Worst-Case Bounds For The Treated Group

Why Bound the Effect Among Treated Units?

The average treatment effect among treated units asks how treatment affected the units that actually received it. For those units, the treated outcome is observed, but their outcome under no treatment is not. If treatment assignment may depend arbitrarily on potential outcomes, the observed outcomes of untreated units do not tell us what the treated units would have experienced without treatment. A known range for the outcome still gives a valid, though often wide, range for the average effect.

The key distinction from bounds for an average effect over the whole population is the target group. Among treated units, the treated potential outcome is observed for every member of the target group. Only their untreated potential outcome is missing. This asymmetry leads to a particularly direct bound.

Definition: Let $D$ equal 1 for a treated unit and 0 for an untreated unit. Let $Y(1)$ and $Y(0)$ denote that unit’s potential outcomes under treatment and no treatment, respectively. The average treatment effect among the treated (ATT) is $$ \operatorname{ATT}=E[Y(1)-Y(0)\mid D=1], $$ provided $P(D=1)>0$. The observed outcome is $Y=D Y(1)+(1-D)Y(0)$. Suppose the outcome is known to satisfy $L\leq Y(1)\leq U$ and $L\leq Y(0)\leq U$ for every unit, where $L$ and $U$ are finite, known limits.

The potential-outcome framework presumes a well-defined treatment and consistency: a unit’s observed outcome equals its potential outcome under the treatment actually received. The bounds below also require that the specified limits genuinely cover both potential outcomes in the target population. They do not require random assignment, conditional exchangeability, or any other assumption connecting treatment status to potential outcomes.

Deriving the Worst-Case ATT Bounds

For a treated unit, consistency gives $Y=Y(1)$. Therefore the treated potential outcome is observed for the target group, and its mean is $E[Y\mid D=1]$. The treated group’s mean untreated potential outcome, $E[Y(0)\mid D=1]$, is not observed. The outcome limits imply that this missing mean must lie between $L$ and $U$. Subtracting it from the observed treated mean yields the bounds.

\begin{aligned} \operatorname{ATT} &=E[Y(1)\mid D=1]-E[Y(0)\mid D=1]\\ &=E[Y\mid D=1]-E[Y(0)\mid D=1],\\ L&\leq E[Y(0)\mid D=1]\leq U,\\ \therefore\quad E[Y\mid D=1]-U &\leq \operatorname{ATT}\leq E[Y\mid D=1]-L. \end{aligned}

Theorem: Under consistency, a nonempty treated group, and the common known support $L\leq Y(0),Y(1)\leq U$, with no assumptions linking treatment assignment to potential outcomes, the sharp identified set for the ATT is $$ \left[E[Y\mid D=1]-U,\;E[Y\mid D=1]-L\right]. $$

“Sharp” means that the observed data and assumptions rule out no value inside this interval and rule out every value outside it. To see why, write $m=E[Y\mid D=1]$. For any candidate $a$ in the interval, set the treated group’s missing untreated potential-outcome mean to $m-a$. The definition of the interval gives $L\leq m-a\leq U$. A missing-outcome mean anywhere in this range is compatible with the support restriction. It can be achieved, for example, by assigning every treated unit the untreated potential outcome $m-a$, without changing any observed outcome. This construction makes the ATT equal to $a$. At the endpoints, assigning $Y(0)=U$ for all treated units gives the lower bound, and assigning $Y(0)=L$ gives the upper bound. Thus neither endpoint can be tightened using only these assumptions.

The untreated group’s observed outcomes do not narrow this interval by themselves. Without a restriction linking treatment assignment to potential outcomes, their outcomes under no treatment need not reveal the distribution of $Y(0)$ among treated units. Additional information or assumptions could tighten the bounds, but they are not part of this worst-case analysis.

Worked Example: A Continuous Outcome on a 0-to-100 Scale

Suppose a hypothetical program has five treated participants whose observed scores are 54, 62, 70, 74, and 80 points. Assume both potential outcomes lie between 0 and 100 points. The target is the average effect for these treated participants, not for all participants.

