Transport with Target Covariates but No Target Outcomes

Why Target Outcomes May Not Be Needed

A study may measure outcomes for people in a source population while a separate sample from the target population provides only covariates. This can happen when outcome follow-up is expensive, the source is a clinical trial, or the target population is represented by a registry without the study outcome. The central question is whether the source outcomes, together with the target covariate distribution, identify the target population’s causal effect.

They can, but only under assumptions connecting the source study to the target population. Target covariates tell us how common measured characteristics are in the target; they do not, by themselves, reveal what the target population’s outcomes would be under either treatment. The source outcomes supply that information, while assumptions justify applying source outcome relationships to target members with the same covariates.

Definition: Let $S=1$ indicate membership in the source study and $S=0$ membership in the target population. Let $A\in\{0,1\}$ be treatment, $X$ the measured covariates, and $Y(a)$ the potential outcome under treatment level $a$. The target average treatment effect is $$ \tau_0 = E\left[Y(1)-Y(0)\mid S=0\right]. $$ It compares the target population’s mean potential outcome under treatment with its mean potential outcome under control. It does not require that target outcomes be observed, but identifying it does require information and assumptions that determine those means.

The covariates $X$ must capture the characteristics needed both to control treatment assignment in an observational source study and to account for differences in causal effects or potential outcomes between the source and target populations. Which role is relevant depends on the design and the scientific setting. A covariate may be needed for one role, both, or neither.

The Transport Formula and Its Assumptions

Write $m_a(x)=E[Y\mid A=a,X=x,S=1]$ for the source population’s observed mean outcome among people with covariates $x$ who received treatment $a$. In an observational source study, interpreting this as a causal mean requires adequate adjustment for treatment assignment. Transporting it to the target requires a separate assumption about comparability across populations.

Identification result: Suppose consistency holds, treatment assignment in the source is conditionally exchangeable given $X$, and the conditional mean potential outcomes are transportable: $$ E[Y(a)\mid X=x,S=1] = E[Y(a)\mid X=x,S=0], \qquad a\in\{0,1\}. $$ If both treatment levels are represented in the source for covariate values occurring in the target, then $$ \tau_0 = \int \left\{ E[Y\mid A=1,X=x,S=1] - E[Y\mid A=0,X=x,S=1] \right\}\,dF(x\mid S=0). $$ Here $F(x\mid S=0)$ is the target population’s covariate distribution. The formula uses source outcomes and target covariates, not target outcomes.

The assumptions have distinct meanings. Consistency says that a person’s observed outcome under the treatment actually received equals the corresponding potential outcome, and that treatment versions are sufficiently well defined for the comparison. Source treatment exchangeability is $Y(a)\perp A\mid X,S=1$: after conditioning on $X$, treatment groups in the source are comparable with respect to their potential outcomes. It is often plausible by design in a randomized trial; in an observational study it requires measuring and adjusting for the relevant treatment-outcome confounders.

The transport condition is different: it says that, at a given $X$, the source and target have the same mean potential outcome under each treatment. It is a conditional mean assumption, not a claim that the populations have identical covariate distributions or identical outcomes overall. It is also not guaranteed by having a large source sample. If a factor that changes potential outcomes across populations is omitted from $X$, the condition may fail.

The support condition in the result means that the source contains information about both treatment-specific outcomes for the covariate profiles represented in the target. This is necessary to use the displayed conditional means there. The condition is stated here because it is part of identification; its practical assessment requires careful attention to the covariate distributions.

Why the Formula Identifies the Target Effect

The reasoning starts by averaging within target covariate profiles. The law of iterated expectation gives the target mean potential outcome under treatment $a$ as an average of its conditional means over the target covariate distribution. The transport assumption replaces each target conditional mean by the corresponding source conditional mean. Source treatment exchangeability and consistency then let observed source outcomes identify those conditional causal means.

\begin{aligned} E[Y(a)\mid S=0] &= \int E[Y(a)\mid X=x,S=0]\,dF(x\mid S=0)\\ &= \int E[Y(a)\mid X=x,S=1]\,dF(x\mid S=0)\\ &= \int E[Y\mid A=a,X=x,S=1]\,dF(x\mid S=0). \end{aligned}

The first equality averages conditional means over target covariates. The second uses conditional mean transportability. For the final equality, source exchangeability gives $E[Y(a)\mid X=x,S=1]=E[Y(a)\mid A=a,X=x,S=1]$; consistency identifies this last quantity with the observed outcome mean among source members receiving $a$. Subtracting the resulting expression for $a=0$ from that for $a=1$ yields the transport formula.

This argument also makes the data requirements clear. Source outcomes must be observed under both treatment levels, alongside treatment and the covariates needed for the assumptions. The target data must characterize the distribution of those covariates. No target outcome is needed for this point identification result, but the assumptions—not the target covariate records—supply the link from source outcomes to target potential outcomes.

Worked Applications

Worked Example: Transporting a Risk Difference from a Randomized Source Study

Suppose a hypothetical randomized source study records outcomes for two covariate groups, low and high baseline risk. Within each group, 100 participants received treatment and 100 received control. The numbers with the outcome were as follows.

Covariate group	Treated outcomes / 100	Control outcomes / 100
Low risk	20	30
High risk	50	70

Assume randomization was implemented as specified, treatment and outcome definitions match the target question, and conditional mean transportability holds given risk group. A hypothetical target covariate sample indicates that 75% of the target population is low risk and 25% is high risk. The target sample has no outcome measurements.

