Common-Support Restriction and the Restricted ATE

Why Define an ATE on Common Support?

The unrestricted average treatment effect (ATE) targets a specified population, whether or not that population contains covariate profiles represented among both treated and control units. In observational data, some profiles may occur almost exclusively in one treatment group. A direct comparison then relies heavily on extrapolation: the data provide little information about what the other treatment group’s outcomes would have been for those units.

A common-support restriction narrows the target population to units whose covariate profiles are represented among both treatment groups. The resulting estimand is not automatically the unrestricted ATE. It is an average causal effect for the units retained by the restriction, and its value can differ when treatment effects vary across covariate profiles.

Definition: Let $A\in\{0,1\}$ denote treatment, $X$ the pre-treatment covariates, and $Y(1)$ and $Y(0)$ the potential outcomes under treatment and control. For a specified region $S$ of covariate profiles, the restricted average treatment effect is $$ \tau_S=E\left[Y(1)-Y(0)\mid X\in S\right]. $$ The region $S$ is a common-support region when both treatment conditions are represented in it sufficiently to support the intended comparison. In a population definition, a basic requirement is $0<P(A=1\mid X=x)<1$ for profiles $x$ in $S$; practical analyses may use a prespecified narrower region to avoid probabilities too close to zero or one.

The definition makes two choices explicit. First, the causal contrast is $Y(1)-Y(0)$. Second, the averaging distribution is the covariate distribution among units with $X\in S$, not necessarily the distribution in the original population. The restriction therefore changes the target population. It does not, by itself, change the meaning of treatment or the individual causal contrast.

A region may be specified by covariate rules or by a range of propensity scores, where $e(x)=P(A=1\mid X=x)$. For example, an analyst might define $S$ as the profiles satisfying $0.10\leq e(x)\leq0.90$, together with any required covariate eligibility criteria. Such cutoffs are choices about the target population, not universal definitions of adequate support. They should be stated and justified rather than presented as a mechanical property of the data.

Identification Within the Restricted Region

The restricted ATE is a potential-outcome estimand. To identify it from observed data, the usual causal assumptions must hold for the units in $S$. Consistency requires that the observed outcome equals the potential outcome under the treatment actually received. There must be no interference between units, so one unit’s treatment does not alter another unit’s potential outcomes. Conditional exchangeability requires that, given the measured pre-treatment covariates, treatment assignment is independent of both potential outcomes. Finally, positivity requires a positive probability of each treatment condition at covariate profiles represented in $S$.

Identification result: If consistency, no interference, conditional exchangeability $\{Y(1),Y(0)\}\perp A\mid X$, and positivity $0<P(A=1\mid X=x)<1$ hold for $x\in S$, then $$ \tau_S = E\left[ E(Y\mid A=1,X)-E(Y\mid A=0,X) \mid X\in S \right]. $$ The inner contrast compares conditional observed-outcome means; the outer expectation averages those contrasts over the covariate distribution of the restricted target population.

The result follows by conditioning on $X$. For profiles in $S$, conditional exchangeability allows the potential-outcome means to be replaced by observed means within treatment groups, and consistency links those observed means to the corresponding potential outcomes:

\begin{aligned} E[Y(1)-Y(0)\mid X\in S] &=E\{E[Y(1)-Y(0)\mid X]\mid X\in S\}\\ &=E\{E[Y(1)\mid X]-E[Y(0)\mid X]\mid X\in S\}\\ &=E\{E[Y\mid A=1,X]-E[Y\mid A=0,X]\mid X\in S\}. \end{aligned}

Positivity matters because the conditional observed mean for each treatment group must be estimable at the profiles being averaged. Restricting to $S$ can improve the plausibility of that requirement, but it cannot repair unmeasured confounding or make poorly measured covariates adequate. Nor does a nonempty empirical overlap guarantee that the causal assumptions hold.

Worked Example: A Restricted ATE From Covariate-Stratum Effects

Consider a hypothetical target population restricted to two covariate strata. The low-risk stratum makes up 60% of this restricted population and the high-risk stratum makes up 40%. In the low-risk stratum, the observed mean outcome is 12 units among treated units and 8 units among controls. In the high-risk stratum, the corresponding means are 25 and 15 units. Assume the treatment probabilities are 0.30 and 0.60 in the two strata, respectively.

Both treatment groups occur in each stratum, so there is empirical positivity for these profiles. Suppose, additionally, that consistency, no interference, and conditional exchangeability given the recorded stratum hold. Under those assumptions, each observed conditional mean difference identifies the stratum-specific average treatment effect.

The estimated conditional effects are $12-8=4$ outcome units in the low-risk stratum and $25-15=10$ units in the high-risk stratum. Average these using the stratum proportions in the restricted target:

\hat{\tau}_S=(0.60)(4)+(0.40)(10)=2.4+4.0=6.4\text{ outcome units}.

Thus, under the stated assumptions, the estimated restricted ATE is 6.4 outcome units. The 60% and 40% weights are the covariate distribution in the restricted target, not the proportions of treated and control observations in the analysis. Using the latter proportions instead would generally define a different average.

How the Restricted and Unrestricted ATEs Differ

The relationship between the two estimands can be made explicit when the restricted and excluded groups partition the population. Let $q=P(X\in S)$, and let $\tau_{\text{out}}=E[Y(1)-Y(0)\mid X\notin S]$, when the outside region is nonempty. Then the unrestricted ATE is a weighted average of the restricted effect and the effect outside the region:

\tau_{\text{ATE}}=q\tau_S+(1-q)\tau_{\text{out}}.

