Defining the Target Population for a Bayesian Estimand

Why the Target Population Is Part of the Question

A treatment effect is not fully described by a number alone. It also matters for whom that effect is intended to apply. A trial may enroll adults with a particular diagnosis and disease stage, while a decision-maker is interested in a broader or narrower group. Even when the outcome and treatment contrast are held fixed, changing the population can change the effect being summarized—especially when treatment effects differ across baseline characteristics.

The target population is therefore not just a description of who happened to enter the trial. It is part of the scientific question. Eligibility criteria, disease stage, and baseline characteristics specify whose outcomes the estimand concerns. The five attributes of a clinical trial estimand include this population attribute; here the focus is on making that attribute precise and understanding its consequences for Bayesian estimation.

Writing the Population Precisely

A useful specification says both who is included and the context in which the definition applies. “Patients with the disease” is usually too broad: it may leave unclear which disease definition, stage, age range, or care setting is intended. A practical description identifies the defining criteria and clarifies whether they apply at screening or another pre-treatment time.

For example, a target population might be adults aged 18–75 with a confirmed diagnosis and moderate disease at baseline, receiving care in outpatient clinics. Each element narrows the group whose treatment effect is being discussed. If eligibility criteria are used to define this target, state them in clinically meaningful terms rather than relying only on a protocol label such as “eligible patients.”

Let \(X\) denote measured baseline characteristics, and let \(F_T\) denote their distribution in the target population. Write \(Y(1)\) and \(Y(0)\) for the outcome that an individual would have under the two treatment conditions being compared. For this tutorial, suppose the outcome is well-defined under both conditions and the estimand is the target-population average difference. The notation \(Y(1)\) and \(Y(0)\) represents potential outcomes; it does not imply that both are observed for an individual.

This expression makes the role of the population explicit: the conditional treatment effects are averaged using the distribution \(F_T\) of baseline characteristics in the target population. If effects vary by \(X\), a different target distribution can yield a different \(\theta_T\). If effects are constant across all relevant values of \(X\), changing the distribution alone does not change this average, though other assumptions may still matter.

In a Bayesian analysis, uncertainty about model parameters and outcomes is represented by a posterior distribution given the data and model. When the target distribution is specified, each posterior draw of the relevant conditional effects can be averaged over that distribution, producing posterior draws of \(\theta_T\). A posterior mean summarizes the target-population effect under the model; a credible interval summarizes posterior uncertainty about that estimand. Neither summary changes what population the estimand refers to.

Baseline Composition Determines the Weighting

Eligibility criteria define the boundaries of a population, while baseline characteristics describe its composition within those boundaries. Characteristics such as disease stage, age, prior treatment, or baseline risk may be effect modifiers. When treatment effects differ across these groups, a population-average effect depends on how common each group is in the target population.

The following calculation is a direct standardization: compute a conditional effect for each subgroup, then average those effects using the target population’s subgroup proportions. It does not assume that the trial’s subgroup proportions are the target proportions.

Bayesian Standardization and Its Assumptions

Let \(\delta(x)=E\{Y(1)\mid X=x\}-E\{Y(0)\mid X=x\}\) denote the conditional effect at baseline profile \(x\). If a model produces posterior draws \(\delta^{(m)}(x)\), for draw \(m\), then a target-population draw can be calculated as

For a target population divided into \(K\) baseline strata, with target proportions \(w_k\), this becomes \(\theta_T^{(m)}=\sum_{k=1}^{K} w_k\delta_k^{(m)}\), where \(w_k\geq 0\) and \(\sum_{k=1}^{K}w_k=1\). If the target distribution is uncertain, that uncertainty can also be represented in the analysis; treating the weights as known is an assumption, not a mathematical necessity.

The calculation is useful only if its inputs and assumptions are credible. A randomized trial supports comparisons between assigned groups within the trial population under the trial’s design and analysis assumptions. Applying conditional effects to a different target population requires additional justification. In particular:

• Population definition: The eligibility and baseline criteria must identify the intended target group clearly and be measured consistently.
• Overlap: Target-population profiles used for standardization should be represented in the trial data. If a target stratum has no relevant trial participants, its effect cannot be learned directly from that trial without additional modeling or external information.
• Conditional transportability: After accounting for the baseline characteristics included in the model, the conditional treatment effects must be applicable to the target population. Unmeasured differences that modify treatment effect can invalidate this assumption.
• Measurement and model adequacy: Baseline characteristics, outcomes, and subgroup definitions must be measured meaningfully, and the model must capture relevant effect modification well enough for the target calculation.

A short justification explains why standardization requires target weights. In a two-stratum population, let \(\delta_1\) and \(\delta_2\) be the stratum-specific effects and let \(w\) be the target proportion in stratum 1. By the law of total expectation, the average effect is the sum of the stratum-specific averages weighted by their probabilities in the target population:

If the trial proportion in stratum 1 is \(q\), using the trial mix instead gives \(q\delta_1+(1-q)\delta_2\). Subtracting the target average yields

Thus, the two weighted averages agree if the stratum proportions agree (\(q=w\)), or if the effects are equal (\(\delta_1=\delta_2\)). Otherwise, using the trial composition generally targets a different average.

Diagnostics, Alternatives, and Limitations

Several checks can reveal when a target-population analysis is becoming unreliable. Compare baseline distributions in the trial and target population, inspect whether each target subgroup appears in the trial, and examine how sensitive the standardized result is to plausible changes in target weights and subgroup effects. Sparse or empty target-relevant trial strata, extreme weights, and large changes under small modeling choices are warnings that the result depends heavily on extrapolation.

When the trial and target populations are already aligned, the unweighted trial-population summary may answer the intended question. When they differ, direct standardization—as in the examples—makes the target composition explicit. Regression modeling can estimate conditional effects and then average them over the target distribution; weighting approaches can instead reweight trial participants toward that distribution. These approaches rely on assumptions about measured covariates and overlap, and their results can differ when the model is misspecified or the target distribution is poorly known. No statistical method repairs a target population that has not been defined or supplies reliable evidence for profiles absent from the trial.

A common mistake is to treat the randomized sample as the target population automatically: “The trial was randomized, so its average effect applies to everyone with the diagnosis.” Randomization supports a treatment comparison in the trial setting; it does not make the enrolled group representative of every population. The correction is to specify the intended population, compare its relevant baseline composition with the trial, and justify any standardization or extrapolation.