Identification of complier and noncomplier average causal effects in the presence of latent missing-at-random (LMAR) outcomes: a unifying view and choices of assumptions

The study of treatment effects is often complicated by noncompliance and missing data. In the one-sided noncompliance setting where of interest are the complier and noncomplier average causal effects, we address outcome missingness of the latent missing at random type (LMAR, also known as latent ignorability). That is, conditional on covariates and treatment assigned, the missingness may depend on compliance type. Within the instrumental variable (IV) approach to noncompliance, methods have been proposed for handling LMAR outcome that additionally invoke an exclusion restriction-type assumption on missingness, but no solution has been proposed for when a non-IV approach is used. This article focuses on effect identification in the presence of LMAR outcomes, with a view to flexibly accommodate different principal identification approaches. We show that under treatment assignment ignorability and LMAR only, effect nonidentifiability boils down to a set of two connected mixture equations involving unidentified stratum-specific response probabilities and outcome means. This clarifies that (except for a special case) effect identification generally requires two additional assumptions: a specific missingness mechanism assumption and a principal identification assumption. This provides a template for identifying effects based on separate choices of these assumptions. We consider a range of specific missingness assumptions, including those that have appeared in the literature and some new ones. Incidentally, we find an issue in the existing assumptions, and propose a modification of the assumptions to avoid the issue. Results under different assumptions are illustrated using data from the Baltimore Experience Corps Trial.

Causal inference for time-to-event data with a cured subpopulation

When studying the treatment effect on time-to-event outcomes, it is common that some individuals never experience failure events, which suggests that they have been cured. However, the cure status may not be observed due to censoring which makes it challenging to define treatment effects. Current methods mainly focus on estimating model parameters in various cure models, ultimately leading to a lack of causal interpretations. To address this issue, we propose 2 causal estimands, the timewise risk difference and mean survival time difference, in the always-uncured based on principal stratification as a complement to the treatment effect on cure rates. These estimands allow us to study the treatment effects on failure times in the always-uncured subpopulation. We show the identifiability using a substitutional variable for the potential cure status under ignorable treatment assignment mechanism, these 2 estimands are identifiable. We also provide estimation methods using mixture cure models. We applied our approach to an observational study that compared the leukemia-free survival rates of different transplantation types to cure acute lymphoblastic leukemia. Our proposed approach yielded insightful results that can be used to inform future treatment decisions.

A BAYESIAN MACHINE LEARNING APPROACH FOR ESTIMATING HETEROGENEOUS SURVIVOR CAUSAL EFFECTS: APPLICATIONS TO A CRITICAL CARE TRIAL

Assessing heterogeneity in the effects of treatments has become increasingly popular in the field of causal inference and carries important implications for clinical decision-making. While extensive literature exists for studying treatment effect heterogeneity when outcomes are fully observed, there has been limited development in tools for estimating heterogeneous causal effects when patient-centered outcomes are truncated by a terminal event, such as death. Due to mortality occurring during study follow-up, the outcomes of interest are unobservable, undefined, or not fully observed for many participants in which case principal stratification is an appealing framework to draw valid causal conclusions. Motivated by the Acute Respiratory Distress Syndrome Network (ARDSNetwork) ARDS respiratory management (ARMA) trial, we developed a flexible Bayesian machine learning approach to estimate the average causal effect and heterogeneous causal effects among the always-survivors stratum when clinical outcomes are subject to truncation. We adopted Bayesian additive regression trees (BART) to flexibly specify separate mean models for the potential outcomes and latent stratum membership. In the analysis of the ARMA trial, we found that the low tidal volume treatment had an overall benefit for participants sustaining acute lung injuries on the outcome of time to returning home but substantial heterogeneity in treatment effects among the always-survivors, driven most strongly by biologic sex and the alveolar-arterial oxygen gradient at baseline (a physiologic measure of lung function and degree of hypoxemia). These findings illustrate how the proposed methodology could guide the prognostic enrichment of future trials in the field.

Principal stratification analysis of noncompliance with time-to-event outcomes

Post-randomization events, also known as intercurrent events, such as treatment noncompliance and censoring due to a terminal event, are common in clinical trials. Principal stratification is a framework for causal inference in the presence of intercurrent events. The existing literature on principal stratification lacks generally applicable and accessible methods for time-to-event outcomes. In this paper, we focus on the noncompliance setting. We specify 2 causal estimands for time-to-event outcomes in principal stratification and provide a nonparametric identification formula. For estimation, we adopt the latent mixture modeling approach and illustrate the general strategy with a mixture of Bayesian parametric Weibull-Cox proportional hazards model for the outcome. We utilize the Stan programming language to obtain automatic posterior sampling of the model parameters. We provide analytical forms of the causal estimands as functions of the model parameters and an alternative numerical method when analytical forms are not available. We apply the proposed method to the ADAPTABLE (Aspirin Dosing: A Patient-Centric Trial Assessing Benefits and Long-Term Effectiveness) trial to evaluate the causal effect of taking 81 versus 325 mg aspirin on the risk of major adverse cardiovascular events. We develop the corresponding R package PStrata.