First calculate their observed treated-outcome mean: $$ \bar{Y}_{D=1}=\frac{54+62+70+74+80}{5} =\frac{340}{5}=68\text{ points}. $$

Here $L=0$ and $U=100$. Substituting into the bounds gives $$ 68-100\leq \operatorname{ATT}\leq68-0, \qquad\text{so}\qquad -32\leq \operatorname{ATT}\leq68\text{ points}. $$

This interval says the assumptions permit an average effect as low as a 32-point decrease or as high as a 68-point increase among treated participants. It does not establish that either endpoint is likely. For the lower endpoint, the hypothetical untreated scores for the five treated participants can all be 100; for the upper endpoint, they can all be 0. Both assignments respect the outcome limits and leave the observed treated scores unchanged.

What the Bounds Can and Cannot Establish

When the treated mean is itself within the assumed outcome range, the lower bound cannot be positive and the upper bound cannot be negative: $$ E[Y\mid D=1]-U\leq 0 \quad\text{and}\quad E[Y\mid D=1]-L\geq 0. $$ Thus zero is always included in the worst-case ATT interval under these assumptions. Without a restriction on how treatment assignment relates to potential outcomes, the bounds alone cannot establish that treatment helped or harmed the treated group on average. They can still limit the possible size of the effect. Their width is $U-L$, the full width of the outcome range, regardless of the observed treated mean.

Worked Example: A Binary Outcome

In a hypothetical evaluation, 12 treated patients are observed, and 5 have a favorable binary outcome, coded 1; the other 7 have outcome 0. Suppose both potential outcomes are binary, so $L=0$ and $U=1$. The ATT is a difference in outcome probabilities, or a risk difference, among treated patients.

The observed treated proportion with a favorable outcome is $$ \hat{p}_{D=1}=\frac{5}{12}=0.4167. $$ Using the treated mean in the worst-case formula gives $$ 0.4167-1\leq \operatorname{ATT}\leq0.4167-0, \qquad\text{so}\qquad -0.5833\leq \operatorname{ATT}\leq0.4167. $$

On the probability scale, the lower limit is a 58.33-percentage-point decrease and the upper limit is a 41.67-percentage-point increase. To attain the lower limit, all 12 treated patients could have had $Y(0)=1$, producing an ATT of $5/12-1$. To attain the upper limit, they could all have had $Y(0)=0$, producing an ATT of $5/12$. The observed treated outcomes are consistent with either completion of the missing outcomes.

These are identification bounds: they describe which population effects are compatible with the assumptions and population-level observed treated mean. In a sample, replacing $E[Y\mid D=1]$ with the sample mean $\bar{Y}_{D=1}$ gives an estimated bound. That calculation does not, by itself, account for sampling variability. A confidence interval for the bound endpoints is a separate inferential task; the identified interval should not be interpreted as a confidence interval.

Worked Example: Checking the Sign and the Scale

Suppose a hypothetical clinical measure is known to range from 10 to 50 units. Four treated patients have observed outcomes of 39, 42, 45, and 48 units. Calculate the ATT bounds and decide whether these assumptions identify a beneficial or harmful average effect.

The treated mean is $$ \bar{Y}_{D=1}=\frac{39+42+45+48}{4} =\frac{174}{4}=43.5\text{ units}. $$ With $L=10$ and $U=50$, the estimated worst-case range is $$ 43.5-50\leq \operatorname{ATT}\leq43.5-10, \qquad\text{or}\qquad -6.5\leq \operatorname{ATT}\leq33.5\text{ units}. $$

Because this interval includes zero, the assumptions do not determine the sign of the effect. The same conclusion follows directly from the endpoint construction: assigning each treated patient an untreated potential outcome equal to that patient’s observed treated outcome is allowed by the 10-to-50 range and gives an ATT of zero. If larger values are clinically worse rather than better, the substantive meaning of positive and negative effects must be interpreted accordingly; the arithmetic bounds do not determine which direction is desirable.

