Population-Adjusted Indirect Comparisons (PAICs)

• Population-adjusted indirect comparisons (PAICs) are methods that reweight or model individual patient data to transport treatment effects to a common target population.
• They ensure valid indirect comparisons in anchored settings by adjusting for effect modifiers and distinguishing between marginal and conditional treatment effects.
• Methodological families such as MAIC, STC, ML-NMR, and parametric G-computation balance covariate differences to improve evidence synthesis in health technology assessments.

Population-adjusted indirect comparisons (PAICs) are indirect comparison methods that use individual patient data from at least one study to reweight or model outcomes so that relative treatment effects can be transported to the population of another trial when cross-trial differences in relevant covariates, especially treatment-effect modifiers, threaten the validity of a naïve indirect comparison. In the common anchored setting, one trial compares $A$ versus $C$, another compares $B$ versus $C$, and the objective is to estimate the $A$ versus $B$ effect in a common target population rather than in two incompatible study populations (Remiro-Azócar et al., 2020, Chandler et al., 19 Feb 2026).

1. Comparison structures and targets of analysis

PAICs are used when head-to-head evidence is unavailable but there remains some basis for indirect comparison. In anchored analyses, two trials share a common comparator, and the inferential task is to place relative effects on a common population basis before combining them. One notation writes the anchored contrast as

$$\hat{d}_{BC(AC)} = \hat{d}_{AC(AC)} - \hat{d}_{AB(AC)},$$

while another writes

$$\hat{\Delta}_{AB}^{(2)} = \hat{\Delta}_{AC}^{(2)} - \hat{\Delta}_{BC}^{(2)}.$$

Both expressions encode the same principle: the indirect comparison is valid only if the component effects are evaluated in a common target population and are compatible as estimands (Remiro-Azócar et al., 2020, Remiro-Azócar et al., 2020).

The distinction between anchored and unanchored designs is foundational. In anchored settings, randomization within each trial protects the within-trial relative effect from pure prognostic imbalance, so the main source of transport bias is imbalance in effect modifiers. In unanchored settings, by contrast, any imbalanced baseline factor can induce bias, so both prognostic variables and effect modifiers matter for identification. This difference explains why anchored PAICs are methodologically less demanding than unanchored comparisons and why unanchored applications are treated much more cautiously in the literature (Remiro-Azócar et al., 2020, Wang, 2021).

PAICs are therefore not merely “adjusted indirect comparisons” in a generic sense. They are transport procedures: they adapt a treatment effect estimated in one study to the covariate distribution of another study or of an explicitly chosen target population. A plausible implication is that the central design choice is not only which studies to compare, but also which population the comparison is meant to represent.

2. Estimands, effect scales, and transportability

The central conceptual distinction in PAICs is between marginal and conditional treatment effects. A marginal effect is a population-average causal contrast, for example comparing

$$\mathbb{E}\!\left[Y^{B} \mid S=AC\right] \quad \text{and} \quad \mathbb{E}\!\left[Y^{A} \mid S=AC\right],$$

whereas a conditional effect compares

$$\mathbb{E}\!\left[Y^{B} \mid X=x, S=AC\right] \quad \text{and} \quad \mathbb{E}\!\left[Y^{A} \mid X=x, S=AC\right]$$

for fixed covariate values, or equivalently the treatment coefficient in a multivariable outcome regression conditional on included covariates. PAIC methodology depends on making that choice explicit, because weighting and regression approaches do not generally target the same estimand on non-collapsible scales (Remiro-Azócar et al., 2020).

For health technology assessment, the argument advanced most strongly in the literature is that the relevant target is usually the marginal, population-average effect, because reimbursement decisions are made for populations rather than covariate-defined individuals. This does not imply that marginal effects must be estimated by unadjusted procedures: covariate-adjusted marginalization, G-computation, standardization, and weighting all remain compatible with a marginal target. The methodological priority is therefore estimand-first: select the decision-relevant estimand, then select the most efficient estimator for that estimand (Remiro-Azócar, 2021).

Effect scale is decisive because transportability properties depend on collapsibility and on the scale on which effect modification is defined. Odds ratios and hazard ratios are treated as practically non-collapsible, so marginal and conditional effects may differ materially even with randomization and no confounding. Mean differences in linear models are collapsible, but even there the inferential framework still matters because uncertainty differs between adjusted and unadjusted analyses. More recent work further shows that shared effect modifier assumptions are not, by themselves, sufficient for direct transportability of marginal effects on common non-collapsible scales; pairwise PAICs usually identify effects in the comparator population, and applying them to another decision population is a further transport step requiring additional assumptions (Chandler et al., 19 Feb 2026).

