Estimands: what they are and why we should use them.

The South China Sea, a semienclosed marginal sea and global hotspot for plastic pollution, presents complex hydrodynamics and pronounced water column stratification, offering a unique setting to investigate the vertical behavior of microplastics. This study examines vertical distribution, environmental drivers, and ecological risks of microplastics using water samples collected from 15 stations across 5 depth layers. Microplastic abundance significantly decreased with depth, accompanied by a compositional shift from low-density, small-sized, fibrous polyethylene and polypropylene at surface to higher-density, larger-sized, granular polyvinyl chloride and polyester in deeper layers. Principal component analysis and stratification index analysis revealed that gradients in temperature, salinity, density, and pressure jointly drive microplastic vertical differentiation. Notably, water column stability─primarily controlled by density gradients─plays a pivotal role in limiting microplastic vertical transport, as evidenced by a strong positive correlation between stratification index and vertical abundance gradient. A species sensitivity distribution model based on representative Chinese coastal species yielded a hazardous concentration for 5% of species of 12.3 items/L and a predicted no-effect concentration of 6.15 items/L. Moderate ecological risks were observed in surface waters, particularly for bivalves and planktonic crustaceans. These findings highlight the critical role of stratification in modulating microplastic exposure risks in marginal seas.

In clinical trials, the observation of participant outcomes may frequently be hindered by death, leading to ambiguity in defining a scientifically meaningful final outcome for those who die. Principal stratification methods are valuable tools for addressing the average causal effect among always-survivors, that is, the average treatment effect among a subpopulation defined as those who would survive regardless of treatment assignment. Although robust methods for the truncation-by-death problem in two-arm clinical trials have been previously studied, their expansion to multi-arm clinical trials remains elusive. In this article, we study the identification of a class of survivor average causal effect estimands with multiple treatments under monotonicity and principal ignorability, and first propose simple weighting and regression approaches for point estimation. As a further improvement, we derive the efficient influence function to motivate doubly robust estimators for the survivor average causal effects in multi-arm clinical trials. We also propose sensitivity methods under violations of key causal assumptions. Extensive simulations are conducted to investigate the finite-sample performance of the proposed methods against the existing methods, and a real data example is used to illustrate how to operationalize the proposed estimators and the sensitivity methods in practice.

Background: Peritoneal metastasis (PM) from colorectal cancer (CRC) carries a poor prognosis. The Peritoneal Cancer Index (PCI) is among the principal prognostic stratification tools, yet the prognostic value of the anatomical distribution of disease beyond total PCI is underexplored. This pilot study evaluated whether quadrant-specific involvement adds prognostic information in patients undergoing cytoreductive surgery (CRS) with hyperthermic intraperitoneal chemotherapy (HIPEC), with a focused analysis of oligometastatic disease (PCI ≤ 6). Methods: A single-institution cohort of 48 CRC-PM patients treated with CRS + HIPEC was analyzed. Primary endpoints were OS, DFS, and PRFS, with a focused evaluation of the oligometastatic subset (PCI ≤ 6). Comparative statistics used Student’s two-sample t test for continuous variables and chi-square or two-sided Fisher’s exact tests for categorical variables. Survival was estimated by Kaplan–Meier with log-rank tests, and prognostic factors were evaluated using Cox regression. Results: Median follow-up was 177 months (IQR 87–224). Outcomes favored PCI ≤ 6: 5-year OS and DFS were 54% and 37.5% versus 6.6% and 0% for PCI > 6, and median OS 64 vs. 29 months (log-rank p = 0.007), median DFS 30 vs. 7 months (p = 0.0002), and median PRFS 26 vs. 8 months (p = 0.0002). In the PCI ≤ 6 subset (n = 27), quadrant 3 (left upper quadrant) was associated with higher recurrence risk and shorter DFS, remaining independently prognostic for DFS (p = 0.005) and PRFS (p = 0.005). For PRFS, quadrants 7 and 8 also showed associations on univariable analysis; Q7 remained independent (p = 0.047), whereas Q8 was borderline (p = 0.077). A histology-related signal at Q8 (p = 0.011) was exploratory due to very small mucinous and signet-ring strata. Sidedness and synchronicity yielded no significant differences in quadrant involvement within PCI ≤ 6. No quadrant effects were observed in PCI > 6. Conclusions: PCI remains the dominant prognostic determinant after CRS + HIPEC, yet in oligometastatic disease, the anatomical distribution adds complementary prognostic information, particularly involvement of Q3 and Q7. These findings are hypothesis-generating and warrant validation in larger, preferably multicenter cohorts with standardized quadrant mapping. If confirmed, quadrant-directed operative planning, including consideration of prophylactic resection in selected high-risk regions, could be prospectively evaluated.