A mixed model approach to estimate the survivor average causal effect in cluster-randomized trials

In many medical studies, the outcome measure (such as quality of life, QOL) for some study participants becomes informatively truncated (censored, missing, or unobserved) due to death or other forms of dropout, creating a nonignorable missing data problem. In such cases, the use of a composite outcome or imputation methods that fill in unmeasurable QOL values for those who died rely on strong and untestable assumptions and may be conceptually unappealing to certain stakeholders when estimating a treatment effect. The survivor average causal effect (SACE) is an alternative causal estimand that surmounts some of these issues. While principal stratification has been applied to estimate the SACE in individually randomized trials, methods for estimating the SACE in cluster-randomized trials are currently limited. To address this gap, we develop a mixed model approach along with an expectation-maximization algorithm to estimate the SACE in cluster-randomized trials. We model the continuous outcome measure with a random intercept to account for intracluster correlations due to cluster-level randomization, and model the principal strata membership both with and without a random intercept. In simulations, we compare the performance of our approaches with an existing fixed-effects approach to illustrate the importance of accounting for clustering in cluster-randomized trials. The methodology is then illustrated using a cluster-randomized trial of telecare and assistive technology on health-related QOL in the elderly.

Stochastic Interventional Vaccine Efficacy and Principal Surrogate Analyses of Antibody Markers as Correlates of Protection against Symptomatic COVID-19 in the COVE mRNA-1273 Trial

The COVE trial randomized participants to receive two doses of mRNA-1273 vaccine or placebo on Days 1 and 29 (D1, D29). Anti-SARS-CoV-2 Spike IgG binding antibodies (bAbs), anti-receptor binding domain IgG bAbs, 50% inhibitory dilution neutralizing antibody (nAb) titers, and 80% inhibitory dilution nAb titers were measured at D29 and D57. We assessed these markers as correlates of protection (CoPs) against COVID-19 using stochastic interventional vaccine efficacy (SVE) analysis and principal surrogate (PS) analysis, frameworks not used in our previous COVE immune correlates analyses. By SVE analysis, hypothetical shifts of the D57 Spike IgG distribution from a geometric mean concentration (GMC) of 2737 binding antibody units (BAU)/mL (estimated vaccine efficacy (VE): 92.9% (95% CI: 91.7%, 93.9%)) to 274 BAU/mL or to 27,368 BAU/mL resulted in an overall estimated VE of 84.2% (79.0%, 88.1%) and 97.6% (97.4%, 97.7%), respectively. By binary marker PS analysis of Low and High subgroups (cut-point: 2094 BAU/mL), the ignorance interval (IGI) and estimated uncertainty interval (EUI) for VE were [85%, 90%] and (78%, 93%) for Low compared to [95%, 96%] and (92%, 97%) for High. By continuous marker PS analysis, the IGI and 95% EUI for VE at the 2.5th percentile (519.4 BAU/mL) vs. at the 97.5th percentile (9262.9 BAU/mL) of D57 Spike IgG concentration were [92.6%, 93.4%] and (89.2%, 95.7%) vs. [94.3%, 94.6%] and (89.7%, 97.0%). Results were similar for other D29 and D57 markers. Thus, the SVE and PS analyses additionally support all four markers at both time points as CoPs.

Solutions for surrogacy validation with longitudinal outcomes for a gene therapy

Valid surrogate endpoints S can be used as a substitute for a true outcome of interest T to measure treatment efficacy in a clinical trial. We propose a causal inference approach to validate a surrogate by incorporating longitudinal measurements of the true outcomes using a mixed modeling approach, and we define models and quantities for validation that may vary across the study period using principal surrogacy criteria. We consider a surrogate-dependent treatment efficacy curve that allows us to validate the surrogate at different time points. We extend these methods to accommodate a delayed-start treatment design where all patients eventually receive the treatment. Not all parameters are identified in the general setting. We apply a Bayesian approach for estimation and inference, utilizing more informative prior distributions for selected parameters. We consider the sensitivity of these prior assumptions as well as assumptions of independence among certain counterfactual quantities conditional on pretreatment covariates to improve identifiability. We examine the frequentist properties (bias of point and variance estimates, credible interval coverage) of a Bayesian imputation method. Our work is motivated by a clinical trial of a gene therapy where the functional outcomes are measured repeatedly throughout the trial.

A Bayesian Approach for Estimating the Survivor Average Causal Effect When Outcomes Are Truncated by Death in Cluster-Randomized Trials

Many studies encounter clustering due to multicenter enrollment and nonmortality outcomes, such as quality of life, that are truncated due to death-that is, missing not at random and nonignorable. Traditional missing-data methods and target causal estimands are suboptimal for statistical inference in the presence of these combined issues, which are especially common in multicenter studies and cluster-randomized trials (CRTs) carried out among the elderly or seriously ill. Using principal stratification, we developed a Bayesian estimator that jointly identifies the always-survivor principal stratum in a clustered/hierarchical data setting and estimates the average treatment effect among them (i.e., the survivor average causal effect (SACE)). In simulations, we observed low bias and good coverage with our method. In a motivating CRT, the SACE and the estimate from complete-case analysis differed in magnitude, but both were small, and neither was incompatible with a null effect. However, the SACE estimate has a clear causal interpretation. The option to assess the rigorously defined SACE estimand in studies with informative truncation and clustering can provide additional insight into an important subset of study participants. Based on the simulation study and CRT reanalysis, we provide practical recommendations for using the SACE in CRTs and software code to support future research.