The source risks are $0.20$ and $0.50$ under treatment, and $0.30$ and $0.70$ under control, in the low- and high-risk groups respectively. Applying the target group proportions gives $$ \begin{aligned} E[Y(1)\mid S=0] &= 0.75(0.20)+0.25(0.50)=0.275,\\ E[Y(0)\mid S=0] &= 0.75(0.30)+0.25(0.70)=0.400,\\ \tau_0 &= 0.275-0.400=-0.125. \end{aligned} $$

The transported risk difference is $-0.125$, or 12.5 percentage points lower risk under treatment than under control, for the target population. This conclusion depends on the randomization supporting the source causal comparisons and on the stated transport condition. The target covariate proportions determine how the two source-group effects are averaged; they do not independently verify the transport condition.

Worked Example: Observational Source Outcomes Need Confounding Control

Consider a hypothetical observational source study in which baseline risk $L$ affects both treatment receipt and outcome. Within each level of $L$, suppose the observed data are:

Baseline risk	Treatment group	Participants	Outcome cases
Low	Treated	100	10
Low	Control	100	20
High	Treated	100	40
High	Control	100	60

Suppose the target population is 20% low risk and 80% high risk. For this calculation to have a causal interpretation, assume source treatment exchangeability conditional on $L$, consistency, and conditional mean transportability given $L$. In particular, the exchangeability assumption requires that $L$ suffice to address confounding in the source; this is an assumption, not something established by the table.

The source outcome risks are $10/100=0.10$ and $20/100=0.20$ among treated and control low-risk participants, and $40/100=0.40$ and $60/100=0.60$ among treated and control high-risk participants. Standardizing to the target distribution gives $$ \begin{aligned} E[Y(1)\mid S=0] &= 0.20(0.10)+0.80(0.40)=0.340,\\ E[Y(0)\mid S=0] &= 0.20(0.20)+0.80(0.60)=0.520,\\ \tau_0 &= 0.340-0.520=-0.180. \end{aligned} $$

Under the assumptions, the estimated target risk difference is $-0.180$, or 18.0 percentage points lower risk under treatment. Comparing the source’s unadjusted treatment and control risks would not implement this target standardization and could mix treatment differences with the differing composition of treatment groups. Measuring target $L$ helps set the target mix, but does not repair unmeasured confounding in the source.

Worked Example: Target Covariates Cannot Replace Missing Source Outcomes

Suppose a hypothetical source has a treated group with 20 outcome cases among 100 people, and a target covariate sample shows that 60% of the target population has a particular measured profile. Assume the profile is the same for everyone in this illustration. If no control outcomes were observed in the source, the target covariate information does not determine the target control risk.

For example, the observed treated risk is $20/100=0.20$. One possible source-and-target relationship has a control risk of $0.10$, implying a risk difference of $0.20-0.10=0.10$. Another has a control risk of $0.40$, implying a risk difference of $0.20-0.40=-0.20$. Both are compatible with the observed treated outcomes and the target covariate proportion, because neither supplies any control outcomes or an assumption that fixes the missing control mean.

Thus, target covariate measurement does not substitute for source outcome information under both treatment levels. A causal contrast needs both potential-outcome means identified; the target covariate distribution only determines how identified conditional means are averaged.

What the Result Does—and Does Not—Establish

With target covariates but no target outcomes, the transported effect is identified through a combination of source outcome information and assumptions about treatment assignment and population comparability. Target outcomes are not required for the formula, but their absence limits what can be checked empirically. In particular, target covariates alone cannot show that the source conditional outcome means apply to the target.

The target covariate data must also represent the target population well enough to characterize $F(x\mid S=0)$, and the covariates used in the analysis must be measured comparably in source and target. If the target sample is not representative, its observed covariate proportions may not be the target distribution required by the estimand. Likewise, if relevant effect modifiers or baseline outcome predictors differ between populations but are omitted, the conditional mean transport assumption may be implausible.

The formula targets the stated causal effect scale. For a binary outcome, the example reports a risk difference because it subtracts two target mean risks. A different effect measure requires defining that estimand and deriving the corresponding target-standardized contrast; it should not be substituted silently for the risk difference.

Common Mistake

Common mistake: “The target covariate sample is large, so the target effect is identified even though no target outcomes were collected.” A large sample can estimate the target covariate distribution precisely, but it cannot establish source treatment exchangeability, transportability of conditional potential-outcome means, or fill in source outcome information that was never observed. The correction is to state the source outcome data required and assess each identifying assumption separately; precise target covariate proportions do not make those assumptions true.

The practical value of this design is that it can extend evidence from a study to a population that was not followed for outcomes. Its credibility rests on a transparent chain: source outcomes identify conditional causal means; measured covariates make those means comparable across populations; and target covariate data determine the target population’s mix of those profiles. If any link is unsupported, the transported result should be interpreted as conditional on that uncertainty rather than as a conclusion established by target covariates alone.

Check Your Understanding

Use the assumptions and distinctions in this tutorial to answer the following questions.

What does the target average treatment effect compare, and why does its definition not require observed target outcomes?
In an observational source study, what role does conditional exchangeability of treatment assignment play in the transport formula?
Which data determine the source conditional outcome means, and which data determine the averaging distribution in the formula?
Why does a large target covariate sample not establish conditional mean transportability?
If the source contains treated outcomes but no control outcomes, what additional information or assumption is missing for identifying a treatment effect?