In clinical trials, patients may discontinue treatments prematurely, breaking the initial randomization. In our motivating study, a randomized controlled trial in oncology, patients assigned the investigational treatment may discontinue it due to adverse events. The ICH E9(R1) Addendum provides guidelines for handling such "intercurrent events." The right strategy to adopt depends on the questions of interest. We propose adopting a principal stratum strategy and decomposing the overall intention-to-treat effect into principal causal effects for groups of patients defined by their potential discontinuation behaviour. We first show how to implement a principal stratum strategy to assess causal effects on a survival outcome in the presence of continuous-time treatment discontinuation, its advantages, and the conclusions that can be drawn. Our strategy allows us to properly handle the time-to-event intermediate variable, which is not defined for patients who would not discontinue, and to account for the fact that the discontinuation time and the primary endpoint are subject to censoring. We employ a flexible model-based Bayesian approach to tackle these complexities, providing easily interpretable results. We apply this Bayesian principal stratification framework to analyze synthetic data of the motivating oncology trial. Supported by a simulation study, we shed light on the role of covariates in this framework. Beyond making structural and parametric assumptions more credible, they lead to more precise inference. Also, they can be used to characterize patients' discontinuation behavior, which could help inform clinical practice and futureprotocols.

ABSTRACTIn clinical trials involving both mortality and morbidity, an active treatment can influence the observed risk of the first nonfatal event either directly, through its effect on the underlying nonfatal event process, or indirectly, through its effect on the death process, or both. Discerning the direct effect of treatment on the underlying first nonfatal event process holds clinical interest. However, with the competing risk of death, the Cox proportional hazards model that treats death as non‐informative censoring and evaluates treatment effects on time to the first nonfatal event provides an estimate of the cause‐specific hazard ratio, which may not correspond to the direct effect. To obtain the direct effect on the underlying first nonfatal event process, within the principal stratification framework, we define the principal stratum hazard and introduce the proportional principal stratum hazards model. This model estimates the principal stratum hazard ratio, which reflects the direct effect on the underlying first nonfatal event process in the presence of death and simplifies to the hazard ratio in the absence of death. The principal stratum membership is identified probabilistically using the shared frailty model, which assumes independence between the first nonfatal event process and the potential death processes, conditional on per‐subject random frailty. Simulation studies are conducted to verify the reliability of our estimators. We illustrate the method using the Carvedilol Prospective Randomized Cumulative Survival trial, which involves heart‐failure events.

Background/Aims:Randomised clinical trials assessing treatment effects on health outcomes (e.g. quality of life) can be affected by data truncation by death, where some patients die before their outcome measure is assessed and their data become undefined after death. The ICH E9(R1) addendum on estimands discusses four strategies for handling such terminal intercurrent events: hypothetical, composite, while-alive, and principal stratum. While the addendum emphasises the importance of aligning statistical methods of analysis (i.e. estimators) with estimands, it does not provide specific guidance and consideration on the choice of estimators in practice. We aim to (1) demonstrate how some statistical methods commonly used in trials can be used to estimate different intercurrent event strategies for handling data truncation by death; and (2) describe how missing outcome data (e.g. due to missed assessments or loss to follow-up) can be handled for each estimator.Method:We use data from SCORAD, a non-inferiority randomised trial comparing single-fraction versus multifraction radiotherapy on ambulatory status at 8 weeks (primary outcome) among patients with spinal canal compression from metastatic cancer. Here, we estimate the effect of radiotherapy on quality of life (secondary outcome), quantified by the difference in mean global health status between the two groups at 8 weeks. We outline the strategies for handling death and describe a selection of commonly used estimators corresponding to each strategy. The handling of missing data is considered and demonstrated as part of the estimation process.Results:The hypothetical strategy, targeting a treatment effect assuming patients had not died, can be estimated using linear mixed models (a likelihood approach) or multiple imputation (a method commonly used for handling missing data). The composite and while-alive strategies relate to the ‘outcome’ attribute of the estimand; the former incorporates death into the definition of the primary outcome, the latter only uses outcome data before death. These can be estimated by re-defining the outcome, for example, assigning a value reflecting poor global health status post-death, or using the last global health status observed before death. The principal stratum strategy, targeting a treatment effect among patients who would not die under either treatment, can be estimated by an analysis of survivors under specific assumptions. Missing data can be handled with linear mixed models or multiple imputation.Conclusions:Regarding death as an intercurrent event in the process of defining the estimand for the trial will help clarify the choice of suitable estimators. When choosing the estimators, it is important to consider the assumptions required by the estimators as well as their plausibility given the setting of the trial.