Using principal stratification in analysis of clinical trials

The ICH E9(R1) addendum (2019) proposed principal stratification (PS) as one of five strategies for dealing with intercurrent events. Therefore, understanding the strengths, limitations, and assumptions of PS is important for the broad community of clinical trialists. Many approaches have been developed under the general framework of PS in different areas of research, including experimental and observational studies. These diverse applications have utilized a diverse set of tools and assumptions. Thus, need exists to present these approaches in a unifying manner. The goal of this tutorial is threefold. First, we provide a coherent and unifying description of PS. Second, we emphasize that estimation of effects within PS relies on strong assumptions and we thoroughly examine the consequences of these assumptions to understand in which situations certain assumptions are reasonable. Finally, we provide an overview of a variety of key methods for PS analysis and use a real clinical trial example to illustrate them. Examples of code for implementation of some of these approaches are given in Supplemental Materials.

Handling parametric assumptions in principal causal effect estimation using Gaussian mixtures

Given the latent stratum membership, principal stratification models with continuous outcomes naturally fit in the parametric estimation framework of Gaussian mixtures. However, with models that are not nonparametrically identified, relying on parametric mixture modeling has been mostly discouraged as a way of identifying principal effects. This study revisits this rather deserted use of parametric mixture modeling, which may open up various possibilities in principal stratification modeling. The main problem with using the parametric mixture modeling approach is that it is hard to assess the quality of principal effect estimates given its reliance on parametric conditions. As a way of assessing the estimation quality in this situation, this study proposes that we use parametric mixture modeling in two different ways, with and without the assurance of nonparametric identification. The key identifying assumption employed in this study is the moving exclusion restriction, a flexible version of the standard exclusion restriction assumption. This assumption is used as a temporary vehicle to help assess the quality of principal effect estimates obtained relying on parametric mixture modeling. The study presents promising results, showing the possibility of using parametric mixture modeling as an accessible tool for causal inference.

Translating questions to estimands in randomized clinical trials with intercurrent events

Intercurrent (post‐treatment) events occur frequently in randomized trials, and investigators often express interest in treatment effects that suitably take account of these events. Contrasts that naively condition on intercurrent events do not have a straight‐forward causal interpretation, and the practical relevance of other commonly used approaches is debated. In this work, we discuss how to formulate and choose an estimand, beyond the marginal intention‐to‐treat effect, from the point of view of a decision maker and drug developer. In particular, we argue that careful articulation of a practically useful research question should either reflect decision making at this point in time or future drug development. Indeed, a substantially interesting estimand is simply a formalization of the (plain English) description of a research question. A common feature of estimands that are practically useful is that they correspond to possibly hypothetical but well‐defined interventions in identifiable (sub)populations. To illustrate our points, we consider five examples that were recently used to motivate consideration of principal stratum estimands in clinical trials. In all of these examples, we propose alternative causal estimands, such as conditional effects, sequential regime effects, and separable effects, that correspond to explicit research questions of substantial interest.

Assessing complier average causal effects from longitudinal trials with multiple endpoints and treatment noncompliance: An application to a study of Arthritis Health Journal

Treatment noncompliance often occurs in longitudinal randomized controlled trials (RCTs) on human subjects, and can greatly complicate treatment effect assessment. The complier average causal effect (CACE) informs the intervention efficacy for the subpopulation who would comply regardless of assigned treatment and has been considered as patient-oriented treatment effects of interest in the presence of noncompliance. Real-world RCTs evaluating multifaceted interventions often employ multiple study endpoints to measure treatment success. In such trials, limited sample sizes, low compliance rates, and small to moderate effect sizes on individual endpoints can significantly reduce the power to detect CACE when these correlated endpoints are analyzed separately. To overcome the challenge, we develop a multivariate longitudinal potential outcome model with stratification on latent compliance types to efficiently assess multivariate CACEs (MCACE) by combining information across multiple endpoints and visits. Evaluation using simulation data shows a significant increase in the estimation efficiency with the MCACE model, including up to 50% reduction in standard errors (SEs) of CACE estimates and 1-fold increase in the power to detect CACE. Finally, we apply the proposed MCACE model to an RCT on Arthritis Health Journal online tool. Results show that the MCACE analysis detects significant and beneficial intervention effects on two of the six endpoints while estimating CACEs for these endpoints separately fail to detect treatment effect on any endpoint.

