Worst-Case Bounds For Bounded Outcomes

Why Bounded Outcomes Give Bounds on the ATE

For a causal effect, each person has two potential outcomes: the outcome under treatment and the outcome without treatment. Only one is observed for each person. When outcomes are known to lie within a fixed interval, that information restricts the unobserved outcomes, even if treatment assignment is confounded. The result is generally an identified interval for the average treatment effect (ATE), rather than a single identified value.

The earlier material on binary outcome bounds develops the special case in which the outcome lies between zero and one. The same logic applies to any known finite outcome range. The scale matters: limits must be defensible for both potential outcomes in the target population, not merely convenient values chosen to produce narrower bounds.

Definition: Let $D\in\{0,1\}$ denote treatment, and let $Y(1)$ and $Y(0)$ be the potential outcomes under treatment and control. The observed outcome is $Y=D Y(1)+(1-D)Y(0)$. The average treatment effect is $\tau=E[Y(1)-Y(0)]$. Suppose both potential outcomes satisfy $L\leq Y(d)\leq U$ for $d=0,1$, where $L$ and $U$ are known finite limits. Write $p=P(D=1)$, $m_1=E[Y\mid D=1]$, and $m_0=E[Y\mid D=0]$. If $0<p<1$, the worst-case bounds are $$ p(m_1-U)+(1-p)(L-m_0)\ \leq\ \tau\ \leq\ p(m_1-L)+(1-p)(U-m_0). $$

The bounds use the observed treated mean $m_1$, the observed control mean $m_0$, and the treatment prevalence $p$. They do not require treatment to be randomized or conditionally independent of potential outcomes. Instead, they allow the unobserved outcomes to take any values in the known range. Consequently, the bounds can be wide.

Assumptions and Derivation

The result applies under a short list of conditions. Treatment has two well-defined states, and consistency holds: a treated person’s observed outcome is $Y(1)$, while a control person’s observed outcome is $Y(0)$. The target population and treatment assignment are defined consistently so that the expectations refer to the same population. Both potential outcomes are bounded by the same known $L$ and $U$. Finally, $0<p<1$, so that both observed group means are defined. No assumption of no unmeasured confounding is imposed.

By splitting each potential-outcome mean according to treatment status, the ATE can be written as

\tau =p\,E[Y(1)\mid D=1] +(1-p)E[Y(1)\mid D=0] -(1-p)E[Y(0)\mid D=0] -p\,E[Y(0)\mid D=1].

Consistency identifies the first conditional mean as $m_1$ and the third as $m_0$. The second term is the mean treated outcome among controls, which is unobserved; its value lies between $L$ and $U$. The fourth term is the mean control outcome among treated people, also unobserved and within the same limits. Because $p$ and $1-p$ are nonnegative, the smallest possible ATE uses the smallest allowed value for the positive missing term and the largest allowed value for the subtracted missing term. The largest possible ATE uses the reverse assignments.

Find the lower endpoint.
Set $E[Y(1)\mid D=0]=L$ and $E[Y(0)\mid D=1]=U$. Substitution gives $p m_1+(1-p)L-(1-p)m_0-pU$, which rearranges to $p(m_1-U)+(1-p)(L-m_0)$.

Find the upper endpoint.
Set $E[Y(1)\mid D=0]=U$ and $E[Y(0)\mid D=1]=L$. This gives $p m_1+(1-p)U-(1-p)m_0-pL=p(m_1-L)+(1-p)(U-m_0)$.

Check attainability.
Each endpoint is attainable by assigning the relevant unobserved potential outcomes to $L$ or $U$, while leaving observed outcomes unchanged. These assignments satisfy the stated support restriction. Thus, without further restrictions, the endpoints are sharp: the data and assumptions do not rule out either endpoint.

The weights $p$ and $1-p$ have a direct interpretation. The missing $Y(0)$ outcomes belong to the treated group, which makes up proportion $p$; the missing $Y(1)$ outcomes belong to the control group, which makes up proportion $1-p$. The observed group means affect the endpoint values, while the group proportions determine how heavily each missing-outcome contribution enters.

Worked Example: Binary Outcomes With Correctly Weighted Contributions

Consider a hypothetical population in which 40% receive treatment. The observed treated mean is $m_1=0.70$, and the observed control mean is $m_0=0.30$. Suppose the outcome and both potential outcomes lie in $[0,1]$. Calculate the worst-case ATE bounds.

Here $p=0.40$, $1-p=0.60$, $L=0$, and $U=1$. The lower endpoint is

$$ 0.40(0.70-1)+0.60(0-0.30) =0.40(-0.30)+0.60(-0.30) =-0.12-0.18=-0.30. $$

The upper endpoint is

$$ 0.40(0.70-0)+0.60(1-0.30) =0.40(0.70)+0.60(0.70) =0.28+0.42=0.70. $$

Thus the identified set is $[-0.30,\,0.70]$, in outcome-probability units. The observed difference in means is $0.70-0.30=0.40$, but it is not by itself the ATE unless additional assumptions justify that interpretation. The bounds include negative, zero, and positive effects, so these data and support restriction alone do not determine the sign of the ATE.

Changing the Outcome Scale

The formula works without rescaling the outcome to zero and one. When the outcome is measured in units such as points, hours, or dollars, the endpoints and means must all use those same units. The resulting ATE bounds are in those outcome units as well.

Worked Example: Outcomes Bounded Between Minus Two and Eight

Suppose a hypothetical outcome is measured in points and both potential outcomes are known to lie between $L=-2$ and $U=8$. Let $p=0.25$, $m_1=5$ points, and $m_0=1$ point. Calculate and interpret the bounds.