3. Methodological families

The principal PAIC methods fall into weighting, outcome-regression, and hybrid or multilevel classes. MAIC is a propensity-score-weighting method; STC and ML-NMR are regression-adjustment methods that typically target conditional estimation; parametric G-computation standardizes a conditional model over the target covariate distribution to recover a compatible marginal effect; and two-stage MAIC adds a treatment-assignment model in the IPD trial to improve precision in some anchored settings (Remiro-Azócar et al., 2020, Remiro-Azócar et al., 2021, Remiro-Azócar, 2022).

Method	Core mechanism	Estimand emphasis
MAIC	Reweight IPD to match target aggregate covariate moments	Marginal effect, often in comparator study sample
STC	Fit outcome model in IPD and predict into target population	Typically conditional unless marginalized
ML-NMR	Joint regression for IPD and AgD in a network	Conditional or marginal, depending on standardization
Parametric G-computation	Fit conditional model, then integrate over target covariate distribution	Compatible marginal effect
2SMAIC	Combine trial-assignment and treatment-assignment weights	Same transported marginal target as MAIC, with precision gains in some scenarios

MAIC is often written through exponential tilting weights such as

$C$0

or its normalized form

$C$1

Under exact mean-balance constraints, MAIC is equivalent to entropy balancing, and, in calibration-estimation terms, it can be viewed as one member of a broader family of balancing-weight estimators. That perspective also clarifies why alternative calibration distances, approximate balance, and model-assisted calibration enter naturally into PAIC methodology (Wang, 2021).

Conventional STC is distinguished from marginalization-based regression approaches by its usual reliance on a treatment coefficient from a conditional model. On non-collapsible scales this creates incompatibility in anchored indirect comparisons, because the adjusted $C$2 versus $C$3 estimate is conditional while the published $C$4 versus $C$5 estimate is usually marginal. Parametric G-computation was proposed precisely to resolve that incompatibility by treating the regression as a nuisance model and then averaging treatment-specific predictions over the target population covariate distribution. In simulation, this marginalized regression-adjusted approach achieved more precise and more accurate estimates than MAIC, particularly when covariate overlap was poor, and conventional outcome regression remained systematically biased because it targeted a conditional rather than marginal effect (Remiro-Azócar et al., 2021).

Two-stage MAIC extends standard MAIC by adding a model for treatment assignment within the IPD trial and combining those weights with the usual trial-assignment weights. In the anchored randomized-trial simulation used as proof of principle, 2SMAIC improved precision and efficiency relative to MAIC while maintaining similarly low bias, with the largest gains when the IPD trial was small; it was less effective when overlap between trial populations was poor, and truncation then produced a bias-variance trade-off rather than a general remedy (Remiro-Azócar, 2022).

A further development concerns target-population flexibility. Pairwise MAIC is restricted to marginal inference in the comparator study sample, whereas ML-NMR can potentially target marginal estimands in any population of interest by integrating predicted outcomes over a chosen covariate distribution. This is one reason ML-NMR is treated as a more flexible framework for evidence synthesis when richer network structure is available (Remiro-Azócar, 2021).

4. Identification assumptions, diagnostics, and inferential stability

PAIC validity rests on a compact set of assumptions that recur across the literature. The first is no unmeasured effect modification across studies in anchored settings: all variables whose imbalance would alter the relative effect between trial populations must be observed and accounted for. The second is correct specification of the weighting or outcome model used for transport. The third is positivity or overlap: the target covariate distribution must be sufficiently represented in the IPD study. The fourth is compatibility of effect measures and estimand type when combining component effects. The fifth is ordinary within-trial randomization, which underpins internal validity of the source trial estimates (Remiro-Azócar et al., 2020, Remiro-Azócar et al., 2020).

In unanchored settings the requirement is stronger. Any imbalanced baseline factor associated with outcome can induce bias, so identification requires adjustment for all prognostic variables and all effect modifiers that differ between populations. This is why unanchored PAICs are regularly characterized as assumption-heavy and why apparently successful adjustment of observed covariates does not by itself establish validity (Cheng et al., 2019).

The principal empirical diagnostic for weighting instability is the effective sample size,

$C$6

Low ESS indicates that the transported estimand is being driven by a small weighted subset of the IPD sample. Practical guidance stresses that presentation and interpretation of ESS are critical, that a substantial decrease signals underlying issues, and that method choice should not rely solely on ESS. Weight distributions, overlap assessment, and sensitivity analyses for alternative balancing sets remain essential complements (Wang, 2021, Baschet et al., 11 May 2026).

Uncertainty estimation is itself nontrivial. In MAIC, naïve weighted variance estimators can be conservative because they ignore the calibration restrictions, while robust sandwich estimators may underestimate variability when effective sample size is very small. Large-sample two-step sandwich estimators, bootstrap procedures, and model-assisted survey-sampling estimators have all been proposed, but finite-sample undercoverage remains a recurrent problem in low-overlap settings. A plausible implication is that PAIC diagnostics should be treated as part of identification analysis rather than as a post hoc reporting formality.

