Why the Target Population Is Part of the Question
A causal effect is not fully specified by naming a treatment and an outcome. The population over which the effect is averaged matters too. A treatment may have different effects for different kinds of people; even if those effects are unchanged across settings, populations with different compositions can therefore have different average treatment effects. A study result for its participants does not automatically answer the corresponding question for every population in which the treatment might be used.
This tutorial makes the population part of the causal question explicit. It distinguishes a source population, associated with the study, from a target population, for which an effect is sought. The distinction is about the estimand—the precise causal quantity of interest—not yet about how to estimate it or what data would identify it.
Potential Outcomes and Population-Specific ATEs
Consider a binary treatment with levels 1 and 0, and a specified outcome measured at a specified time. For each unit, \(Y(1)\) denotes the outcome that would occur under treatment 1, and \(Y(0)\) the outcome that would occur under treatment 0. The unit-level causal effect is \(Y(1)-Y(0)\). Only one of these potential outcomes is observed for a unit under a single treatment assignment, but both define the causal contrast of interest.
Write these estimands as
Here \(\mathbb{E}_{S}\) and \(\mathbb{E}_{T}\) mean averages over the source and target populations, respectively. The algebraic equality follows from linearity of expectation. The treatment contrast and outcome must be defined consistently in both quantities: for example, “offer a six-week program” versus “usual care,” with symptom score measured at eight weeks. If the intervention, comparator, outcome, or measurement time changes, that is a different causal question, even if the populations remain the same.
An estimand is not an estimator. The estimand is the population quantity being sought; an estimator is a rule for calculating an estimate from observed data. Nor does defining a target ATE show that it can be identified from available data. Identification requires additional assumptions and information, and estimation requires an appropriate procedure. Those are separate questions from specifying which population’s average effect is wanted.
Generalization and Transport
The terms generalization and transport both concern applying causal evidence beyond the individuals directly studied, but they describe different population relationships. Usage can vary somewhat across fields; the distinction here is based on whether the desired population is the one associated with the study or a different population.
For instance, a trial’s enrolled participants may be a subset of the eligible patients in the health system that conducted the trial. Asking for the effect among all eligible patients is a generalization question. Asking for the effect among patients in a different health system is a transport question. The labels do not, on their own, guarantee that the effect is the same across populations or that the target effect can be learned from the source data.
A study can also report an effect for its own participants, while decision-makers need an effect for a larger or different group. These are distinct estimands even when they use the same treatment contrast and outcome. The first step is to state the population precisely; only then can one judge whether a study’s result addresses the intended question.
How Population Composition Changes an Average
Suppose a population consists of a finite set of groups, such as two age categories. If the average treatment effect within group \(k\) is \(\tau_k\), and the proportion of the population in that group is \(w_k\), then the population ATE is the group-weighted average of those effects. The weights are the proportions in the population whose ATE is being defined.
This is a statement about the estimand, not a recipe for obtaining valid estimates from a particular dataset. It follows by partitioning the population into mutually exclusive groups that cover it. Within each group, the mean individual effect is \(\tau_k\); averaging those group means across the population weights each one by that group’s share.
The derivation is the law of total expectation applied to the individual effect. If the group-specific effects are the same in two populations but the group proportions differ, their population ATEs can differ. Conversely, different population compositions do not necessarily imply different ATEs: if all \(\tau_k\) are equal, the weighted average is that same effect for any set of weights. These possibilities are about the definition and value of the average; they do not establish that subgroup effects are known or estimable.
Worked Example: Defining the Source-Population ATE
In a hypothetical trial, the source population is 500 adults eligible for a defined exercise program. The outcome is the number of symptom-free days during the four weeks after the program period. Suppose the average potential outcome under the program is 24.0 days and the average potential outcome under usual care is 21.5 days in this source population. What is the source-population ATE?
The intervention, comparator, outcome, follow-up period, and population are specified, so the causal contrast is clear. By definition,
The source-population ATE is 2.5 additional symptom-free days per adult over four weeks, comparing the program with usual care. This is a hypothetical population quantity, not necessarily the estimate calculated from a trial sample. It says nothing by itself about the effect in a different target population or about whether the potential-outcome means can be identified from observed data.