Diagnostics, Failure Modes, and Alternatives

The calculation is simple, but its validity depends on specifying the target and the outcome scale correctly. Before reporting bounds, check the following:

Verify the support limits. $L$ and $U$ must bound both potential outcomes for every unit in the target population, not just the observed treated outcomes. Limits chosen for convenience, or limits that exclude plausible values, can produce invalid bounds.
Keep the target group fixed. The treated mean must be calculated among units with $D=1$, because the estimand is conditional on treatment received. A mean over all units answers a different question.
Confirm consistency and complete treated outcomes. The observed treatment and outcome definitions must correspond to the potential outcomes being discussed. If treated outcomes are missing for some treated units, the simple expression does not apply without addressing that missingness.
Report the scale and units. For an outcome measured in points, the bounds are in points; for a binary outcome, they are risk differences. Reversing a scale or changing its coding changes the interpretation.
Separate identification from precision. A small treated sample makes the sample mean uncertain. The population bound formula does not make that uncertainty disappear.

The worst-case bounds are a useful benchmark when little is known about treatment assignment, but their width can make them uninformative. A design that justifies random assignment or exchangeability can identify an ATT more tightly by linking observed and counterfactual outcomes. Matching and weighting approaches rely on such identifying assumptions; they do not obtain a narrower causal conclusion merely by rearranging observed data. The worst-case interval makes the cost of having no treatment-selection assumptions explicit.

Common Mistakes

Using the untreated group’s mean as the treated group’s counterfactual. That substitution assumes a link between treatment assignment and potential outcomes. The worst-case bounds make no such assumption.
Reversing the endpoints. The smallest ATT pairs the largest possible missing untreated mean with the observed treated mean: $E[Y\mid D=1]-U$. The largest ATT subtracts $L$.
Calling the identified interval a 95% confidence interval. It is a range of population effects compatible with the specified assumptions. It does not express a probability that the parameter lies in the interval and does not incorporate sampling uncertainty.
Claiming a positive effect because the treated outcome is high. A high treated mean alone cannot establish a positive effect: the treated group might also have had high outcomes without treatment.

Key takeaway: With outcomes in $[L,U]$, the treated group’s observed mean identifies one part of the ATT, while the missing untreated counterfactual mean can range from $L$ to $U$. The sharp worst-case bounds are $E[Y\mid D=1]-U$ and $E[Y\mid D=1]-L$.

Check Your Understanding

Answer each question using the outcome limits and target group specified. Solutions follow the questions.

Outcomes are bounded between 0 and 20 units, and the treated mean is 13. What are the worst-case ATT bounds?
For a binary outcome, the treated group’s favorable-outcome proportion is 0.70. Calculate the bounds and state whether they establish a positive effect.
Why can observed outcomes among untreated units not, by themselves, narrow the worst-case ATT bounds?
Suppose the treated mean equals the upper outcome limit $U$. What are the bounds, and what does that imply about the ATT under these assumptions?
A researcher reports bounds using limits that cover observed outcomes but may not cover all potential outcomes. What condition has not been justified?

Solutions. (1) The lower bound is $13-20=-7$, and the upper bound is $13-0=13$, so the interval is $[-7,13]$ units. (2) With $L=0$ and $U=1$, the bounds are $0.70-1=-0.30$ and $0.70-0=0.70$. Zero is included, so the assumptions do not establish a positive effect. (3) Without an assumption linking treatment status to potential outcomes, the untreated group’s observed $Y(0)$ values do not constrain treated units’ missing $Y(0)$ values beyond the stated support. (4) If the treated mean is $U$, the bounds are $U-U=0$ and $U-L$. The ATT must be nonnegative under the assumptions, though it need not be positive. (5) The researcher has not justified that both potential outcomes for every target-population unit lie within the specified limits; the bounds require that support condition, not merely that observed outcomes do.