Bounds For Continuous Outcomes On A Known Scale

Why a Known Outcome Scale Matters

For each person, only one treatment status is observed, so one of the two potential outcomes is missing. When the outcome is known to lie between finite limits, those limits constrain the missing values, even without assuming that treated and untreated people are comparable. The resulting bounds can be wide, but they give a defensible range of average treatment effects under relatively weak assumptions.

This tutorial concerns the average treatment effect in a target population. It uses the treatment proportion and the observed outcome means in the treated and untreated groups. It does not assume that treatment assignment is random or that the observed group difference is causal. The outcome scale must have substantively justified lower and upper limits; an arbitrary choice of scale limits does not create valid information about unobserved outcomes.

Write \(p=P(D=1)\), \(m_1=E[Y\mid D=1]\), and \(m_0=E[Y\mid D=0]\), with \(0<p<1\). By consistency, \(m_1=E[Y(1)\mid D=1]\) and \(m_0=E[Y(0)\mid D=0]\). These are observed group means. The means \(E[Y(1)\mid D=0]\) and \(E[Y(0)\mid D=1]\), in contrast, are not observed.

Deriving the Worst-Case Bounds

The population mean under treatment combines the treated group, for whom \(Y(1)\) is observed, with the untreated group, for whom \(Y(1)\) is missing. The outcome-range assumption says that the missing values in that group have mean somewhere from \(L\) to \(U\). The same reasoning applies to the population mean under control.

To get a lower bound on \(\tau=E[Y(1)]-E[Y(0)]\), subtract the largest allowed value of \(E[Y(0)]\) from the smallest allowed value of \(E[Y(1)]\). For an upper bound, subtract the smallest allowed control mean from the largest allowed treatment mean. Thus:

An identified set is the set of parameter values compatible with the observed information and assumptions. These endpoints are sharp: neither can generally be improved using only the treatment proportion, observed group means, and known outcome limits. For the lower endpoint, assign the missing \(Y(1)\) values in the untreated group the value \(L\), and the missing \(Y(0)\) values in the treated group the value \(U\). This gives \(\tau=\tau_L\). Reversing those assignments, using \(U\) for missing \(Y(1)\) and \(L\) for missing \(Y(0)\), gives \(\tau=\tau_U\). Both assignments respect the scale limits and leave the observed data unchanged. Intermediate choices can produce intermediate effects, so the observed data alone do not select one value within the interval.

Worked Calculations and Interpretation

Diagnostics, Comparisons, and Limitations

Before interpreting a result, check that the outcome limits apply to both potential outcomes for every person in the target population, not merely to the observed sample. Confirm the coding direction and units, verify that the observed means lie within the stated scale, and use the treatment proportion from the same target population as those means. A sign error is especially easy to make: the bounds above are for \(Y(1)-Y(0)\), so a negative effect means treatment lowers the outcome.

The width of the identified set has a direct interpretation:

For this unconditional worst-case interval, the width equals the full scale width. This follows because the upper and lower endpoints differ by \((1-p)(U-L)+p(U-L)=U-L\). Thus, changing the proportion treated or the observed means shifts the interval but does not change its width. The result can be uninformative when the known outcome scale is broad relative to the effect of interest. The earlier material on covariate-specific bounds explains how observed covariates can, under appropriate conditions, tighten bounds; the formulas here are the unconditional calculation.

These bounds are not a confidence interval. They describe uncertainty about the causal effect that remains even if the observed quantities \(p,m_1,m_0\) were known exactly. In an actual finite sample, those quantities are estimated, adding sampling uncertainty. A confidence procedure for the bounds would need to account for that uncertainty; reporting the plug-in endpoints alone does not do so.

The assumptions also matter. If the asserted range is incorrect, the bounds need not contain the true effect. If treatment or outcome definitions are inconsistent, consistency does not link the observed outcome to the intended potential outcome. And if interference is present, one person’s outcome may depend on other people’s treatment, so the two-potential-outcome setup used here may not describe the study. These bounds do not establish that a treatment is beneficial merely because the treated group’s observed mean is better.

Compared with a point estimate based on an exchangeability assumption, worst-case bounds require less about treatment assignment but usually provide less precise conclusions. If the whole identified set is below zero, the effect is negative for every causal model compatible with the assumptions and observed information; if the set includes zero, the sign is not determined by these assumptions. At \(p=1\), there is no untreated group mean and the effect bounds reduce to \(m_1-U\) and \(m_1-L\). At \(p=0\), they reduce to \(L-m_0\) and \(U-m_0\). The main formulas requiring both group means apply when \(0<p<1\).

Common Pitfalls

• Using the observed mean difference as the causal effect. The difference \(m_1-m_0\) compares different groups. Without a treatment-assignment assumption, it need not equal \(E[Y(1)-Y(0)]\).
• Bounding only one missing potential outcome. Both \(Y(1)\) for untreated people and \(Y(0)\) for treated people are unobserved. Each must be included when bounding the average treatment effect.
• Reversing the endpoints when subtracting. A lower bound on a difference is the lower possible first term minus the upper possible second term; an upper bound uses the reverse combination.
• Treating the outcome range as a convenient analysis choice. The finite limits must be justified for the potential outcomes in the target population, not selected to make the interval narrow.
• Calling the identified set a confidence interval. The set expresses partial identification under assumptions. It does not by itself quantify sampling error.