5. Extensions, generalizations, and special settings

Several methodological branches extend PAIC logic beyond the standard anchored binary-outcome case. One line concerns external controls and single-arm studies with survival outcomes and a cure fraction. There, calibration weighting can be combined with pseudo-observations so that censored time-to-event quantities become approximately individual-level outcomes, yielding estimators such as

$C$7

for the control cure rate in the target population. This extends MAIC-style adjustment to settings in which hazard-ratio-based summaries are especially unattractive (Wang et al., 2023).

A second extension addresses unmeasured shifted effect modifiers. Proximal indirect comparison introduces an adjustment proxy $C$8 observed in both trials and a reweighting proxy $C$9 observed in the source trial, and identifies the target effect through bridge functions rather than ordinary measured-effect-modifier adjustment. The resulting estimator is doubly robust against misspecification of the bridge functions and asymptotically normal under mild consistency conditions for the bridge estimators (Su et al., 2024).

A third extension addresses target-population incoherence in standard MAIC. “Arbitrated indirect treatment comparisons” were proposed to resolve the “MAIC paradox,” in which opposite-direction analyses target different study populations and may therefore produce contradictory conclusions. The proposed remedy is to define a common overlap population using trial-membership propensity scores $B$0, with overlap weights

$B$1

so that both sides estimate effects for the same target population (Fang et al., 20 Oct 2025).

A fourth development places PAIC-style weighting inside a causal meta-analysis framework. Inverse-weighting methods can standardize multiple trials to multiple target populations using estimands of the form

$B$2

and then decompose total heterogeneity into case-mix heterogeneity $B$3 and beyond case-mix heterogeneity $B$4. This suggests that population adjustment removes only the component of heterogeneity due to observed effect modifiers, not all between-study variation (Vo et al., 7 Mar 2025).

A fifth branch concerns disconnected evidence. Multilevel unanchored meta-regression (ML-UMR) extends ML-NMR to unanchored settings by jointly modeling IPD and AgD within a unified Bayesian likelihood, enabling estimation of treatment-specific outcomes and both marginal and conditional effects across multiple treatments, studies, and target populations. The method does not weaken the strong assumptions required for unanchored inference, including conditional exchangeability, correct specification of the outcome model, and cross-treatment assumptions such as the shared prognostic factor assumption, but it makes them explicit within a single synthesis framework (Chandler et al., 18 Jun 2026).

6. Role in health technology assessment and continuing controversies

In health technology assessment, PAICs occupy a specific place in an evidence hierarchy. When a connected randomized evidence network exists and transitivity is acceptable, network meta-analysis is generally preferred. When effect-modifier imbalance threatens a conventional anchored indirect comparison, ML-NMR is often considered before pairwise PAICs such as MAIC or STC. When randomized comparison is not feasible at all, external-control comparisons may be used, but unanchored PAICs are treated as low-evidence analyses and are highly discouraged (Baschet et al., 11 May 2026).

This hierarchy reflects two enduring controversies. The first concerns which estimand should be targeted. A strong position in the recent literature is that HTA submissions should target marginal effects in an explicitly defined decision-relevant population, not merely the comparator study sample, because pairwise methods are often “sample-adjusted” rather than genuinely population-adjusted. Real-world data sources such as registries, cohort studies, insurance claims, and hospital electronic health records have therefore been proposed as more appropriate target-population definitions for some applications (Remiro-Azócar, 2021).

The second controversy concerns whether a PAIC estimate can be used outside the population in which it was identified. Pairwise PAIC approaches typically identify the treatment effect in the comparator population. If that estimate is then inserted into a cost-effectiveness model for another jurisdictional or trial population, an additional transport step has occurred, often implicitly. For hazard ratios and odds ratios in particular, direct transportability of marginal effects generally fails even when shared effect modifiers are assumed, so a PAIC result should not automatically be treated as population-free (Chandler et al., 19 Feb 2026).

The practical literature on single-arm trials and external controls adds a further caution. Validity depends first on defining the estimand, then on selecting external controls with aligned eligibility criteria, endpoints, treatment timing, and usual care, and only then on choosing an adjustment method. Because no indirect comparison with external controls can guarantee the evidentiary level of a well-conducted randomized trial, post-market evaluation is even more important in these settings (Lambert et al., 2022).

PAICs are therefore best understood as an estimand-driven transportability enterprise rather than a fixed set of branded algorithms. Their methodological development has moved from pairwise weighting toward generalized calibration, marginalization-based outcome regression, multilevel synthesis, proxy-based identification, and explicit target-population arbitration. Across these developments, the durable lesson is consistent: population adjustment is only meaningful when the target population, effect scale, estimand type, identifying assumptions, and compatibility of component effects are all made explicit.