Using Auxiliary Data To Tighten Bounds

Why Auxiliary Information Can Tighten Bounds

Worst-case bounds leave unobserved potential outcomes as unrestricted as the outcome scale permits. That caution is useful when nothing else is known, but sometimes a study has valid auxiliary information: for example, a measurement of a potential outcome in a relevant validation sample, or a credible range for a population-wide outcome mean. Such information can rule out some counterfactual values without assuming that treatment assignment is unconfounded.

The key is to use auxiliary information as a constraint on the same population quantities that define the causal effect. A measurement from a different population, a proxy with an unverified relationship to the outcome, or a noisy estimate treated as if it were an exact truth does not automatically provide a valid constraint. The analysis must state why the auxiliary quantity corresponds to the target population and how its uncertainty is represented.

Set Up the Bounds Before Adding Information

Let \(D\) indicate treatment, with \(D=1\) for treated units and \(D=0\) for untreated units. Let \(Y(1)\) and \(Y(0)\) be the outcomes each unit would have under treatment and no treatment, respectively. The observed outcome is \(Y=DY(1)+(1-D)Y(0)\). The target here is the average treatment effect, \(E[Y(1)-Y(0)]\). No assumption of random assignment or conditional exchangeability is imposed.

For a binary outcome, define \(p=P(D=1)\), \(m_1=E[Y\mid D=1]\), and \(m_0=E[Y\mid D=0]\). Also write \(a=E[Y(1)\mid D=0]\) and \(b=E[Y(0)\mid D=1]\). The observed data identify \(p,m_1,m_0\), but not \(a\) or \(b\). Since a binary outcome has conditional means between zero and one, \(a,b\in[0,1]\). Decomposing each population potential-outcome mean by treatment status gives

With no other restrictions, the smallest value sets \(a=0\) and \(b=1\); the largest sets \(a=1\) and \(b=0\). Thus the worst-case interval is

An auxiliary population mean can constrain these missing means. For example, if \(M_1=E[Y(1)]\), then \(M_1=p m_1+(1-p)a\). A valid range for \(M_1\) therefore restricts \(a\). Similarly, \(M_0=E[Y(0)]=p b+(1-p)m_0\) links an auxiliary range for \(M_0\) to \(b\). The narrowed effect interval follows by minimizing and maximizing the effect expression over the remaining feasible values.

One Margin Can Tighten Only Part of the Uncertainty

Auxiliary information need not identify both population potential-outcome means. A constraint on just one of them can still improve one or both endpoints. It is important to leave the other missing mean unrestricted rather than silently treating it as known.

When Is the Tightened Interval Sharp?

A bound is sharp when every value inside it is compatible with the stated restrictions and its endpoints can be attained by at least one data-generating arrangement satisfying them. For the calculations above, the feasible set is described by independent intervals for the two missing conditional means, together with the observed means. The effect is linear in those missing means, so its minimum and maximum occur at feasible endpoints. If no restrictions are imposed on the joint distribution of \(Y(1)\) and \(Y(0)\), binary potential outcomes with any specified marginal probabilities can be jointly distributed; for example, the two potential outcomes can be generated independently conditional on treatment status. This allows the endpoint conditional means to be realized without adding a cross-world assumption.

That justification depends on the absence of additional constraints. If auxiliary data also restrict how potential outcomes co-occur, or if several reported margins must hold simultaneously, taking separate endpoint ranges may combine values that cannot jointly occur. In that case, the bounds must be computed over the joint feasible set rather than by optimizing each restriction in isolation. A claim of sharpness requires checking joint feasibility, not merely showing that each auxiliary range looks plausible by itself.

Diagnostics, Alternatives, and Limits

A first diagnostic is feasibility. If a proposed auxiliary range implies \(a<0\), \(a>1\), \(b<0\), or \(b>1\) for binary outcomes, the restrictions conflict with the observed data and outcome scale. Other conflicts can arise when multiple auxiliary facts overlap. An empty feasible set is a warning that at least one input, population definition, measurement assumption, or data source needs review; it is not evidence for an unusually precise effect.

A second diagnostic is how much each constraint changes the endpoints. A very small change may mean the auxiliary information is weak, redundant with the observed data, or attached to a missing mean that barely affects the bound. A large change deserves careful scrutiny of transportability, outcome measurement, and whether the range really covers the target population quantity. Wider or narrower should not be confused with more or less credible: credibility comes from the assumptions supporting the constraint.

The relevant alternative depends on what information is genuinely available. The worst-case bound uses only the observed data and the outcome scale, so it is robust but can be wide. A point estimate based on a stronger assignment or modeling assumption may be more precise, but answers the question only if that assumption is justified. Auxiliary constraints occupy a middle ground: they add information while retaining a set of compatible effects. They do not, by themselves, establish unconfoundedness or turn the interval into a point estimate.

An auxiliary estimate also has sampling uncertainty. If it is estimated from a finite validation sample, its uncertainty must be accounted for when constructing an inferential procedure; substituting a sample estimate as an exact population equality generally understates uncertainty. Further, a direct measurement of a potential outcome is not interchangeable with a proxy. A proxy can impose a constraint only when a defensible measurement or calibration relationship links it to the target potential outcome. These issues are separate from the algebra of identification.