Why Identification Requires Both Treatment Levels
Covariate adjustment estimates an average treatment effect by comparing treated and untreated outcomes within covariate values, then averaging those comparisons over a target population. That comparison cannot be made at a covariate value where everyone receives the same treatment: one of the two outcome means is absent from the observed data. This is not merely a computational inconvenience. Without additional assumptions, the missing mean is not determined by the observed data, so the target effect is not identified.
The key condition is positivity: every covariate value occurring in the target population must have a nonzero probability of receiving either treatment. The condition is about the population assignment mechanism, not whether a particular finite sample happens to include both treatment groups in every covariate subgroup.
Assumptions and the Covariate-Adjustment Formula
The formula applies to a specified target population and requires finite expectations for the outcomes being averaged. Its causal assumptions are consistency, conditional exchangeability, and conditional positivity. Consistency means the observed outcome equals the potential outcome corresponding to the treatment actually received: \(Y=AY(1)+(1-A)Y(0)\). Conditional exchangeability means \((Y(0),Y(1))\perp A\mid X\): after conditioning on \(X\), treatment assignment is independent of both potential outcomes. Positivity requires both treatment probabilities to be positive for almost every covariate value represented in the target population.
For \(a\in\{0,1\}\), define the observed conditional outcome mean \(m_a(x)=E[Y\mid A=a,X=x]\), where it is defined by observed data. Under consistency, exchangeability, and positivity, these means equal the conditional potential-outcome means in the target population. The ATE is therefore identified by averaging their difference over the target covariate distribution:
The first equality is the law of iterated expectations. The second uses linearity of expectation. For the third, conditional exchangeability gives \(E[Y(a)\mid X]=E[Y(a)\mid A=a,X]\); consistency then gives \(E[Y(a)\mid A=a,X]=E[Y\mid A=a,X]\). Positivity matters at this step: if \(P(A=a\mid X=x)=0\), the conditional observed mean \(E[Y\mid A=a,X=x]\) is not available from the observed-data distribution at that \(x\). Thus the expression cannot be evaluated over the full target population without imposing further assumptions.
Worked Example: Adjustment When Both Treatment Levels Occur
Suppose a hypothetical target population has two covariate strata, \(X=L\) and \(X=H\), with probabilities \(P(X=L)=0.6\) and \(P(X=H)=0.4\). In each stratum, half the population receives treatment and half receives control. Assume consistency and conditional exchangeability, and suppose the observed conditional outcome means are as follows:
| Stratum | Target proportion | \(e(x)\) | \(m_1(x)\) | \(m_0(x)\) |
|---|---|---|---|---|
| \(L\) | 0.6 | 0.5 | 8 | 4 |
| \(H\) | 0.4 | 0.5 | 10 | 8 |
Both treatment probabilities are positive in both strata, so positivity holds. The stratum-specific adjusted differences are \(8-4=4\) outcome units for \(L\) and \(10-8=2\) outcome units for \(H\). Averaging over the stated target proportions gives
Under the assumptions, the covariate-adjustment formula identifies an ATE of 3.2 outcome units for this target population. The calculation is not the unweighted average of the two stratum effects: the target population contains more individuals in stratum \(L\), so that stratum receives more weight.
What Fails at a Structural Zero
A structural positivity violation occurs when treatment is impossible for a covariate value in the target population. If \(e(x)=0\), nobody with covariates \(x\) is treated, so the observed data reveal \(m_0(x)\) but not \(m_1(x)\). If \(e(x)=1\), nobody with those covariates is untreated, so the observed data reveal \(m_1(x)\) but not \(m_0(x)\). Either case leaves a term in the ATE formula unspecified.
The following example shows that this is a failure of identification, not simply a missing entry that can be filled in by calculation. Two causal models can agree on all observed data and on the assumptions, yet have different ATEs.
Worked Example: ATE Not Identified When Treatment Probability Is Zero
In a hypothetical target population, half the individuals have \(X=0\) and half have \(X=1\). Everyone with \(X=0\) has probability \(0.5\) of treatment; everyone with \(X=1\) is untreated, so \(e(1)=0\). Suppose the observed conditional means are \(m_1(0)=8\), \(m_0(0)=5\), and \(m_0(1)=4\). There are no treated observations with \(X=1\).
Consider two possible assignments of the unobserved treated potential outcome at \(X=1\). In Model A, let \(E[Y(1)\mid X=1]=6\); in Model B, let \(E[Y(1)\mid X=1]=10\). Both models have the same observed outcomes: individuals with \(X=1\) are untreated and have mean 4. Both can satisfy conditional exchangeability: treatment is randomized within \(X=0\), and treatment is constant within \(X=1\). Consistency also holds in both models.
The effect within \(X=0\) is identified as \(8-5=3\). In Model A the effect within \(X=1\) is \(6-4=2\), giving
In Model B the effect within \(X=1\) is \(10-4=6\), giving
The observed data and assumptions are the same, but the implied ATEs differ. Consequently, the full-population ATE is not identified from those data under these assumptions. The adjustment formula makes the gap explicit: \(m_1(1)\) is needed in \(E\{m_1(X)-m_0(X)\}\), but the observed data do not determine it.
