its endpoints must be estimated and their uncertainty handled. A plug-in interval between estimated endpoints is not automatically a confidence region for the effect or a simultaneous confidence region for the entire identified set. Those are distinct coverage goals. Statistical uncertainty can shrink as sampling improves while the population identified set remains wide, so a report should distinguish the two rather than calling both an error bar.

For ATT define δ₀ as the difference between the treated group’s untreated potential-outcome mean and the observed untreated group’s mean. Add and subtract the first quantity in the observed mean contrast to obtain ATT equal to that contrast minus δ₀. Exchangeability sets this particular bias parameter to zero, while a scientifically chosen range expresses plausible departures. A restriction on δ₀ alone concerns ATT; it does not generally identify ATE, whose treated counterfactual outcome among controls also matters.

Choose the range before inspecting which values preserve a preferred conclusion, and check that it is compatible with the outcome bounds. If the observed control mean is μ₀, the treated group’s untreated mean must remain in [0,1], restricting δ₀ to [−μ₀,1−μ₀]. A sensitivity interval is conditional on that maintained range, rather than a posterior distribution or a frequentist confidence interval by itself. Similar trend-deviation parameters can be used for DID, but the parameter and target must be redefined for that design.