Clarifying selection bias in cluster randomized trials

BACKGROUND

In cluster randomized trials, patients are typically recruited after clusters are randomized, and the recruiters and patients may not be blinded to the assignment. This often leads to differential recruitment and consequently systematic differences in baseline characteristics of the recruited patients between intervention and control arms, inducing post-randomization selection bias. We aim to rigorously define causal estimands in the presence of selection bias. We elucidate the conditions under which standard covariate adjustment methods can validly estimate these estimands. We further discuss the additional data and assumptions necessary for estimating causal effects when such conditions are not met.

METHODS

Adopting the principal stratification framework in causal inference, we clarify there are two average treatment effect (ATE) estimands in cluster randomized trials: one for the overall population and one for the recruited population. We derive analytical formula of the two estimands in terms of principal-stratum-specific causal effects. Furthermore, using simulation studies, we assess the empirical performance of the multivariable regression adjustment method under different data generating processes leading to selection bias.

RESULTS

When treatment effects are heterogeneous across principal strata, the average treatment effect on the overall population generally differs from the average treatment effect on the recruited population. A naïve intention-to-treat analysis of the recruited sample leads to biased estimates of both average treatment effects. In the presence of post-randomization selection and without additional data on the non-recruited subjects, the average treatment effect on the recruited population is estimable only when the treatment effects are homogeneous between principal strata, and the average treatment effect on the overall population is generally not estimable. The extent to which covariate adjustment can remove selection bias depends on the degree of effect heterogeneity across principal strata.

CONCLUSION

There is a need and opportunity to improve the analysis of cluster randomized trials that are subject to post-randomization selection bias. For studies prone to selection bias, it is important to explicitly specify the target population that the causal estimands are defined on and adopt design and estimation strategies accordingly. To draw valid inferences about treatment effects, investigators should (1) assess the possibility of heterogeneous treatment effects, and (2) consider collecting data on covariates that are predictive of the recruitment process, and on the non-recruited population from external sources such as electronic health records.

Incorporating baseline covariates to validate surrogate endpoints with a constant biomarker under control arm

A surrogate endpoint S in a clinical trial is an outcome that may be measured earlier or more easily than the true outcome of interest T. In this work, we extend causal inference approaches to validate such a surrogate using potential outcomes. The causal association paradigm assesses the relationship of the treatment effect on the surrogate with the treatment effect on the true endpoint. Using the principal surrogacy criteria, we utilize the joint conditional distribution of the potential outcomes T, given the potential outcomes S. In particular, our setting of interest allows us to assume the surrogate under the placebo, S ( 0 ) , is zero-valued, and we incorporate baseline covariates in the setting of normally distributed endpoints. We develop Bayesian methods to incorporate conditional independence and other modeling assumptions and explore their impact on the assessment of surrogacy. We demonstrate our approach via simulation and data that mimics an ongoing study of a muscular dystrophy gene therapy.

Bayesian phase II clinical trial design with noncompliance

Noncompliance issue is common in early phase clinical trials; and may lead to biased estimation of the intent-to-treat effect and incorrect conclusions for the clinical trial. In this work, we propose a Bayesian approach for sequentially monitoring the phase II randomized clinical trials that takes account for the noncompliance information. We adopt the principal stratification framework and propose to use Bayesian additive regression trees for selecting useful baseline covariates and estimating the complier average causal effect (CACE) for both efficacy and toxicity outcomes. The decision of early termination or not is then made adaptively based on the estimated CACE from the accumulated data. Simulation studies have confirmed the excellent performance of the proposed design in the presence of noncompliance.

Incorporating partial adherence into the principal stratification analysis framework

Participants in pragmatic clinical trials often partially adhere to treatment. However, to simplify the analysis, most studies dichotomize adherence (supposing that subjects received either full or no treatment), which can introduce biases in the results. For example, the popular approach of principal stratification is based on the concept that the population can be separated into strata based on how they will react to treatment assignment, but this framework does not include strata in which a partially adhering participant would belong. We expanded the principal stratification framework to allow partial adherers to have their own principal stratum and treatment level. The expanded approach is feasible in pragmatic settings. We have designed a Monte Carlo posterior sampling method to obtain the relevant parameter estimates. Simulations were completed under a range of settings where participants partially adhered to treatment, including a hypothetical setting from a published simulation trial on the topic of partial adherence. The inference method is additionally applied to data from a real randomized clinical trial that features partial adherence. Comparison of the simulation results indicated that our method is superior in most cases to the biased estimators obtained through standard principal stratification. Simulation results further suggest that our proposed method may lead to increased accuracy of inference in settings where study participants only partially adhere to assigned treatment.

Implementation of tripartite estimands using adherence causal estimators under the causal inference framework

Intercurrent events (ICEs) and missing values are inevitable in clinical trials of any size and duration, making it difficult to assess the treatment effect for all patients in randomized clinical trials. Defining the appropriate estimand that is relevant to the clinical research question is the first step in analyzing data. The tripartite estimands, which evaluate the treatment differences in the proportion of patients with ICEs due to adverse events, the proportion of patients with ICEs due to lack of efficacy, and the primary efficacy outcome for those who can adhere to study treatment under the causal inference framework, are of interest to many stakeholders in understanding the totality of treatment effects. In this manuscript, we discuss the details of how to estimate tripartite estimands based on a causal inference framework and how to interpret tripartite estimates through a phase 3 clinical study evaluating a basal insulin treatment for patients with type 1 diabetes.