ABSTRACT Over the past decades, the primary interest in vaccine efficacy evaluation has mostly been on the effect observed in trial participants complying with the protocol requirements (per protocol analysis). The ICH E9 (R1) addendum provides a structured framework to formulate the clinical questions of interest and formalize them as estimands. In this paper, the estimand framework is retrospectively implemented in a human papillomavirus (HPV) phase 3 trial, where the vaccine efficacy was originally estimated on the per protocol set. We focus on two strategies for dealing with the presence of intercurrent events: the hypothetical and the principal stratum strategies. We address the interpretation of these two estimands, their estimation as well as articulation of the underlying identifiability assumptions. Finally, we leverage the results of the HPV application to formulate general considerations regarding the implementation of the ICH E9 (R1) addendum in vaccine efficacy studies.

Abstract Post-treatment variables often complicate causal inference. They appear in many scientific problems, including non-compliance, truncation by death, mediation, and surrogate endpoint evaluation. Principal stratification is a strategy to address these challenges by adjusting for the potential values of the post-treatment variables, defined as the principal strata. It allows for characterizing treatment effect heterogeneity across principal strata and unveiling the mechanism of the treatment’s impact on the outcome related to post-treatment variables. However, the existing literature has primarily focused on binary post-treatment variables, leaving the case with continuous post-treatment variables largely unexplored. This gap persists due to the complexity of infinitely many principal strata, which present challenges to both the identification and estimation of causal effects. We fill this gap by providing nonparametric identification and semiparametric estimation theory for principal stratification with continuous post-treatment variables. We propose to use working models to approximate the underlying causal effect surfaces and derive the efficient influence functions of the corresponding model parameters. Based on the theory, we construct doubly robust estimators and implement them in the R package continuousPCE.

Immortal time bias due to post-treatment definition of eligibility criteria can affect experimental and observational studies, and yet, in contrast to the extensive literature on the classical form of immortal time bias, it has seldom been the focus of methodological discussions. Here, we propose an account of eligibility-related immortal time bias that uses the principal stratification framework to explain the non-comparability of treatment arms (or exposure groups) conditional on selection. In particular, we show that the statistical estimand that conditions on observed eligibility after time zero of follow-up can be interpreted using partially overlapping principal strata. Further, we show that, under this perspective, as the timing of eligibility approaches time zero of follow-up, the probabilities of the outcome for eligible individuals monotonically approach the corresponding unconditional (in absence of selection) expected potential outcomes under different treatment levels. Our study provides a potential outcomes-based explanation of eligibility-related immortal time bias, and indicates that, in addition to the target trial emulation framework, principal effects might, for some studies, be useful causal estimands.

Epidemiologic analyses that aim to quantify exposure effects on disease progression are not uncommon. Understanding the implications of these studies, however, is complicated, in part because different causal estimands could, at least in theory, be the target of such analyses. Here, to facilitate interpretation of these studies, we describe different settings in which causal questions related to disease progression can be asked, and consider possible estimands. For clarity, our discussion is structured around settings defined based on two factors: whether the disease occurrence is manipulable or not, and the type of outcome. We describe relevant causal structures and sets of response types, which consist of joint potential outcomes of disease occurrence and disease progression, and argue that settings where interventions to manipulate disease occurrence are not plausible are more common, and that, in this case, principal stratification might be an appropriate framework to conceptualize the analysis. Further, we suggest that the precise definition of the outcome of interest, in particular of what constitutes its permissible levels, might determine whether potential outcomes linked to disease progression are definable in different strata of the population. Our hope is that this paper will encourage additional methodological work on causal analysis of disease progression, as well as serve as a resource for future applied studies.

Abstract We consider assessing causal mediation in the presence of a posttreatment event (examples include noncompliance, a clinical event, or death). We identify natural mediation effects for the entire study population and for each principal stratum characterized by the joint potential values of the posttreatment event. We derive the efficient influence function for each mediation estimand, which motivates a set of multiply robust estimators for inference. The multiply robust estimators are consistent under four types of misspecifications and are efficient when all nuisance models are correctly specified. We also develop a nonparametric efficient estimator that leverages data-adaptive machine learners to achieve efficient inference and discuss sensitivity methods to address key identification assumptions. We illustrate our methods via simulations and two real data examples.