The lower endpoint is

0.25(5-8)+0.75(-2-1) =0.25(-3)+0.75(-3) =-0.75-2.25=-3\text{ points}.

The upper endpoint is

0.25(5-(-2))+0.75(8-1) =0.25(7)+0.75(7) =1.75+5.25=7\text{ points}.

The ATE is therefore bounded by $-3$ and $7$ points. The observed difference in means is $5-1=4$ points, but without further assumptions the average causal effect could be below or above zero. The calculation also shows why inserting the outcome limits matters: replacing $[-2,8]$ with $[0,1]$ would not describe this outcome scale and would produce invalid bounds.

How Treatment Prevalence Changes the Contributions

The group proportions determine which missing potential outcomes receive greater weight. This does not mean that a larger treatment group necessarily gives tighter ATE bounds. Under these worst-case assumptions, the total width is

\begin{aligned} &\big[p(m_1-L)+(1-p)(U-m_0)\big] -\big[p(m_1-U)+(1-p)(L-m_0)\big]\\ &\qquad=p(U-L)+(1-p)(U-L)=U-L. \end{aligned}

Thus, for fixed outcome limits, the width is exactly the range $U-L$, regardless of $p$, $m_1$, and $m_0$. Prevalence changes how that width is distributed around the endpoints, but does not reduce the total width in this unconditional worst-case calculation.

Worked Example: Comparing Two Treatment Prevalences

For illustration, suppose $L=0$, $U=10$, $m_1=7$, and $m_0=4$. Compare the bounds when treatment prevalence is $p=0.20$ and $p=0.80$. These are hypothetical values used to show the weighting, not a claim that the two populations are otherwise identical.

When $p=0.20$, the treated group contributes weight $0.20$, and the control group contributes weight $0.80$. The bounds are

\begin{aligned} \text{Lower}&=0.20(7-10)+0.80(0-4)=-0.60-3.20=-3.80,\\ \text{Upper}&=0.20(7-0)+0.80(10-4)=1.40+4.80=6.20. \end{aligned}

When $p=0.80$, the weights reverse:

\begin{aligned} \text{Lower}&=0.80(7-10)+0.20(0-4)=-2.40-0.80=-3.20,\\ \text{Upper}&=0.80(7-0)+0.20(10-4)=5.60+1.20=6.80. \end{aligned}

Both intervals have width $10$, the outcome range. The endpoint locations differ because the observed means are weighted differently at each prevalence. A change in prevalence alone therefore does not provide a general guarantee of a narrower worst-case interval.

Diagnostics, Comparisons, and Limitations

Before reporting these bounds, check that the outcome limits are known and apply to both potential outcomes for everyone in the target population. Observed values falling between $L$ and $U$ are necessary but do not, by themselves, establish that unobserved potential outcomes obey the same limits. Check also that treatment is coded as zero or one, that $p$ is calculated for the intended population, and that the treated and control means use the corresponding observed groups. A misplaced group proportion or a mean computed on a different scale changes the result.

These are identification bounds, not a confidence interval. They describe the range of ATE values compatible with the stated population information and assumptions. In an actual sample, estimates of $p$, $m_1$, and $m_0$ have sampling uncertainty; quantifying uncertainty around estimated bounds requires additional inferential methods. The numerical interval alone does not express that sampling uncertainty.

Worst-case bounds are useful when a point estimate would require assumptions that cannot be defended, but their width can make them uninformative about the sign or size of the effect. A point-identifying approach, such as adjustment under an appropriate no-unmeasured-confounding assumption, can yield a more precise target only if its additional assumptions are credible. Binary outcome bounds are the special case $L=0$, $U=1$. Restrictions beyond the outcome range may tighten bounds, but they require substantive justification rather than arithmetic convenience.

Common pitfall: Do not treat each weighted term as though it were the unweighted difference inside its parentheses. For example, with $p=0.40$, $m_1=0.70$, $m_0=0.30$, and limits $[0,1]$, the lower contributions are $0.40(-0.30)=-0.12$ and $0.60(-0.30)=-0.18$, not $-0.30$ each. Their sum is $-0.30$. Keep the prevalence weights visible through the arithmetic.

Key takeaway: Under consistency and known bounds $L\leq Y(d)\leq U$, the observed group means identify the observed parts of the ATE, while the treatment prevalence weights the unknown parts. The resulting sharp worst-case interval has width $U-L$; a narrower causal conclusion needs additional information or assumptions.

Check Your Understanding

Answer each question, then compare your work with the solutions below.

For outcomes in $[0,1]$, let $p=0.50$, $m_1=0.60$, and $m_0=0.40$. What are the lower and upper ATE bounds?
Why does the lower endpoint assign $L$ to the missing treated outcome among controls and $U$ to the missing control outcome among treated people?
If the outcome range is $[-5,5]$, what is the width of the worst-case ATE bounds, assuming both treatment groups are observed?
Do bounds calculated from sample means account automatically for sampling uncertainty? Explain.

Solutions. 1. The lower endpoint is $0.50(0.60-1)+0.50(0-0.40)=-0.20-0.20=-0.40$. The upper endpoint is $0.50(0.60-0)+0.50(1-0.40)=0.30+0.30=0.60$, so the bounds are $[-0.40,0.60]$. 2. The missing treated outcome enters the ATE with a positive sign, so choosing its smallest allowed value lowers the ATE; the missing control outcome is subtracted, so choosing its largest allowed value lowers the ATE. 3. The width is $U-L=5-(-5)=10$ outcome units. 4. No. The bounds describe identification under the assumptions. Sample estimates of their inputs are subject to sampling variation, which requires separate inferential treatment.