Worked Example: Treatment Probability One Also Leaves a Missing Mean
Suppose a hypothetical target population has \(P(X=L)=0.4\) and \(P(X=H)=0.6\). In the \(L\) stratum, treatment probability is \(0.5\), with observed means \(m_1(L)=7\) and \(m_0(L)=4\). In the \(H\) stratum, everyone is treated, so \(e(H)=1\), and the observed treated mean is \(m_1(H)=9\). Assume consistency and conditional exchangeability. The effect for \(L\) is identified as \(7-4=3\) outcome units, but the untreated mean for \(H\) is not observed.
If the unobserved \(m_0(H)\) were 6, the ATE would be \(0.4(3)+0.6(9-6)=3.0\). If instead it were 8, the ATE would be \(0.4(3)+0.6(9-8)=1.8\). These different values are both compatible with the observed means. The full target-population ATE is therefore not identified. The effect for the restricted population with \(X=L\), however, is identified as 3 outcome units, provided that this is the target population of interest and the assumptions hold there.
This illustrates an important distinction: restricting attention to covariate values with observed support can yield an identifiable effect for that restricted population, but it does not identify the original full-population ATE. Changing the target population changes the estimand.
Zero, Near-Zero, and Finite-Sample Gaps
An exact zero and a very small positive probability have different implications. If \(e(x)=0\) or \(e(x)=1\) on a set with positive probability in the target population, the nonparametric adjustment formula lacks one conditional mean there. If \(e(x)\) is positive but close to zero, the formula can still be identified, but data may contain few treated individuals with those covariates. Estimates of the corresponding mean can then be imprecise and sensitive to the observed outcomes. Small probabilities create a practical data problem even when they do not create the same formal identification failure as a structural zero.
Conversely, a finite sample can have no treated observations in a subgroup even when the population treatment probability is positive. That sample gap is not, by itself, proof that the population probability is exactly zero. Positivity is a condition on the population mechanism; assessment must distinguish substantive impossibility from limited sample information.
In practice, warning signs include treatment assignment rules that exclude a group by design, covariate strata with only one treatment level, and estimated treatment probabilities near a boundary. These signs call for careful scrutiny of the data and design. An observed lack of a treatment group can make an estimate unavailable or unstable, but it does not alone establish the population probability. The earlier material on Common Score Support Versus Covariate Support is relevant here: the condition in this derivation concerns treatment availability at covariate values in the target population, not merely the appearance of a score range.
Why Other Estimation Strategies Do Not Remove the Requirement
Covariate adjustment, inverse-probability weighting, and subclassification are different ways of using observed treatment and covariate information. Under the same causal assumptions, they target effects through the same underlying comparison of treatment and control outcomes. Weighting expresses the comparison by reweighting observed outcomes; adjustment expresses it through conditional outcome means; subclassification aggregates comparisons across groups. None of these operations reveals an outcome mean for a treatment level that never occurs at a target covariate value.
An outcome model may still produce a numerical prediction in an unsupported stratum. That prediction relies on how the model extrapolates beyond the treatment-group data there; it is not identified by the observed data alone. Such a result may be defensible if additional functional-form or scientific assumptions are stated and justified, but it should not be presented as nonparametric identification from positivity and exchangeability. Likewise, changing the target population to exclude unsupported strata can make a different effect identifiable, but that change must be explicit.
Common Mistake
- Wrong reasoning: “We can fit a regression for the outcome, so the model supplies the missing treated or untreated mean and the original ATE is identified.”
- Correction: A model can generate an extrapolated prediction, but the observed data do not determine its value in a stratum with no observations at that treatment level. Identification then depends on extra modeling assumptions, not on the covariate-adjustment formula alone.
A related mistake is to treat a very small estimated propensity score as proof that the ATE is mathematically unidentified. Near-boundary probabilities can create severe instability, but formal nonidentification under the stated assumptions arises when a treatment probability is zero for a set of positive probability in the target population.
Check Your Understanding
Try each problem, then check the solutions below.
- State the three assumptions used to identify the ATE by covariate adjustment.
- A target population has two equally common strata. In the first, \(m_1=9\) and \(m_0=5\); in the second, \(m_1=8\) and \(m_0=6\). If positivity and the other assumptions hold, calculate the ATE.
- In one target-population stratum, \(e(x)=0\). Which conditional outcome mean is absent from the observed data, and why does that matter for the ATE?
- Does observing no treated individuals in a small sample subgroup prove that the population treatment probability is zero? Explain.
- An analyst changes the target to a covariate subgroup where both treatment levels occur and estimates an effect there. Has the analyst identified the original full-population ATE? Explain.
Solutions. (1) Consistency, conditional exchangeability given \(X\), and conditional positivity for both treatment levels almost everywhere in the target population. (2) The stratum effects are \(9-5=4\) and \(8-6=2\); the equally weighted ATE is \(0.5(4)+0.5(2)=3\) outcome units. (3) \(m_1(x)=E[Y\mid A=1,X=x]\) is absent because no one with those covariates is treated. Its value is needed for the conditional treatment effect and cannot be recovered from the observed data without additional assumptions. (4) No. A finite-sample gap can occur when the population probability is positive; it is not proof of a structural zero. (5) No. The subgroup effect pertains to a restricted target population. It does not identify the original ATE unless further assumptions connect the subgroup effect to the effects in excluded strata.