Abstract Principal stratification is essential for revealing causal mechanisms involving post-treatment intermediate variables, in real-world applications like surrogate marker evaluation. Principal stratification analysis with continuous intermediate variables is increasingly common but challenging due to the infinite principal strata and the nonidentifiability and nonregularity of principal causal effects (PCEs). Inspired by recent research, we resolve these challenges by first using a flexible copula-based principal score model to identify PCE under weak principal ignorability. We then target the local functional substitute of PCE, which is statistically regular and can accurately approximate PCE with vanishing bandwidth. We simplify the full efficient influence function of the local functional substitute by considering its oracle-scenario alternative. This leads to a computationally efficient and straightforward estimator for the local functional substitute and PCE with vanishing bandwidth. We prove the double robustness of our proposed estimator, and derive its asymptotic normality for inferential purposes. With a vanishing bandwidth, our method attains minimax optimality for the nonparametric estimation of the PCE. With a fixed bandwidth, it achieves semiparametric efficiency in estimating its local functional substitute. We demonstrate the strong performance of our proposed estimator through simulations and apply it to surrogate analysis of short-term CD4 count in ACTG 175.

Here, we posit that studies comparing outcomes of patients hospitalized with COVID-19 by vaccination status are important descriptive epidemiologic studies, but they contrast two groups that are not comparable with regard to causal analyses. We use the principal stratification framework to show that these studies can estimate a causal vaccine effect only for the subgroup of individuals who would be hospitalized with or without vaccination. Further, we describe the methodology for, and present sensitivity analyses of, this effect. Using this approach can change the interpretation of studies only reporting the standard analyses that condition on observed hospital admission status—that is, analyses comparing outcomes for all hospitalized COVID-19 patients by vaccination status.

Continuous outcome measurements truncated by death present a challenge for the estimation of unbiased treatment effects in randomized controlled trials (RCTs). One way to deal with such situations is to estimate the survivor average causal effect (SACE), but this requires making nontestable assumptions. Motivated by an ongoing RCT in very preterm infants with intraventricular hemorrhage, we performed a simulation study to compare an SACE estimator with complete case analysis (CCA) and analysis after multiple imputation of missing outcomes. We set up nine scenarios combining positive, negative, and no treatment effect on the outcome (cognitive development) and on survival at 2 years of age. Treatment effect estimates from all methods were compared in terms of bias, mean squared error, and coverage with regard to two true treatment effects: the treatment effect on the outcome used in the simulation and the SACE, which was derived by simulation of both potential outcomes per patient. Despite targeting different estimands (principal stratum estimand, hypothetical estimand), the SACE-estimator and multiple imputation gave similar estimates of the treatment effect and efficiently reduced the bias compared to CCA. Also, both methods were relatively robust to omission of one covariate in the analysis, and thus violation of relevant assumptions. Although the SACE is not without controversy, we find it useful if mortality is inherent to the study population. Some degree of violation of the required assumptions is almost certain, but may be acceptable inpractice.

In randomized clinical trials, stopping study medication, use of rescue treatment, and other intercurrent events can complicate interpretation of results. The ICH E9(R1) Addendum on estimands stimulated important cross-disciplinary discussions on trial objectives. Unfortunately, with influence of the Addendum, many trials have proposed analyzing primary endpoints using "while on treatment", "hypothetical", or "principal stratum" strategies that handle intercurrent events in ways that use post-randomization outcomes to exclude information from randomized participants and don't preserve integrity of randomization, or that don't reliably capture the intervention's meaningful net effects. These approaches have inherent limitations in ability to draw scientifically rigorous inference on clinically relevant causal effects important for decisions about adopting interventions. We describe advantages of trials with standard-of-care control arms targeting estimands using "treatment policy" approaches for intercurrent events, while potentially incorporating other meaningful intercurrent events, such as death, into the primary endpoint applying a composite strategy. Well-designed and -conducted trials targeting such estimands achieve scientifically rigorous causal inference through analyzes that protect the integrity of randomization. Such estimands also provide meaningful information relevant to real-world settings because they (1) are unconditional in nature i.e., they don't condition on post-treatment circumstances that might not be many participants' experiences; and (2) properly evaluate the experimental intervention within a regimen that includes possible ancillary care that would be clinically appropriate in real-world settings. We hope to add clarity about appropriate strategies for intercurrent events and how to improve design, conduct, and analysis of clinical trials to address questions of greatest clinical importance reliably.