Assessing the ratio of means as a causal estimand in clinical endpoint bioequivalence studies in the presence of intercurrent events

In clinical endpoint bioequivalence studies, the observed per-protocol (PP) population (compliers and completers in general) is usually used in the primary analysis for equivalence assessment. However, intercurrent events, ie, missingness and noncompliance, are not properly handled. The resulting estimand is not causal. Previously, we proposed the first causal framework to assess equivalence in the presence of missing data and noncompliance. We proposed a causal survivor average causal effect (SACE) estimand for the difference of means (DOM). In equivalence assessment, DOM is not as widely used as the ratio of means (ROM). However, no existing formula links the observed PP estimand to the SACE estimand for ROM as exists for DOM. Herein, we propose a similar causal framework for ROM using the principal stratification approach, one of the strategies recommended by the International Conference on Harmonisation (ICH) E9 R1 addendum. We quantify the bias of the observed ROM PP estimand for the SACE estimand, which provides a basis to identify three conditions under which the two estimands are equal. We propose a sensitivity analysis method to evaluate the robustness of the current PP estimator to estimate the SACE estimand. We extend Fieller's confidence interval for the SACE estimand using ROM, which can be applied to many settings. Simulation demonstrates that the PP estimator is biased in either directions and may inflate type 1 error and/or change power when the three identified conditions are violated. Our work can be applied to comparative clinical biosimilar studies.

Analyzing patient-reported outcome data when completion differs between arms in open-label trials: an application of principal stratification

PURPOSE

Cancer trials are often open-label and include patient-reported outcomes (PROs). Previous work has demonstrated that patients may complete PRO assessments less frequently in the control arm compared with the experimental arm in open-label trials. Such differential completion may affect PRO results. This paper sought to explore principal stratification methodology to address potential bias caused by the posttreatment intermediate variable of questionnaire completion.

METHODS

We evaluated six randomized trials (five open-label and one double-blind) of anticancer therapies with varying levels of PRO completion submitted to the Food and Drug Administration (FDA). We applied complete case analysis (CCA), multiple imputation (MI), and principal stratification to evaluate PRO results for quality of life (QOL) and the domains of physical, role, and emotional function (PF, RF, and EF). Assignment to potential principal strata was by the expectation maximization algorithm using patient baseline characteristics.

RESULTS

Completion rates in the experimental arm ranged from 66% to 94% and 51% to 95% in the control arm. Four trials had negligible completion differences between arms (1%-2%), and two had large differences favoring the experimental arm (15%-17%). For trials with negligible completion differences, principal stratification results were similar to CCA and MI results for all domains. Notable differences in point estimates may be observed in trials with large differences in completion rates. However, in the examined trials, the confidence intervals for the principal stratification estimates overlapped with the ones obtained using CCA.

CONCLUSIONS

The principal stratification estimand may be a useful additional analysis, especially if PRO completion differs between arms.

Simple methods for the estimation and sensitivity analysis of principal strata effects using marginal structural models: Application to a bone fracture prevention trial

In randomized clinical trials, it is often of interest to estimate the effect of treatment on quality of life (QOL), in addition to those on the event itself. When an event occurs in some patients prior to QOL score assessment, investigators may compare QOL scores between patient subgroups defined by the event after randomization. However, owing to postrandomization selection bias, this analysis can mislead investigators about treatment efficacy and result in paradoxical findings. The recent Japanese Osteoporosis Intervention Trial (JOINT-02), which compared the benefits of a combination therapy for fracture prevention with those of a monotherapy, exemplifies the case in point; the average QOL score was higher in the combination therapy arm for the unfractured subgroup but was lower for the fractured subgroup. To address this issue, principal strata effects (PSEs), which are treatment effects estimated within subgroups of individuals stratified by potential intermediate variable, have been discussed in the literature. In this paper, we describe a simple procedure for estimating the PSEs using marginal structural models. This procedure utilizes SAS code for the estimation. In addition, we present a simple sensitivity analysis method for examining the resulting estimates. The analyses of JOINT-02 data using these methods revealed that QOL scores were higher in the combination therapy arm than in the monotherapy arm for both subgroups.

