Population-Level Estimands Overview

• Population-Level Estimands are clearly defined targets that quantify average treatment or exposure effects across entire populations.
• They require explicit specification of the target population, intervention details, endpoints, and handling of intercurrent events to yield reliable causal estimates.
• Specialized methodologies such as weighting, marginalization, and covariate adjustment ensure accurate estimation in observational, experimental, and dynamic time-to-event settings.

Population-level estimands are formal, well-defined targets of inference that quantify treatment effects or exposure impacts for an entire population or a specified subpopulation, rather than for subgroups or specific individuals. They play a central role in modern causal inference, epidemiology, and policy-oriented empirical research, especially where the average effect across a broad population guides regulatory, clinical, or public health decisions. Population-level estimands are characterized not only by clear mathematical definitions but also by explicit specification of the target population, intervention, endpoint, handling of intercurrent events, and summary measure. They are distinct from both sample-level and conditional estimands and require specialized methodologies across experimental, observational, and complex paper designs.

1. Definition and Conceptual Scope

Population-level estimands represent summary causal effects averaged over a reference population, often identified by the factual or target distribution of covariates or units. In the potential outcomes framework, classical estimands such as the Average Treatment Effect (ATE) are defined as

$\text{ATE} = E[Y(1) - Y(0)],$

where the expectation is over the population of interest. Variations include Average Treatment Effect on the Treated (ATT), Average Treatment Effect on the Untreated (ATU), and effects defined within an ‘overlap’ or equipoise population (Greifer et al., 2021).

Beyond binary or continuous outcomes, estimands generalize to settings involving ordinal non-numeric outcomes (Volfovsky et al., 2015) (using joint distributions of potential outcomes), time-to-event data with dynamic exposures and competing risks (Cube et al., 2019, Cube et al., 2019), and networks or clusters with interference (Papadogeorgou et al., 2017, Cai et al., 2021). Population-level estimands also encompass measures like the population-attributable fraction (PAF), the risk difference, marginal odds ratio, hazard ratio, or more general functionals of first moments under the target distribution (Boughdiri et al., 19 May 2025).

In real-world evidence (RWE) and pragmatic settings, population-level estimands formalize clinical or policy questions by explicitly describing the population, intervention, endpoint, intercurrent events, and summary metric, following the ICH E9(R1) framework (Chen et al., 2023).

2. Methodological Foundations and Statistical Properties

Population-level estimands are linked to the target distribution via summation or integration over the population covariate distribution $f(X)$. In the presence of confounder imbalance, non-representative sampling, or limited overlap between paper groups, estimands are mapped to specific weighting functions $h(x)$ so that the targeted estimand is

$$\tau_h = \frac{\int \tau(x) h(x) f(x) dx}{\int h(x) f(x) dx},$$

where $\tau(x)$ is the conditional average treatment effect (Barnard et al., 15 Oct 2024, Wang et al., 19 Feb 2025). Choices of $h(x)$ define ATE ($h(x) = 1$), ATT ($h(x)$ proportional to Pr[T=1|X=x]), ATU, ATO (overlap weights), or more general functionals for integrated or overlap populations.

In interference or cluster-based designs, methods extend to average over cluster-specific distributions or marginalize conditional on allocation programs (Papadogeorgou et al., 2017, Cai et al., 2021), requiring design-based or model-integrated estimators. For marginal population-level direct and indirect (spillover) effects under interference, estimands may aggregate expected outcomes over entire networks or distributions of exposure programs (Papadogeorgou et al., 2017).

Collapsibility is a key methodological concept: population-level (marginal) effects for non-collapsible measures (odds ratio, hazard ratio) do not generally coincide with conditional effects, even in the absence of effect modification (Huang et al., 2021, Remiro-Azócar, 2022, Phillippo et al., 15 Oct 2024). Marginal estimands are necessary for population-level interpretation, particularly in health technology assessment and evidence synthesis.

3. Population-Level Estimands in Time-to-Event and Dynamic Settings

Population-level estimands generalize naturally to complex temporal and dynamic settings. For time-to-event data with internal time-dependent exposures and competing events, classical estimands are extended via multi-state models:

• Descriptive population-attributable fraction (PAF):

$$\text{PAF}_o(t) = \frac{P(D(t)=1) - P(D(t)=1|E(t)=0)}{P(D(t)=1)}$$

• Causal PAF (counterfactual estimand):

$$\text{PAF}_c(t) = \frac{P(D(t)=1) - P(D_0(t)=1)}{P(D(t)=1)}$$

where $D(t)$ is event by time $t$, $E(t)$ is time-dependent exposure, and $D_0(t)$ is the potential outcome under absent exposure (Cube et al., 2019). Landmark-based approaches (Cube et al., 2019) refine the estimand to temporal windows and “at-risk” subpopulations at each landmark, enhancing clinical interpretability in dynamic settings.

