Three Different Requirements for Identification
A causal effect is not identified merely because treatment and outcomes have both been observed. Identification asks whether the observed-data distribution, together with stated assumptions, determines the causal quantity of interest. For a binary treatment \(A\), let \(Y(1)\) and \(Y(0)\) denote the outcomes that would occur under the two specified treatment interventions. The average treatment effect (ATE) in a target population is
Consistency, exchangeability, and positivity address different obstacles to learning this quantity from observed data. Consistency connects an individual's observed outcome to the potential outcome for the treatment actually received. Exchangeability addresses whether treated and untreated individuals are comparable with respect to their potential outcomes, possibly after conditioning on covariates \(X\). Positivity requires that each treatment option occur with positive probability in the covariate groups represented in the target population. None of these assumptions is a substitute for another.
These definitions presuppose that the treatment interventions and potential outcomes are well-defined, and that the unit-level potential-outcome framework is appropriate. In particular, consistency requires that the recorded treatment correspond to the intervention represented by \(Y(a)\); the earlier material on treatment versions and interference describes why those conditions matter. If there are no covariates, exchangeability can be stated unconditionally as \(\{Y(0),Y(1)\}\perp A\).
How the Assumptions Work Together
Under conditional exchangeability, consistency, and positivity, each mean potential outcome can be expressed using observed conditional outcome means and then averaged over the target population's covariate distribution. The argument has three distinct steps.
By iterated expectation, \(E[Y(a)]=E_X\{E[Y(a)\mid X]\}\).
Conditional exchangeability implies \(E[Y(a)\mid X]=E[Y(a)\mid A=a,X]\), wherever that treatment level is represented.
Consistency gives \(E[Y(a)\mid A=a,X]=E[Y\mid A=a,X]\). Positivity ensures this observed conditional mean exists on the target covariate support for each \(a\).
Combining the steps gives the g-formula for the mean potential outcome and the ATE:
The derivation shows why the assumptions have separate jobs. Exchangeability licenses the comparison within covariate groups; consistency connects those potential-outcome means to observed outcomes; and positivity makes the required observed comparisons available across the target population. If a required step fails, the formula is not justified just because it can still be calculated.
Worked Example: All Three Assumptions Support Standardization
Consider a hypothetical target population in which half the individuals have low baseline risk and half have high baseline risk. Suppose the conditional mean outcomes, measured in symptom-score points, are as follows. Assume the treatment is a single well-defined intervention, consistency holds, and treatment is conditionally exchangeable given baseline risk.
| Baseline risk \(X\) | \(E[Y\mid A=1,X]\) | \(E[Y\mid A=0,X]\) | \(P(A=1\mid X)\) |
|---|---|---|---|
| Low | 8 | 5 | 0.80 |
| High | 12 | 10 | 0.20 |
Both treatment probabilities are strictly between 0 and 1, so positivity holds in both risk groups. Under conditional exchangeability and consistency, the low-risk treatment effect is \(8-5=3\) points and the high-risk treatment effect is \(12-10=2\) points. Standardizing to the target population gives
Thus the estimated mean effect, under the assumptions, is a 2.5-point higher symptom score under treatment than under no treatment. The crude observed comparison need not equal this value. For example, if treated individuals are concentrated in the low-risk group and untreated individuals in the high-risk group, their outcome means mix different covariate distributions. Standardization compares treatment options within risk groups and then averages those contrasts using the target population's group proportions.
When Exchangeability Fails: Confounding Despite Overlap
Consistency and positivity do not guarantee exchangeability. In an observational study, treatment selection may depend on prognosis, leaving treated and untreated people different in their potential outcomes even within the covariates that were measured. The next hypothetical example has overlap and a well-defined treatment, but treatment assignment is related to baseline potential outcomes.
Worked Example: Treatment Selection Produces a Confounded Difference
Suppose four hypothetical individuals have no measured covariate differences, and their potential outcomes are:
| Individual | \(Y(0)\) | \(Y(1)\) | Observed \(A\) | Observed \(Y\) |
|---|---|---|---|---|
| 1 | 0 | 1 | 0 | 0 |
| 2 | 1 | 2 | 0 | 1 |
| 3 | 8 | 9 | 1 | 9 |
| 4 | 9 | 10 | 1 | 10 |
Treatment is well-defined and the observed outcome equals the potential outcome for the received treatment, so consistency holds. Both treatment levels occur in the population: \(P(A=1)=2/4=0.50\) and \(P(A=0)=2/4=0.50\), so positivity holds. Each individual's treatment effect is \(Y(1)-Y(0)=1\), giving a true ATE of 1 outcome unit.
However, treatment was given to the two individuals with the largest untreated potential outcomes. Therefore \(A\) is not independent of the potential outcomes, and exchangeability fails. The observed treated mean is \((9+10)/2=9.5\) units, while the observed untreated mean is \((0+1)/2=0.5\) units. Their difference is \(9.5-0.5=9.0\) units, not the true ATE of 1 unit. Having both treatment groups represented does not remove the systematic prognostic difference between them.
When Positivity Fails: A Missing Comparison
Exchangeability and consistency can hold while positivity fails. In that case, treatment and control outcomes may be comparable in principle within covariate groups, but one of the groups is absent for some covariate values. Observed data then cannot supply the conditional mean needed for standardization over those parts of the target population.