Simultaneous Inference of Treatment Effect Modification by Intermediate Response Endpoint Principal Strata with Application to Vaccine Trials

In randomized clinical trials, researchers are often interested in identifying an inexpensive intermediate study endpoint (typically a biomarker) that is a strong effect modifier of the treatment effect on a longer-term clinical endpoint of interest. Motivated by randomized placebo-controlled preventive vaccine efficacy trials, within the principal stratification framework a pseudo-score type estimator has been proposed to estimate disease risks conditional on the counter-factual biomarker of interest under each treatment assignment to vaccine or placebo, yielding an estimator of biomarker conditional vaccine efficacy. This method can be used for trial designs that use baseline predictors of the biomarker and/or designs that vaccinate disease-free placebo recipients at the end of the trial. In this article, we utilize the pseudo-score estimator to estimate the biomarker conditional vaccine efficacy adjusting for baseline covariates. We also propose a perturbation resampling method for making simultaneous inference on conditional vaccine efficacy over the values of the biomarker. We illustrate our method with datasets from two phase 3 dengue vaccine efficacy trials.

Estimating causal effects of treatment in RCTs with provider and subject noncompliance

Subject noncompliance is a common problem in the analysis of randomized clinical trials (RCTs). With cognitive behavioral interventions, the addition of provider noncompliance further complicates making causal inference. As a motivating example, we consider an RCT of a motivational interviewing (MI)-based behavioral intervention for treating problem drug use. Treatment receipt depends on compliance of both a therapist (provider) and a patient (subject), where MI is received when the therapist adheres to the MI protocol and the patient actively participates in the intervention. However, therapists cannot be forced to follow protocol and patients cannot be forced to cooperate in an intervention. In this article, we (1) define a causal estimand of interest based on a principal stratification framework, the average causal effect of treatment among provider-subject pairs that comply with assignment or ACE(cc); (2) explore possible assumptions that identify ACE(cc); (3) develop novel estimators of ACE(cc); (4) evaluate estimators' statistical properties via simulation; and (5) apply our proposed methods for estimating ACE(cc) to data from our motivating example.

Identifying treatment responders using counterfactual modeling and potential outcomes

Individualizing treatment according to patients' characteristics is central for personalized or precision medicine. There has been considerable recent research in developing statistical methods to determine optimal personalized treatment strategies by modeling the outcome of patients according to relevant covariates under each of the alternative treatments, and then relying on so-called predicted individual treatment effects. In this paper, we use potential outcomes and principal stratification frameworks and develop a multinomial model for left and right-censored data to estimate the probability that a patient is a responder given a set of baseline covariates. The model can apply to RCT or observational study data. This method is based on the monotonicity assumption, which implies that no patients would respond to the control treatment but not to the experimental one. We conduct a simulation study to evaluate the properties of the proposed estimation method. Results showed that the predictions of the probability of being a responder were well calibrated even if we observed variability and a small bias when many parameters were estimated. We finally applied the method to a cohort study on the selection of patients for additional radiotherapy after resection of a soft-tissue sarcoma.

Links between causal effects and causal association for surrogacy evaluation in a gaussian setting

Two paradigms for the evaluation of surrogate markers in randomized clinical trials have been proposed: the causal effects paradigm and the causal association paradigm. Each of these paradigms rely on assumptions that must be made to proceed with estimation and to validate a candidate surrogate marker (S) for the true outcome of interest (T). We consider the setting in which S and T are Gaussian and are generated from structural models that include an unobserved confounder. Under the assumed structural models, we relate the quantities used to evaluate surrogacy within both the causal effects and causal association frameworks. We review some of the common assumptions made to aid in estimating these quantities and show that assumptions made within one framework can imply strong assumptions within the alternative framework. We demonstrate that there is a similarity, but not exact correspondence between the quantities used to evaluate surrogacy within each framework, and show that the conditions for identifiability of the surrogacy parameters are different from the conditions, which lead to a correspondence of these quantities.

Power/sample size calculations for assessing correlates of risk in clinical efficacy trials

In a randomized controlled clinical trial that assesses treatment efficacy, a common objective is to assess the association of a measured biomarker response endpoint with the primary study endpoint in the active treatment group, using a case-cohort, case-control, or two-phase sampling design. Methods for power and sample size calculations for such biomarker association analyses typically do not account for the level of treatment efficacy, precluding interpretation of the biomarker association results in terms of biomarker effect modification of treatment efficacy, with detriment that the power calculations may tacitly and inadvertently assume that the treatment harms some study participants. We develop power and sample size methods accounting for this issue, and the methods also account for inter-individual variability of the biomarker that is not biologically relevant (e.g., due to technical measurement error). We focus on a binary study endpoint and on a biomarker subject to measurement error that is normally distributed or categorical with two or three levels. We illustrate the methods with preventive HIV vaccine efficacy trials and include an R package implementing the methods. Copyright © 2016 John Wiley & Sons, Ltd.

A Bayesian approach to estimating causal vaccine effects on binary post-infection outcomes

To estimate causal effects of vaccine on post-infection outcomes, Hudgens and Halloran (2006) defined a post-infection causal vaccine efficacy estimand VEI based on the principal stratification framework. They also derived closed forms for the maximum likelihood estimators of the causal estimand under some assumptions. Extending their research, we propose a Bayesian approach to estimating the causal vaccine effects on binary post-infection outcomes. The identifiability of the causal vaccine effect VEI is discussed under different assumptions on selection bias. The performance of the proposed Bayesian method is compared with the maximum likelihood method through simulation studies and two case studies - a clinical trial of a rotavirus vaccine candidate and a field study of pertussis vaccination. For both case studies, the Bayesian approach provided similar inference as the frequentist analysis. However, simulation studies with small sample sizes suggest that the Bayesian approach provides smaller bias and shorter confidence interval length.