In mediated and missingness-prone scenarios (e.g., public health interventions with incomplete measurement), counterfactual strata effects are employed as population-level estimands, targeting total effects among the relevant (potentially affected) population (Nakato et al., 6 Jun 2025).

4. Covariate Adjustment, Transportability, and Real-World Data

The specification and estimation of population-level estimands under covariate distributional shift or external data integration are formalized within transportability and generalization frameworks. Methods include

• Density ratio (importance) weighting across source and target populations,
• Outcome regression (G-formula) and semiparametric estimators using efficient influence function (EIF) corrections,
• Post-residualization or covariate adjustment using predictive modeling to reduce estimator variance (Huang et al., 2021, Boughdiri et al., 19 May 2025).

The causal estimand in the target population is typically

$\tau_P = \Phi(E_P[Y^{(1)}], E_P[Y^{(0)}])$

for a chosen effect-measure function $\Phi$, with identification generally requiring exchangeability of conditional means or effect measures, and estimators derived from either reweighting or model-based strategies. Correction for covariate imbalance is essential when populations differ, and the impact of non-collapsibility must be addressed for certain summary measures (Huang et al., 2021, Remiro-Azócar, 2022, Boughdiri et al., 19 May 2025).

In RWE and health technology assessments, population-level estimands must be precisely defined considering population heterogeneity, real-world treatment regimes, and complex intercurrent events, with causal identification provided by a potential outcomes framework and careful data curation (Chen et al., 2023).

5. Challenges Associated with Effect Modification, Non-Collapsibility, and Estimand Selection

Effect modification and non-collapsibility pose major challenges: marginal and conditional estimands may diverge in both magnitude and treatment ranking, especially for measures such as odds ratios or hazard ratios (Phillippo et al., 15 Oct 2024, Remiro-Azócar, 2021). This can result in conflicting policy or clinical recommendations, as the marginal estimand addresses minimizing population event risk, while the conditional estimand averages individual-level relative effects. ML-NMR methods are uniquely capable of producing both conditional and marginal estimates for any target population, and explicit pre-specification of the estimand tied to the research or decision question is recommended (Remiro-Azócar, 2021, Phillippo et al., 15 Oct 2024). In evidence synthesis, choosing directly collapsible measures (e.g., risk difference) facilitates transportability and population-level interpretation (Remiro-Azócar, 2022).

Further, when overlap between treated and control covariate distributions is poor, tradeoffs between targeting the scientific estimand (e.g., the ATE) and reducing statistical bias/variance must be balanced, and estimand selection procedures have been developed to navigate this tension using design-based metrics (Barnard et al., 15 Oct 2024).

6. Bayesian and Empirical Likelihood Perspectives

Bayesian inference for population-level estimands differs fundamentally from sample-level estimands in terms of what is identified, modeled, and how the posterior is constructed. For population-level estimands (such as the Population Average Treatment Effect, PATE), inference relies purely on the posterior over model parameters; missing counterfactuals are integrated out. This is in contrast to sample-level estimands like the SATE or ITE, which require explicit imputation of missing outcomes and cross-world assumptions (Oganisian, 20 Aug 2025). Correct specification of the marginal in the posterior, first-principles logic, and correct computation—often via g-formula-based Monte Carlo integration over the covariate distribution—are essential to avoid common implementation errors.

Empirical likelihood methods incorporating design information and population-level constraints (as side-constraints or augmented estimating equations) further enhance efficiency and enable principled estimation of population-level parameters under complex survey designs or partially observed joint populations (Chaudhuri et al., 2022).

7. Practical Impact and Applications

Population-level estimands formally bridge the gap between statistical analyses, causal inference, and policy or clinical translation. Examples of applications include:

• Educational interventions with ordinal outcomes, where scale-free estimands are constructed via conditional distributions (Volfovsky et al., 2015).
• Health policy and regulatory science, employing population-level summaries (e.g., the PATE, marginal odds ratio, hazard ratio) directly relevant for reimbursement and public health decisions (Chen et al., 2023, Remiro-Azócar, 2021).
• Cluster-randomized designs with post-randomization selection, where principal stratification and augmented data collection are necessary for unbiased estimation (Li et al., 2021).
• Infectious disease interventions in networks, where estimands defined via marginalization over exposure histories yield interpretable causal contrasts under complex interference (Cai et al., 2021).
• Efficient transport of RCT results to broader populations by combining RCT and external control data using balancing and overlap weights, with careful choice of target estimand for each scientific, policy, or data context (Wang et al., 19 Feb 2025, Huang et al., 2021).

In summary, population-level estimands provide the rigorous foundation upon which valid, interpretable, and policy-relevant causal inference is constructed, with formal linkages to methodological decisions, statistical efficiency, covariate adjustment, and real-world application challenges. There is a consensus across the methodological literature that their precise definition, identification, and empirical estimation are essential for reliable evidence synthesis and actionable decision-making in modern quantitative sciences.