Why Transporting an Observational Effect Requires Two Adjustments
An observational study can have treatment groups that differ in their prognosis, so comparing their observed outcomes may not identify a causal effect. Even after adjusting for this confounding within the study, the adjusted effect need not apply to a target population whose covariate distribution differs. Transporting an effect therefore involves two distinct questions: can the source study identify causal effects within covariate groups, and are those group-specific effects informative about the target?
Let \(S=1\) indicate membership in the observational source study and \(S=0\) membership in the target population. Let \(A\) be a binary treatment, \(Y\) the outcome, and \(X\) a set of measured pre-treatment covariates. Write \(Y^a\) for the outcome that would be observed under treatment level \(a\), where \(a\in\{0,1\}\). The target estimand in this tutorial is the target-population average treatment effect on the mean-difference scale.
The covariates \(X\) must be chosen for both parts of the problem. They must suffice to control confounding of treatment and outcome in the source study. They must also account for differences between source and target that matter for the causal contrast being transported. A covariate can be important for one role, both roles, or neither.
Identification Conditions
The following conditions give a standard sufficient route to identifying the target effect. They are assumptions about the data-generating process, not facts established merely by fitting a model.
- Consistency and a well-defined intervention: if a person receives \(A=a\), then \(Y=Y^a\). Treatment versions must be sufficiently well defined for this equality to make sense, and one person’s treatment must not alter another person’s outcome.
- Source treatment exchangeability: \(Y^a\perp A\mid X,S=1\) for \(a=0,1\). Conditional on \(X\), treatment assignment in the source is independent of the potential outcomes.
- Conditional transportability: \(E(Y^a\mid X=x,S=0)=E(Y^a\mid X=x,S=1)\) for \(a=0,1\) and covariate values relevant to the target. A stronger, commonly used assumption is \(Y^a\perp S\mid X\).
- Source treatment positivity: for every \(x\) occurring in the target, the source has a positive probability of each treatment: \(P(A=a\mid X=x,S=1)>0\).
- Sampling positivity: every covariate value occurring in the target has a positive chance of appearing in the source, equivalently \(P(S=1\mid X=x)>0\) over the target’s covariate support.
Conditional transportability is sometimes called conditional exchangeability across study participation. It does not require source and target populations to have the same covariate distribution. Instead, it requires the relevant potential-outcome mean, within a given \(X\) stratum, to agree across populations. The covariate distribution can differ and is explicitly accounted for in the target standardization.
Under source treatment exchangeability, consistency, and treatment positivity, the source conditional mean among people receiving \(a\) identifies the source potential-outcome mean:
The first equality follows from conditional exchangeability: within \(X=x\) in the source, restricting to \(A=a\) does not change the mean of \(Y^a\). The second follows because \(Y=Y^a\) for people who receive \(a\), by consistency. Conditional transportability then equates this identified source mean to the target potential-outcome mean at the same \(x\). Averaging over the target’s distribution of \(X\) gives the identification formula:
For continuous \(X\), replace the sums with integrals over the target covariate distribution. The formula has two logically separate pieces: estimate treatment-specific outcome means within source covariate strata, then average their contrast using the target—not the source—covariate distribution.
Worked Example: Confounding Adjustment Followed by Target Standardization
Worked Example: Confounding Adjustment Followed by Target Standardization
Suppose a hypothetical observational study compares a treatment for a binary outcome, with \(X\) indicating low or high baseline risk. The following conditional outcome risks and population proportions are assumed to be known for illustration. In the source, the low-risk proportion is 0.30 and the high-risk proportion is 0.70. In the target, the corresponding proportions are 0.75 and 0.25.
| Baseline risk group | Source risk if untreated | Source risk if treated | Target proportion |
|---|---|---|---|
| Low | 0.10 | 0.30 | 0.75 |
| High | 0.30 | 0.60 | 0.25 |
Assume consistency, source treatment exchangeability conditional on baseline risk, conditional transportability of both potential-outcome means, and positivity for treatment and study participation in both risk groups. These assumptions are substantive: for example, exchangeability requires that measured baseline risk captures the common causes of treatment and outcome in the source. The table alone cannot verify that no unmeasured confounding remains.
Within the low-risk group, the source risk difference is \(0.30-0.10=0.20\). Within the high-risk group, it is \(0.60-0.30=0.30\). Standardizing the treated risk to the target gives \(0.75(0.30)+0.25(0.60)=0.225+0.150=0.375\). The corresponding untreated risk is \(0.75(0.10)+0.25(0.30)=0.075+0.075=0.150\).
Thus, the identified target risk difference is \(0.375-0.150=0.225\), or 22.5 percentage points. Under the stated assumptions, treatment increases the mean binary outcome by 0.225 in the target population. This is not the crude difference between observed treatment groups in the source.
For comparison, standardizing the same conditional effects to the source distribution gives \(0.30(0.20)+0.70(0.30)=0.06+0.21=0.27\). The target effect differs because the risk groups have different effects and their proportions differ across populations. If source treatment assignment is also associated with baseline risk, the crude source comparison can differ from both standardized quantities.
