Bounds From Restrictions On Treatment Effect Heterogeneity

Why Restrict the Range of Individual Effects?

An average treatment effect can be difficult to identify when treatment assignment is related to potential outcomes. One useful source of information is a substantive restriction on how large or small any person's treatment effect can be. Such a restriction need not say that everyone benefits, or that effects are equal. It only says that each effect lies within a specified range.

That restriction alone puts the average effect in the same range, but the observed outcomes can make some effects within that range impossible. For example, if outcomes are known to lie between 0 and 10, a treated person's observed outcome of 1 cannot be consistent with a treatment effect of 4: the implied untreated outcome would be below 0. The central task is to combine the treatment-effect restriction with the known outcome scale, person by person, and then aggregate the resulting limits.

Definition: Let \(D\in\{0,1\}\) indicate treatment, and let \(Y(1)\) and \(Y(0)\) be the potential outcomes under treatment and control. The individual treatment effect is \(\tau=Y(1)-Y(0)\). The average treatment effect (ATE) is \(E[\tau]\). A restriction on treatment-effect heterogeneity specifies known constants \(L\le U\) such that \(L\le\tau\le U\) for every individual.

The effect restriction permits heterogeneity: different individuals can have different \(\tau\). It does not assert that treatment is randomly assigned, or that observed treated and control outcomes are directly comparable. We also require a known common support for both potential outcomes: \(a\le Y(1)\le b\) and \(a\le Y(0)\le b\). The endpoints \(a\) and \(b\) must be justified as limits on possible outcomes, not merely the smallest and largest values seen in a sample.

Deriving the Outcome-Specific Restrictions

Consistency means that the observed outcome is \(Y=Y(1)\) for a treated individual and \(Y=Y(0)\) for a control. For a treated individual, the observed outcome is \(Y(1)\), so the missing outcome is \(Y(0)=Y(1)-\tau\). Requiring this missing outcome to lie in \([a,b]\) further restricts the possible effect. For a control, the missing outcome is \(Y(1)=Y(0)+\tau\), and its support restriction likewise limits the effect.

\begin{aligned} D=1:\quad &a\le Y(1)-\tau\le b &&\Longrightarrow\quad Y(1)-b\le\tau\le Y(1)-a,\\ D=0:\quad &a\le Y(0)+\tau\le b &&\Longrightarrow\quad a-Y(0)\le\tau\le b-Y(0). \end{aligned}

Intersect each of these intervals with the assumed range \([L,U]\). For an observed treated outcome \(y\), the compatible effects range from \(l_1(y)\) to \(u_1(y)\); for an observed control outcome \(y\), they range from \(l_0(y)\) to \(u_0(y)\).

\begin{aligned} l_1(y)&=\max\{L,y-b\},& u_1(y)&=\min\{U,y-a\},\\ l_0(y)&=\max\{L,a-y\},& u_0(y)&=\min\{U,b-y\}. \end{aligned}

Sharp-bound result: Suppose consistency holds, both potential outcomes lie in the known support \([a,b]\), and every individual effect lies in \([L,U]\). Let \(q=P(D=1)\). If the compatible intervals are nonempty for almost every observed outcome, then the identified set for the ATE is

\left[ qE\{l_1(Y)\mid D=1\}+(1-q)E\{l_0(Y)\mid D=0\},\ qE\{u_1(Y)\mid D=1\}+(1-q)E\{u_0(Y)\mid D=0\} \right].

Here “sharp” means that, without additional assumptions, each endpoint can be attained by some assignment of the unobserved potential outcomes consistent with the restrictions and the observed data. The result does not require treatment to be independent of potential outcomes.

Why the Bounds Hold—and When They Are Sharp

By the law of total expectation, the ATE is the treatment-share-weighted mean effect among treated individuals plus the corresponding mean among controls. Each observed outcome restricts that person's effect to the interval just derived. Therefore the effect mean in each group is at least the average of its lower limits and at most the average of its upper limits. Weighting the group limits by \(q\) and \(1-q\) gives the stated ATE interval.