In clinical trials, principal stratification analysis is commonly employed to address the issue of truncation by death, where a subject dies before the outcome can be measured. However, in practice, many survivor outcomes may remain uncollected or be missing not at random, posing a challenge to standard principal stratification analysis. In this article, we explore the identification, estimation, and bounds of the average treatment effect within a subpopulation of individuals who would potentially survive under both treatment and control conditions. We show that the causal parameter of interest can be identified by introducing a proxy variable that affects the outcome only through the principal strata, while requiring that the treatment variable does not directly affect the missingness mechanism. Subsequently, we propose an approach for estimating causal parameters and derive nonparametric bounds in cases where identification assumptions are violated. We illustrate the performance of the proposed method through simulation studies and a real dataset obtained from a human immunodeficiency virus study.

The estimand framework, introduced in the ICH E9 (R1) Addendum, provides a structured approach for defining precise research questions in randomised clinical trials. It suggests five strategies for addressing intercurrent events (ICE). This case study examines the principal stratum strategy, highlighting its potential for estimating causal treatment effects in specific subpopulations and the challenges involved. The occurrence of anti-drug antibodies (ADAs) and their potential clinical impact are important factors in evaluating biosimilars. Typically, analyses focus on subgroups of patients who develop ADAs during the study. However, conducting subgroup analyses based on post-randomisation variables, such as immunogenicity, can introduce substantial bias into treatment effect estimates and is therefore methodologically not optimal. The principal stratum strategy provides a statistical pathway for estimating treatment effects in subpopulations that cannot be anticipated at baseline. By leveraging counterfactuals to assess treatment outcomes, with and without the incidence of intercurrent events (ICEs), this approach can be implemented through a missing data perspective. We demonstrate the implementation of the principal stratum strategy in a phase 3 equivalence trial of a biosimilar for the treatment of rheumatoid arthritis. Using a multiple imputation approach, we leverage longitudinal measurements to create analysis datasets for subpopulations who develop ADAs as ICE. Our results highlight the principal stratum strategy's potential and challenges, emphasising its reliance on unobserved ICE states and the need for complex and rigorous modelling. This study contributes to a nuanced understanding and practical implementation of the principal stratum strategy within the ICH E9 (R1) framework.

Abstract Fuzzy regression-discontinuity evaluations of college remediation often find negative and null estimates of local average treatments effects (LATEs), but with substantial heterogeneity. We find that a remedial quantitative skills course at Wellesley College has a modestly positive LATE on participation in mathematically intensive fields of study—including the sciences, mathematics, and economics courses. Yet, LATEs are a weighted average of average causal effects (at the passing cutoff) in two principal strata: students who voluntarily comply with remediation, and those who are coerced to comply after scoring below the cutoff on an optional retest. In the retest sample, we show that average causal effects are close to zero among (1) coerced compliers, and (2) never-takers. By implication, there are even larger effects among a smaller group of voluntary compliers at the cutoff. The results help interpret the mixed findings in the literature, in which compliance varies widely, and demonstrate methods for assessing external validity in fuzzy-discontinuity designs.

Recently, the International Conference on Harmonisation finalized an estimand framework for randomized trials that was adopted by regulatory bodies worldwide. The framework introduced five strategies for handling post-randomization events; namely the treatment policy, composite variable, while on treatment, hypothetical and principal stratum estimands. We describe an illustrative example to elucidate the difference between these five strategies for handling intercurrent events and provide an estimation technique for each. Specifically, we consider the intercurrent event of treatment discontinuation and introduce potential outcome notation to describe five estimands and corresponding estimators: (1) an intention-to-treat estimator of the total effect of a treatment policy; (2) an intention-to-treat estimator of a composite of the outcome and remaining on treatment; (3) a per-protocol estimator of the outcome in individuals observed to remain on treatment; (4) a g-computation estimator of a hypothetical scenario that all individuals remain on treatment; and (5) a principal stratum estimator of the treatment effect in individuals who would remain on treatment under the experimental condition. Additional insight is provided by defining situations where certain estimands are equal, and by studying the while on treatment strategy under repeated outcome measures. We highlight relevant causal inference literature to enable adoption in practice.