Effect Transport Can Be Weaker Than Outcome-Mean Transport
The formula above transports each treatment-specific potential-outcome mean. That is a convenient sufficient condition, but for a risk difference it can be stronger than necessary. To identify the target risk difference, it is enough to assume that the conditional risk difference itself transports: \(E(Y^1-Y^0\mid X=x,S=0)=E(Y^1-Y^0\mid X=x,S=1)\). Source exchangeability and consistency identify the source conditional contrast; averaging that contrast over target \(X\) then identifies the target risk difference. This weaker condition allows baseline outcome risks to differ across populations within \(X\), provided the conditional difference remains the same.
Worked Example: Transporting a Risk Difference Despite Baseline Shifts
Consider a second hypothetical setting with two covariate groups. In the source, the untreated and treated risks are 0.10 and 0.30 in group 1, and 0.30 and 0.60 in group 2. The corresponding conditional risk differences are 0.20 and 0.30. Suppose the target untreated risks are instead 0.20 and 0.40, while the target treated risks are 0.40 and 0.70. The target conditional differences are still 0.20 and 0.30, although neither target potential-outcome mean equals its source counterpart.
Assume source treatment exchangeability and positivity identify the source conditional differences, and assume those differences—not necessarily the two outcome means separately—are invariant across populations within each group. If the target proportions are 0.60 in group 1 and 0.40 in group 2, then the target risk difference is \(0.60(0.20)+0.40(0.30)=0.12+0.12=0.24\).
The stronger mean-transport formula is not justified by these numbers: for example, the group 1 untreated risk is 0.10 in the source but 0.20 in the target. The effect-specific assumption nevertheless identifies the target risk difference as 0.24. This distinction is scale-dependent. An assumption that a risk difference transports does not automatically imply that a risk ratio, odds ratio, or other effect measure transports.
Worked Example: Omitting an Effect Modifier Can Give the Wrong Target Effect
Worked Example: Omitting an Effect Modifier Can Give the Wrong Target Effect
Suppose \(Z\) is a binary pre-treatment effect modifier. In a hypothetical source population, half the people have \(Z=0\) and half have \(Z=1\). After appropriate adjustment for source treatment confounding, suppose the conditional risk differences are 0.10 for \(Z=0\) and 0.50 for \(Z=1\). The source-average difference is \(0.50(0.10)+0.50(0.50)=0.05+0.25=0.30\).
Now suppose 20% of the target has \(Z=0\) and 80% has \(Z=1\). If conditional risk differences transport given \(Z\), the target risk difference is \(0.20(0.10)+0.80(0.50)=0.02+0.40=0.42\). Simply carrying over the source-average difference of 0.30 would understate the target effect by 0.12, or 12 percentage points.
The source and target differ in the distribution of a covariate that predicts how the treatment effect varies. A transport analysis that omits \(Z\) cannot justify averaging the source effects using the target composition. If \(Z\) is not measured in the target, this particular standardization cannot be carried out without additional information or assumptions.
What the Conditions Do—and Do Not—Guarantee
The identification formula is not a recipe that removes bias automatically. Source treatment exchangeability is a causal assumption about treatment assignment after conditioning on \(X\); it can fail if relevant confounders are unmeasured, poorly measured, or incorrectly timed. Conditional transportability is a separate assumption about how potential outcomes or effects relate to study participation after conditioning on \(X\). Measuring confounders in the source does not, by itself, establish transportability.
Both positivity conditions also matter. Treatment positivity requires that the source contain treated and untreated people at covariate values represented in the target. Sampling positivity requires that source information exist for target covariate values. If either condition fails, the displayed formula calls for conditional means that the source cannot identify from observed data. A fitted model may still produce numerical predictions there, but the predictions rely on extrapolation rather than the stated identification conditions.
The target distribution \(P(X\mid S=0)\) must also be known or estimable from suitable target-population information. Knowing the source’s covariate distribution is not a substitute when source and target compositions differ. Finally, the intervention represented by \(A=1\) and \(A=0\) must be meaningful in the target: a nominally identical treatment that entails different versions of care may not support the same causal interpretation.
Common Mistake
- Wrong reasoning: “I adjusted for confounding in the observational study, so its adjusted effect applies to the target.”
- Correction: Source adjustment identifies source conditional causal effects only under source exchangeability, consistency, and treatment positivity. Applying them to the target additionally requires an appropriate transport assumption, sampling positivity, and averaging over the target covariate distribution. If effects vary with covariates, the target average can differ from the source average even when conditional effects transport perfectly.
Check Your Understanding
Use the assumptions and formulas above to answer the following.
- State the target-population average treatment effect using potential outcomes.
- Which assumption addresses confounding of treatment in the source study, and which addresses differences between source and target?
- Why does the standardization formula use \(P(X\mid S=0)\) rather than \(P(X\mid S=1)\)?
- In the first worked example, what would the target untreated risk be if the target proportions were 0.50 low-risk and 0.50 high-risk?
- Explain how conditional risk differences could transport even when conditional untreated risks differ between populations.
- What does a failure of source treatment positivity mean for the identification formula at a target covariate value?