Defining Target Populations for Subgroup Effects

Why a Subgroup Effect Needs a Population Definition

A subgroup causal effect is not determined by a subgroup label alone. “The effect among older patients,” for example, leaves open who counts as older, which patients are eligible for treatment, and which population of older patients is the target. Those choices determine whose potential outcomes are averaged and therefore what causal quantity is being described.

This matters even when the treatment and outcome are unchanged. If treatment effects differ across age groups, an average over a population with many younger members can differ from an average over one with many older members. A precise subgroup estimand therefore names both the membership rule and the target population whose members contribute to the average.

Here \(Q\) describes the population to which the average refers; it might be a particular trial-eligible population or a defined clinical population. The conditional expectation means that the average is taken among members of that target population who are eligible and satisfy the subgroup rule. If the outcome is a binary event, \(\tau_g(Q)\) is a risk difference. For a continuous outcome, it has the outcome’s measurement units.

Eligibility, Membership, and Target Composition

These three decisions serve different purposes. Eligibility says who is in scope for the causal question. Subgroup membership says which eligible people are included in a particular subgroup. Target composition says how characteristics vary among the target population’s members within that subgroup. Each should be stated in operational terms rather than left implicit.

• Eligibility: Specify the inclusion and exclusion criteria, ideally using characteristics determined before treatment. Changing eligibility changes the population being described.
• Membership: Define the subgroup rule, including thresholds, categories, and timing. For example, “age 65 years or older at enrollment” is more precise than “older patients.”
• Target composition: State which population’s distribution of relevant characteristics is used in the average. Two target populations can have different proportions of age, disease severity, or other baseline characteristics even when the subgroup label is identical.

For a set of subgroups that partitions the eligible population, each person belongs to exactly one category. Some subgroup definitions instead overlap: a person may meet more than one rule. Overlapping subgroups can each have their own estimand, but their effects should not be treated as pieces of a single partition or combined as if they were mutually exclusive.

Subgroup membership should generally be defined using pre-treatment information when the aim is to compare treatment effects in baseline subgroups. If membership is determined after treatment, treatment itself may affect who enters the subgroup. Comparing those selected groups does not, without further causal definitions and assumptions, answer the same question as comparing baseline-defined subgroups.

How Population Composition Changes an Average

The composition of the target population matters when treatment effects vary across characteristics within a subgroup. Suppose \(X\) divides the subgroup into a finite set of categories. The subgroup effect for a target population is the weighted average of the category-specific effects, where each weight is that category’s proportion in the target subgroup.

This expression follows by partitioning the subgroup according to the possible values of \(X\). Within each category \(x\), the average effect is multiplied by the proportion of the target subgroup in that category; summing those contributions recovers the overall subgroup average. The weights must describe the target population, not automatically the trial sample or another convenient data set.

The formula also clarifies an important distinction: a change in an overall subgroup effect need not mean that any category-specific effect changed. The weighted average can change solely because the target population contains different proportions of the categories. If the effect is identical in every category, changing the weights alone cannot change the average.

Changing a Subgroup Rule Changes Who Is Averaged

A threshold is not merely a convenient label. Moving it can add people with different potential outcomes and treatment effects, thereby changing the population contrast. When a rule is broadened, the resulting estimand is an average for the newly defined group; it is not simply the old estimand with a more inclusive description.

Defining an Estimand Is Not the Same as Identifying It

An estimand states the causal quantity of interest; it does not guarantee that observed data identify that quantity. In a randomized trial, random assignment can support identification of a treatment contrast within a baseline-defined subgroup, provided both treatment conditions are represented in that subgroup and the observed outcomes correspond to the assigned treatment conditions. In an observational study, identification additionally requires adequate control of confounding within the target subgroup, as well as treatment positivity: each treatment condition must be possible for the relevant covariate profiles.

There is a further distinction when the target population differs from the population that supplied the data. Estimating a trial subgroup contrast does not, by itself, establish that the same average applies to a different target population. Transporting the result requires information or assumptions that link the trial participants’ effects to those in the target population. The target composition must also be available or otherwise justified if the goal is a standardized average for that target.

A useful discipline is to write the target population into the estimand before selecting an estimator. If the data only support an effect among trial participants, report that target. If the scientific question concerns a different eligible population, make that distinction explicit rather than silently substituting the trial’s composition.

Common Mistake: Treating a Subgroup Label as a Complete Definition

A tempting but incomplete argument is: “The subgroup is patients with high disease severity, so its treatment effect is the high-severity effect.” This leaves the severity threshold, eligibility criteria, timing of measurement, outcome scale, and target composition unspecified. It also assumes that all populations labelled “high severity” have the same distribution of other characteristics relevant to their average effects.

The correction is to state the membership rule and the target population explicitly, then define the contrast over that population. For example: “Among eligible participants with baseline severity score at least 8, estimate the average difference in 12-month symptom score under assignment to treatment versus control, standardized to the age distribution of the specified target population.” This identifies who is averaged and what the average concerns. It does not, on its own, resolve whether the available data identify that estimand.