Worked Example: Different Population Shares, Different ATEs
Consider a hypothetical treatment whose outcome is symptom-free days over four weeks. There are two mutually exclusive groups. The average effect is 3.0 days in Group A and 0.0 days in Group B. Suppose Group A makes up 70% of the source population and 30% of the target population; the remaining proportions are in Group B. For this calculation, assume these group-specific effects apply in both populations.
For the source population, the group-weighted average is
For the target population, the corresponding calculation uses the target population’s proportions:
Thus the same group-specific effects yield different population ATEs: 2.10 days in the source population and 0.90 days in the target population. The difference comes from the defined populations’ group proportions, not from a change in the treatment contrast or outcome. The assumption that group-specific effects apply in both populations is essential to this arithmetic example; it is not established merely by knowing the proportions.
Worked Example: A Broader Source Population Versus a Separate Target
Suppose a hypothetical study includes two groups of participants and estimates average effects of 4.0 mmHg in Group A and 1.0 mmHg in Group B on systolic blood pressure after a specified treatment period. Assume for illustration that these group-specific effects are the same in each population considered. Group A is 60% of the study participants, 30% of the wider eligible population in the study’s health system, and 10% of patients in a separate health system.
The average for the study participants is
If the question is the ATE in the wider eligible population of the study’s own health system, the target is that broader population. The implied average is
This is a generalization question under the distinction used here. If the question instead concerns the separate health system, its population proportions give
That is a transport question. The three values—2.80, 1.90, and 1.30 mmHg—answer different population-specific questions, despite using the same treatment contrast and outcome. They are illustrative calculations, not conclusions that the study data alone establish these effects for the broader or separate populations.
Specifying the Estimand Before Interpreting a Result
A useful estimand statement names the population, treatment and comparator, outcome, and time point. For example: “the mean difference in symptom-free days over four weeks under the exercise program versus usual care among all eligible adults in the target health system.” That wording makes clear which population average is intended and what contrast is being averaged.
The target population is not merely a description added after an effect has been calculated. It determines the individuals over whom the mean is taken and therefore can determine the value of the estimand. If effects differ between groups, changing their proportions can change the overall average. If effects are constant across groups, the population change may leave the ATE unchanged—but that possibility should not be assumed without justification.
Finally, distinguish three claims that are often blurred together: a population-specific causal effect is defined; a study has estimated an effect for some population; and that estimate identifies or informs the effect in a different population. The first is an estimand question, the second concerns estimation, and the third requires support beyond the definition. Specifying the target ATE is necessary for a transport or generalization analysis, but it is not itself evidence that the desired effect has been recovered.
Common Mistake: Treating the Study Average as Universal
- Wrong reasoning: “The study estimated an ATE, so that is the treatment effect for any population.”
- Correction: An ATE is an average over a named population. A different target population can have different subgroup proportions and thus a different average, even under the same treatment contrast.
- Wrong reasoning: “Generalization and transport are just different names for the same target.”
- Correction: Generalization, as used here, concerns a broader population associated with the study; transport concerns a distinct population. In both cases, name the actual target rather than relying on the label.
- Wrong reasoning: “Defining the target ATE proves the study data can estimate it.”
- Correction: Definition, identification, and estimation are separate. A well-specified estimand does not establish the assumptions or data needed to learn its value.
Check Your Understanding
Use potential outcomes and population-specific averages to answer these questions.
- In one sentence, define the source-population ATE and the target-population ATE.
- A trial enrolls patients from one hospital, and the question concerns all eligible patients in that same hospital system. Is this generalization or transport under the distinction in this tutorial? Explain.
- Two groups have average effects of 5 and 2 units. Group A is 40% of the source population and 70% of the target population. Calculate both population ATEs.
- Can the ATE differ between two populations if the group-specific effects are identical across populations? State what else would have to differ.
- Why does a precise definition of the target ATE not, by itself, show that a study identifies or estimates it?