Evaluating surrogate endpoints, prognostic markers, and predictive markers: Some simple themes

BACKGROUND

A surrogate endpoint is an endpoint observed earlier than the true endpoint (a health outcome) that is used to draw conclusions about the effect of treatment on the unobserved true endpoint. A prognostic marker is a marker for predicting the risk of an event given a control treatment; it informs treatment decisions when there is information on anticipated benefits and harms of a new treatment applied to persons at high risk. A predictive marker is a marker for predicting the effect of treatment on outcome in a subgroup of patients or study participants; it provides more rigorous information for treatment selection than a prognostic marker when it is based on estimated treatment effects in a randomized trial.

METHODS

We organized our discussion around a different theme for each topic.

RESULTS

"Fundamentally an extrapolation" refers to the non-statistical considerations and assumptions needed when using surrogate endpoints to evaluate a new treatment. "Decision analysis to the rescue" refers to use the use of decision analysis to evaluate an additional prognostic marker because it is not possible to choose between purely statistical measures of marker performance. "The appeal of simplicity" refers to a straightforward and efficient use of a single randomized trial to evaluate overall treatment effect and treatment effect within subgroups using predictive markers.

CONCLUSION

The simple themes provide a general guideline for evaluation of surrogate endpoints, prognostic markers, and predictive markers.

Bayesian sequential monitoring design for two-arm randomized clinical trials with noncompliance

In early-phase clinical trials, interim monitoring is commonly conducted based on the estimated intent-to-treat effect, which is subject to bias in the presence of noncompliance. To address this issue, we propose a Bayesian sequential monitoring trial design based on the estimation of the causal effect using a principal stratification approach. The proposed design simultaneously considers efficacy and toxicity outcomes and utilizes covariates to predict a patient's potential compliance behavior and identify the causal effects. Based on accumulating data, we continuously update the posterior estimates of the causal treatment effects and adaptively make the go/no-go decision for the trial. Numerical results show that the proposed method has desirable operating characteristics and addresses the issue of noncompliance.

Estimating the health benefit of reducing indoor air pollution in a randomized environmental intervention

Recent intervention studies targeted at reducing indoor air pollution have demonstrated both the ability to improve respiratory health outcomes and to reduce particulate matter (PM) levels in the home. However, these studies generally do not address whether it is the reduction of PM levels specifically that improves respiratory health. In this paper we apply the method of principal stratification to data from a randomized air cleaner intervention designed to reduce indoor PM in homes of children with asthma. We estimate the health benefit of the intervention amongst study subjects who would experience a substantial reduction in PM in response to the intervention. For those subjects we find an increase in symptom-free days that is almost three times as large as the overall intention-to-treat effect. We also explore the presence of treatment effects amongst those subjects whose PM levels would not respond to the air cleaner. This analysis demonstrates the usefulness of principal stratification for environmental intervention trials and its potential for much broader application in this area.

Design and inference for the intent-to-treat principle using adaptive treatment

Nonadherence to assigned treatment jeopardizes the power and interpretability of intent-to-treat comparisons from clinical trial data and continues to be an issue for effectiveness studies, despite their pragmatic emphasis. We posit that new approaches to design need to complement developments in methods for causal inference to address nonadherence, in both experimental and practice settings. This paper considers the conventional study design for psychiatric research and other medical contexts, in which subjects are randomized to treatments that are fixed throughout the trial and presents an alternative that converts the fixed treatments into an adaptive intervention that reflects best practice. The key element is the introduction of an adaptive decision point midway into the study to address a patient's reluctance to remain on treatment before completing a full-length trial of medication. The clinical uncertainty about the appropriate adaptation prompts a second randomization at the new decision point to evaluate relevant options. Additionally, the standard 'all-or-none' principal stratification (PS) framework is applied to the first stage of the design to address treatment discontinuation that occurs too early for a midtrial adaptation. Drawing upon the adaptive intervention features, we develop assumptions to identify the PS causal estimand and to introduce restrictions on outcome distributions to simplify expectation-maximization calculations. We evaluate the performance of the PS setup, with particular attention to the role played by a binary covariate. The results emphasize the importance of collecting covariate data for use in design and analysis. We consider the generality of our approach beyond the setting of psychiatric research.