To see why no narrower interval follows from these assumptions alone, construct potential outcomes at the limits. For each treated individual, retain their observed \(Y(1)\) and set \(Y(0)=Y(1)-l_1(Y(1))\) to attain the lower effect limit. The intersection defining \(l_1\) ensures that this missing outcome is in \([a,b]\); the construction for controls is \(Y(1)=Y(0)+l_0(Y(0))\). These choices preserve the observed outcomes and satisfy the restrictions, so they attain the lower ATE endpoint. Replacing each lower limit by its upper limit constructs the upper endpoint in the same way. No cross-person restriction in the assumptions prevents these endpoint constructions.

If the intervals are empty for outcomes that occur with positive probability, the assumptions cannot all hold for that population. If the outcome support is not known, observed values alone do not establish a valid population support. In that case, the effect-range restriction still implies \(L\le E[\tau]\le U\), but the observed outcome distributions cannot tighten it by this argument.

Worked Example: Outcome Extremes Tighten the Effect Range

Consider a hypothetical population with \(q=0.5\). The treated group's outcomes are 1 and 2 with equal probability; the control group's outcomes are 8 and 9 with equal probability. Assume both potential outcomes lie in \([0,10]\) and every individual effect lies in \([-2,4]\). Calculate the sharp ATE bounds.

For treated outcomes, \(l_1(y)=\max\{-2,y-10\}\) and \(u_1(y)=\min\{4,y\}\). At \(y=1\), the compatible effect range is \([-2,1]\); at \(y=2\), it is \([-2,2]\). Thus the treated-group mean effect lies between \[ ( -2 + -2 )/2=-2 \quad\text{and}\quad (1+2)/2=1.5. \]

For control outcomes, \(l_0(y)=\max\{-2,-y\}\) and \(u_0(y)=\min\{4,10-y\}\). At \(y=8\), the range is \([-2,2]\); at \(y=9\), it is \([-2,1]\). The control-group mean effect therefore also lies between \(-2\) and \(1.5\).

Weighting by the equal group shares gives \[ 0.5(-2)+0.5(-2)=-2,\qquad 0.5(1.5)+0.5(1.5)=1.5. \] The ATE is bounded by \([-2,1.5]\), in outcome units. The original individual-effect restriction allowed effects as high as 4, but the observed outcomes near the edges of the known support rule out such large effects for these individuals.

Worked Example: The Distribution Matters, Not Just the Means

Take \(a=0\), \(b=10\), \(L=-2\), \(U=4\), and equal treatment and control shares. Suppose first that everyone in both groups has an observed outcome of 5. For either group, the observed outcome imposes no additional restriction: the compatible effect interval is \([-2,4]\). The ATE bounds are therefore \([-2,4]\).

Now consider a different hypothetical population in which each group has observed outcomes 1 and 9 with equal probability. Each group still has mean 5. Among treated individuals, outcome 1 gives the interval \([-2,1]\), and outcome 9 gives \([-1,4]\). The mean lower and upper limits are \((-2-1)/2=-1.5\) and \((1+4)/2=2.5\).

Among controls, outcome 1 gives \([-1,4]\), and outcome 9 gives \([-2,1]\). Their average limits are also \(-1.5\) and \(2.5\). Weighting the groups gives ATE bounds of \([-1.5,2.5]\). Thus two populations with the same observed group means can have different bounds: applying the restrictions to each outcome before averaging preserves information that the means discard.

Worked Example: Checking Compatibility Before Reporting Bounds

Suppose outcomes are known to lie in \([0,10]\), effects are restricted to \([2,4]\), and a treated individual has observed outcome 1. Since \(Y(0)=Y(1)-\tau\), this person's untreated outcome would have to be between \(1-4=-3\) and \(1-2=-1\). Neither value is in \([0,10]\). Equivalently, \[ l_1(1)=\max\{2,1-10\}=2,\qquad u_1(1)=\min\{4,1-0\}=1, \] so \(l_1(1)>u_1(1)\). The assumptions are incompatible with this observed outcome; it would be incorrect to report an ATE interval using an empty compatible range.