Worked Example: Deterministic Treatment by Covariate
Suppose half a hypothetical population has \(X=0\) and half has \(X=1\). Everyone with \(X=0\) receives \(A=0\), and everyone with \(X=1\) receives \(A=1\). The observed outcomes are 2 units for \(X=0\) and 5 units for \(X=1\). The treatment is well-defined and consistency holds.
Conditional exchangeability holds in the formal sense: within each value of \(X\), treatment is constant, so it is independent of the potential outcomes conditional on \(X\). But positivity fails because \(P(A=1\mid X=0)=0\) and \(P(A=0\mid X=1)=0\). The observed data reveal \(Y(0)=2\) for the \(X=0\) group and \(Y(1)=5\) for the \(X=1\) group, but not the other potential outcome in either group.
To see the identification problem, consider two hypothetical causal worlds that produce exactly these observed data. In the first, let the missing outcomes be \(Y(1)=3\) when \(X=0\) and \(Y(0)=4\) when \(X=1\). The effects are 1 unit in each group, so the ATE is \(0.5(1)+0.5(1)=1\) unit. In the second world, let the missing outcomes be \(Y(1)=100\) when \(X=0\) and \(Y(0)=-20\) when \(X=1\). The effects are then 98 and 25 units, and the ATE is \(0.5(98)+0.5(25)=61.5\) units. Both worlds agree on every observed treatment and outcome, yet their ATEs differ. The ATE is therefore not identified by these observed data and assumptions.
When Consistency Fails: The Treatment Is Not One Intervention
Consistency is not a claim that treatment groups are comparable. It is a link between the treatment variable and the potential outcomes: the observed outcome must equal the potential outcome for the intervention represented by the observed treatment value. If a binary label combines materially different versions, that link may fail for any single intervention the analyst intends to study.
Worked Example: A Treatment Label Combines Different Versions
Imagine a hypothetical randomized study in which \(A=1\) is recorded whenever a participant receives a pain-relief drug, whether as a low-dose oral formulation or a high-dose injection. Let the causal question concern a specified low-dose oral intervention versus no drug. Suppose assignment to the binary label is randomized, and both \(A=0\) and \(A=1\) occur with positive probability. Exchangeability and positivity for the randomized label therefore hold.
Now suppose some participants classified as \(A=1\) receive the injection rather than the specified oral intervention, and their outcomes can differ because of the version received. For those participants, the observed outcome is not necessarily the potential outcome \(Y(1)\) for the specified oral intervention. Consistency for that causal question fails. Randomization of the broad label does not make the observed mean among all \(A=1\) participants equal to the mean under the particular oral intervention. The analyst must define a coherent treatment contrast and ensure that the observed treatment variable represents it before using the identification formula.
Diagnostics, Alternatives, and Limits
These assumptions differ not only in their roles but also in how much the observed data can tell us about them.
- Positivity: estimated treatment probabilities and covariate distributions can reveal sparse or empty treatment groups. Very small estimated probabilities warn that comparisons may be unstable, even when strict positivity is not demonstrably violated. A sample with no observed group in a covariate region is a practical warning, not proof about every possible population.
- Exchangeability: measured covariate balance and knowledge of treatment assignment can inform an assessment, but balance on measured covariates does not establish independence from unmeasured causes of treatment and outcome. Exchangeability is a substantive assumption, not a property confirmed by one diagnostic statistic.
- Consistency: treatment definitions, protocols, and recorded versions should be examined. Outcome data alone generally cannot establish that distinct versions are causally interchangeable or that a label corresponds to a single intervention.
If exchangeability is doubtful, a different adjustment set or study design may be needed; adjustment cannot automatically remove unmeasured confounding. If positivity fails for the target population, one may need to change the target population to a region with overlap, while explicitly recognizing that this changes the estimand. If consistency fails, the treatment must be defined more precisely, or the effects of distinct versions must be treated as different causal questions. These are not interchangeable repairs: each addresses a different failure.
A common mistake is to say, “Both treatment groups appear in the data, so the causal effect is identified.” That reasoning checks only a crude form of overlap. It does not establish exchangeability within covariate groups, and it does not show that the treatment label represents one well-defined intervention. Conversely, randomization can support exchangeability but does not by itself solve a treatment-definition problem. Always ask which assumption performs each step of the identification argument.
Check Your Understanding
For each question, identify the assumption at issue and explain what follows.
- A study has both treated and untreated people at every measured covariate value, but treatment is preferentially given to people with worse unmeasured prognosis. Which assumption is threatened?
- In a covariate stratum, everyone receives treatment. Which assumption fails, and what comparison is unavailable?
- A treatment indicator combines two versions with different effects, but the causal question concerns only one version. Which assumption needs scrutiny?
- Suppose all three assumptions hold for a specified intervention and target population. What observed-data expression identifies the ATE?
Solutions. (1) Conditional exchangeability is threatened because assignment depends on prognosis not controlled by the measured covariates; observed group differences may remain confounded despite overlap. (2) Positivity fails because the probability of control in that stratum is zero; the conditional outcome mean under control is not observed there. (3) Consistency needs scrutiny because the recorded treatment value may not correspond to the specific intervention represented by the potential outcome. (4) The ATE is \(E_X\{E[Y\mid A=1,X]-E[Y\mid A=0,X]\}\), with the expectation taken over the target population's covariate distribution.