This identity is a consequence of conditioning the population mean effect on whether $X\in S$. It shows why the restricted and unrestricted effects coincide if the outside region has no population mass, or if its mean effect equals $\tau_S$. Otherwise, the two effects generally differ. Even if restricting support makes estimation more credible, it does not establish that the restricted estimate applies to excluded units.

Worked Example: Quantifying the Change in Target

Suppose a hypothetical population contains 100 units. Eighty have profiles in the specified common-support region, where the average treatment effect is 5 days. The remaining 20 units have profiles outside that region, where the average effect is 11 days. These are hypothetical population effects, not quantities that would both be directly observed for each unit in an actual study.

The restricted ATE is $\tau_S=5$ days. The unrestricted ATE averages effects over all 100 units:

\tau_{\text{ATE}} =\frac{80}{100}(5)+\frac{20}{100}(11) =4+2.2 =6.2\text{ days}.

The difference, $6.2-5=1.2$ days, arises because the excluded units have a larger average effect in this example. The restricted result is a valid answer to the question about units in $S$; it is not an estimate of the all-population ATE unless additional assumptions justify extending it to those outside $S$.

Worked Example: A More Restrictive Region Changes the Estimate Again

Consider a second hypothetical population with three profile groups. A central-support group contains 700 of 1,000 units and has an average treatment effect of 4 outcome units. A near-edge group contains 200 units and has an average effect of 9 units. The remaining 100 units are outside adequate common support and have an average effect of 15 units.

If the specified region includes the central-support and near-edge groups, it contains 900 units. Its restricted effect is

\tau_{S_1} =\frac{700(4)+200(9)}{900} =\frac{2{,}800+1{,}800}{900} =5.111\text{ outcome units}.

If the analyst instead specifies a narrower region containing only the central-support group, the restricted effect is $\tau_{S_2}=4$ outcome units. The unrestricted ATE is

\tau_{\text{ATE}} =\frac{700(4)+200(9)+100(15)}{1{,}000} =\frac{6{,}100}{1{,}000} =6.100\text{ outcome units}.

All three values refer to different target populations: the narrower central region, the broader 900-unit region, and the full population. A difference between them is not automatically a contradiction or a sign of bias; it can arise from genuine treatment-effect heterogeneity and different averaging weights.

Choosing and Assessing the Restriction

A useful common-support restriction should have a clear target-population meaning. For example, it might retain profiles for which both treatment options are plausible according to subject-matter knowledge and observed treatment probabilities. The analyst should report the rule defining $S$, the fraction and types of units retained, and the population to which the resulting effect refers. If a cutoff is selected after inspecting the data, that selection should be disclosed: the chosen region may depend on the sample and may add uncertainty not reflected in a calculation that treats $S$ as fixed.

Several warning signs deserve attention. A region that retains almost no units may produce an estimand too narrow to answer the practical question. A region that retains profiles with treatment probabilities near zero or one may still have weak empirical support. Large differences in covariate composition between the original and restricted samples indicate that the target has shifted substantially; they do not by themselves show that the restriction is wrong. Finally, if a rule is based on post-treatment variables, it may change the causal question in ways not captured by the intended baseline-population estimand.

Common-support restriction is one way to address limited comparability, not a substitute for examining the design. The unrestricted ATE retains the original target but can require unsupported extrapolation. The restricted ATE narrows that target to $S$. The ATT instead averages effects for treated units, and the ATC averages effects for controls; either may include units outside a chosen common-support region unless the target is explicitly restricted. The overlap-population effect targets a population weighted toward profiles with treatment equipoise, which is conceptually related but is not simply the ATE conditional on an arbitrary region $S$. State the estimand before comparing numerical estimates.

Common Mistakes and Practical Interpretation

Calling the restricted estimate the ATE without qualification. If units were excluded, the estimate targets $E[Y(1)-Y(0)\mid X\in S]$, not necessarily the full-population ATE. Name the restricted target.
Assuming restriction removes confounding. Common support addresses a lack of treatment-group representation across profiles. Conditional exchangeability remains necessary for a causal interpretation.
Treating any observed overlap as sufficient. A few units in each group may not provide stable information, and a probability near zero or one can still imply weak support. The substantive adequacy of the region must be assessed.
Assuming the effect generalizes to discarded units. That requires further evidence or assumptions, such as a defensible model for effect patterns beyond $S$. The restricted ATE alone does not supply it.
Comparing two estimates without comparing their populations. A change in the estimate may reflect different covariate weights and effect heterogeneity, rather than a change in the underlying individual-level causal effects.

Key takeaway: A restricted ATE is defined by both a causal contrast and an explicit target region. Restriction may improve the credibility of comparisons where treatment and control units are represented, but the resulting effect applies to that region, and only to that region without additional assumptions.

Check Your Understanding

Try each question, then compare your reasoning with the solutions below.

A common-support region contains 75% of a population. Its ATE is 3 days, and the mean effect among excluded units is 7 days. What is the unrestricted ATE?
In a specified region, treatment and control units both occur at each covariate profile, but an important confounder is unmeasured. Does common support alone identify the restricted ATE?
A researcher reports an effect for units with propensity scores between 0.15 and 0.85. What additional information is needed to interpret the result?
Why might restricted and unrestricted ATEs differ even if both are estimated without bias for their respective targets?

Solutions. (1) The unrestricted effect is $(0.75)(3)+(0.25)(7)=2.25+1.75=4$ days. (2) No. Common support does not establish conditional exchangeability; unmeasured confounding can still make the observed conditional contrast differ from the causal contrast. (3) The researcher should describe the target population retained by the rule, the covariates or propensity-score model used to define it, and how many and which kinds of units were excluded. The assumptions for causal identification also need to be stated. (4) They average individual effects over different populations. If effects vary across profiles, changing the population weights changes the average even when each estimate correctly targets its own estimand.