For a compatible illustration with the same assumptions, let treated outcomes be 3 and 4, and control outcomes be 7 and 8, again with equal shares and equal probabilities within each group. Treated outcomes 3 and 4 give effect intervals \([2,3]\) and \([2,4]\), whose average lower and upper limits are 2 and 3.5. Control outcomes 7 and 8 give \([2,3]\) and \([2,2]\), with average limits 2 and 2.5. The ATE bounds are \[ 0.5(2)+0.5(2)=2,\qquad 0.5(3.5)+0.5(2.5)=3. \] The resulting interval is \([2,3]\), in outcome units.

Diagnostics, Alternatives, and Limitations

A useful diagnostic is to calculate \(l_1(y)\), \(u_1(y)\), \(l_0(y)\), and \(u_0(y)\) across the observed outcome distributions before aggregating. Check that each lower limit is no greater than its upper limit. Also check the provenance of \([a,b]\) and \([L,U]\): both are substantive assumptions, not quantities automatically validated by the observed data. If the support is chosen too narrowly, or the effect range is implausibly restrictive, the resulting interval can appear precise while resting on questionable premises.

These bounds differ from point identification under conditional exchangeability or random assignment, which can identify an ATE under their own assumptions. They also differ from bounds based only on a known outcome support: here the treatment-effect range adds a person-level restriction, and the support can further restrict that range according to each observed outcome. The earlier material on sensitivity curves for treatment effect bounds considers how bounds vary as a restriction changes; here the emphasis is the derivation and role of the observed outcome distributions at a specified restriction.

The method does not recover the distribution of individual effects, reveal which particular individuals benefit, or correct confounding by itself. It yields a population interval under explicit restrictions. If \([a,b]\) is unavailable, or if the compatible intervals are wide, the bounds may be uninformative. Finally, these are population identification bounds: estimates computed from a finite sample require separate inference to account for sampling uncertainty.

Common Pitfalls

Using the effect range as if it were a constant effect. The restriction \(L\le\tau\le U\) allows effects to vary. It does not set every individual's effect to one common value.
Using sample extrema as a known outcome support. The largest observed outcome is not necessarily a population maximum. A support restriction needs substantive justification.
Clipping an empty interval and continuing. If \(l_d(y)>u_d(y)\), that outcome contradicts the joint assumptions; it is not a valid interval to average.
Using only group means. The limits involve maxima and minima applied to each outcome, so the full outcome distributions can matter even when group means agree.
Calling the interval a confidence interval. It describes identification under assumptions, not sampling uncertainty. Statistical inference is a separate task.

Check Your Understanding

Answer each question, then compare with the solutions below.

With outcomes in \([0,8]\) and effects in \([-1,3]\), find the compatible effect interval for a treated individual with observed outcome 2.
Under the same restrictions, find the compatible effect interval for a control with observed outcome 7.
Why can two populations with identical treated and control outcome means have different ATE bounds?
If an observed treated outcome implies \(l_1(y)>u_1(y)\), what does that tell you?

Solutions. (1) \(l_1(2)=\max\{-1,2-8\}=-1\) and \(u_1(2)=\min\{3,2-0\}=2\), so the interval is \([-1,2]\). (2) \(l_0(7)=\max\{-1,0-7\}=-1\) and \(u_0(7)=\min\{3,8-7\}=1\), giving \([-1,1]\). (3) The outcome-specific limits apply nonlinear maxima and minima before averaging, so differences in the distributions, including how much mass lies near support edges, can change the bounds without changing the means. (4) The observed outcome cannot be reconciled with the assumed outcome support and effect range; check the assumptions or data rather than averaging the incompatible limits.