Defining Target Populations for Causal Effects

Why the Target Population Is Part of the Causal Question

A causal effect is not fully described by naming a treatment and an outcome. “Does this program improve health?” leaves open whose health is being considered, what the comparison treatment is, which health outcome matters, and how outcomes will be summarized. Those choices define the question before any statistical method is selected.

This matters in propensity score matching and other observational designs because the people retained or emphasized by an analysis may not represent everyone about whom a decision-maker wants to draw a conclusion. A result for patients who could plausibly receive either treatment, for example, need not answer the question about all eligible patients. The target population is therefore part of the estimand, not a label to add after an analysis.

Four Components of an Estimand

Let \(A\) denote a binary treatment, with \(A=1\) for the treatment of interest and \(A=0\) for its comparator. For each unit, let \(Y(1)\) and \(Y(0)\) denote the potential outcomes under those respective treatment conditions. Only one of these potential outcomes is observed for a unit, but the two potential outcomes define the causal contrast of interest.

The target population is the set of units, or a specified distribution of units, to which the question refers. It should be described in substantive terms: eligibility criteria, setting, and relevant time period. It might be all patients meeting specified criteria in a health system, rather than merely the patients whose records happened to be available. A finite target population can be described by its members; a broader population can be described by the distribution of characteristics among its members. Write \(E_Q\) for an expectation over the specified target population \(Q\).

The treatment contrast identifies the two conditions being compared. “Treatment versus control” may be too vague if the treatment has multiple doses, delivery methods, or versions. The comparator also needs to be explicit: no treatment, usual care, or a different active treatment can define different questions.

The outcome must name what is measured and when. “Health” is not a sufficient outcome definition. Examples include a symptom score at 12 weeks, death within one year, or total hospital days during six months. The summary measure determines how outcomes under the two conditions are compared. A mean difference, risk difference, and risk ratio are different quantities, even when they use the same population, treatment, and event.

For example, one possible estimand is the target-population mean difference in a numerical outcome:

This expression defines a population-level contrast on the outcome’s scale. If the outcome is measured in days, the difference is in days. It does not say how the quantity will be estimated, and it does not by itself assert that an observed treatment-group difference is causal.

Population Composition Can Change the Answer

If effects vary across kinds of people, changing the target population can change the population-level contrast. This follows from how a population summary averages the effects of its members. A simple two-group derivation makes the point without requiring a particular estimation method.

Suppose a target population consists of groups \(G=1\) and \(G=0\). Let \(q=P_Q(G=1)\), the proportion in group 1 in the target population. If the group-specific mean effects are \(\delta_1\) and \(\delta_0\), then partitioning the population gives:

The last line is a weighted average because the two groups exhaust the target population and their proportions sum to one. If the group-specific effects differ, changing \(q\) generally changes the population average. If \(\delta_1=\delta_0\), this particular change in group composition does not change the mean difference. Thus, an estimand shift is possible, not inevitable.

Keep the Outcome Summary Explicit

Even with the same target population and treatment contrast, different summary measures can describe different aspects of the outcome distribution. For a binary event, a risk difference is the treatment-condition risk minus the comparator risk; a risk ratio is the treatment-condition risk divided by the comparator risk. The difference is on the probability scale, while the ratio is unitless. Neither should be reported simply as “the effect” without identifying the measure.

Defining a Target That Data Can Support

An estimand can be defined even if the available data cannot identify it. Keep those tasks separate. Defining the estimand states the scientific target; identification asks whether the observed data and assumptions are sufficient to recover that target; estimation applies a statistical procedure to the data.

For an observational comparison, the connection between observed outcomes and potential outcomes typically requires consistency, conditional exchangeability, and positivity. Consistency means that a unit receiving treatment \(a\) has observed outcome \(Y=Y(a)\), with treatment versions defined clearly enough for this equality to be meaningful. Conditional exchangeability means that, given the measured pretreatment covariates used for adjustment, treatment assignment is independent of the potential outcomes. Positivity means that each treatment condition has a positive probability at covariate values represented in the target population.

If the data come from a source population different from the target, another issue arises: the outcome relationships learned in the source must apply to the target, conditional on the variables used to connect them, and the necessary covariate information must be available for the target. If the target includes groups absent from the source data, their effects cannot be recovered merely by changing weights or naming a broader target.

Before analyzing data, check that the target population is stated in terms that can be applied, that the treatment and comparator are well defined, and that the outcome and measurement time are fixed. Then compare the target’s covariate support with the observed data. In propensity score analyses, near-zero or near-one treatment probabilities can signal that some target-population units have few comparable units under the other treatment. Such a problem is not just a computational inconvenience: it may mean the data do not support the intended comparison for those units.

Common Mistakes and Practical Comparisons

• Treating the analysis sample as the target by default. The included records may be a convenience sample or may exclude eligible people. Name the population the result is intended to describe.
• Changing the population without acknowledging a changed question. A result for a restricted group does not automatically describe the broader eligible population. When the effects vary across groups, the population average can change with the group proportions.
• Leaving the outcome summary implicit. A 10-percentage-point risk reduction and a 50% relative reduction are not interchangeable descriptions. State which quantity is being reported.
• Assuming a causal label makes an observed comparison causal. The estimand is a causal target, but its identification from observational data still depends on assumptions. An adjusted association is not automatically that target.
• Using a target that has no data support. A population definition may be scientifically important but not identifiable from the available source data. A narrower, supported target can be more defensible, provided the change is explicit.

Choosing between candidate estimands is a substantive decision. A population-wide question may be most relevant to policy, while a question about a clearly defined eligible subgroup may better match the available evidence or a clinical decision. Likewise, an absolute risk difference can aid planning for event counts, whereas a risk ratio can communicate relative change. These are choices about the question and its usefulness, not merely choices among statistical formulas.