Inference for environmental intervention studies using principal stratification

Previous research has found evidence of an association between indoor air pollution and asthma morbidity in children. Environmental intervention studies have been performed to examine the role of household environmental interventions in altering indoor air pollution concentrations and improving health. Previous environmental intervention studies have found only modest effects on health outcomes and it is unclear if the health benefits provided by environmental modification are comparable with those provided by medication. Traditionally, the statistical analysis of environmental intervention studies has involved performing two intention-to-treat analyses that separately estimate the effect of the environmental intervention on health and the effect of the environmental intervention on indoor air pollution concentrations. We propose a principal stratification approach to examine the extent to which an environmental intervention's effect on health outcomes coincides with its effect on indoor air pollution. We apply this approach to data from a randomized air cleaner intervention trial conducted in a population of asthmatic children living in Baltimore, Maryland, USA. We find that among children for whom the air cleaner reduced indoor particulate matter concentrations, the intervention resulted in a meaningful improvement of asthma symptoms with an effect generally larger than previous studies have shown. A key benefit of using principal stratification in environmental intervention studies is that it allows investigators to estimate causal effects of the intervention for sub-groups defined by changes in the indoor air pollution concentration.

A principal stratification approach for evaluating natural direct and indirect effects in the presence of treatment-induced intermediate confounding

Recently, several authors have shown that natural direct and indirect effects (NDEs and NIEs) can be identified under the sequential ignorability assumptions, as long as there is no mediator-outcome confounder that is affected by the treatment. However, if such a confounder exists, NDEs and NIEs will generally not be identified without making additional identifying assumptions. In this article, we propose novel identification assumptions and estimators for evaluating NDEs and NIEs under the usual sequential ignorability assumptions, using the principal stratification framework. It is assumed that the treatment and the mediator are dichotomous. We must impose strong assumptions for identification. However, even if these assumptions were violated, the bias of our estimator would be small under typical conditions, which can be easily evaluated from the observed data. This conjecture is confirmed for binary outcomes by deriving the bounds of the bias terms. In addition, the advantage of our estimator is illustrated through a simulation study. We also propose a method of sensitivity analysis that examines what happens when our assumptions are violated. We apply the proposed method to data from the National Center for Health Statistics.

Identification and estimation of survivor average causal effects

In longitudinal studies, outcomes ascertained at follow-up are typically undefined for individuals who die prior to the follow-up visit. In such settings, outcomes are said to be truncated by death and inference about the effects of a point treatment or exposure, restricted to individuals alive at the follow-up visit, could be biased even if as in experimental studies, treatment assignment were randomized. To account for truncation by death, the survivor average causal effect (SACE) defines the effect of treatment on the outcome for the subset of individuals who would have survived regardless of exposure status. In this paper, the author nonparametrically identifies SACE by leveraging post-exposure longitudinal correlates of survival and outcome that may also mediate the exposure effects on survival and outcome. Nonparametric identification is achieved by supposing that the longitudinal data arise from a certain nonparametric structural equations model and by making the monotonicity assumption that the effect of exposure on survival agrees in its direction across individuals. A novel weighted analysis involving a consistent estimate of the survival process is shown to produce consistent estimates of SACE. A data illustration is given, and the methods are extended to the context of time-varying exposures. We discuss a sensitivity analysis framework that relaxes assumptions about independent errors in the nonparametric structural equations model and may be used to assess the extent to which inference may be altered by a violation of key identifying assumptions. © 2014 The Authors. Statistics in Medicine published by John Wiley & Sons, Ltd.

Fold rise in antibody titers by measured by glycoprotein-based enzyme-linked immunosorbent assay is an excellent correlate of protection for a herpes zoster vaccine, demonstrated via the vaccine efficacy curve

BACKGROUND

The phase III Zostavax Efficacy and Safety Trial of 1 dose of licensed zoster vaccine (ZV; Zostavax; Merck) in 50-59-year-olds showed approximately 70% vaccine efficacy (VE) to reduce the incidence of herpes zoster (HZ). An objective of the trial was to assess immune response biomarkers measuring antibodies to varicella zoster virus (VZV) by glycoprotein-based enzyme-linked immunosorbent assay as correlates of protection (CoPs) against HZ.

METHODS

The principal stratification vaccine efficacy curve framework for statistically evaluating immune response biomarkers as CoPs was applied. The VE curve describes how VE against the clinical end point (HZ) varies across participant subgroups defined by biomarker readout measuring vaccine-induced immune response. The VE curve was estimated using several subgroup definitions.

RESULTS

The fold rise in VZV antibody titers from the time before immunization to 6 weeks after immunization was an excellent CoP, with VE increasing sharply with fold rise: VE was estimated at 0% for the subgroup with no rise and at 90% for the subgroup with 5.26-fold rise. In contrast, VZV antibody titers measured 6 weeks after immunization did not predict VE, with similar estimated VEs across titer subgroups.

CONCLUSIONS

The analysis illustrates the value of the VE curve framework for assessing immune response biomarkers as CoPs in vaccine efficacy trials.

CLINICAL TRIALS REGISTRATION

NCT00534248.

Causal inference in longitudinal comparative effectiveness studies with repeated measures of a continuous intermediate variable

We propose a principal stratification approach to assess causal effects in nonrandomized longitudinal comparative effectiveness studies with a binary endpoint outcome and repeated measures of a continuous intermediate variable. Our method is an extension of the principal stratification approach originally proposed for the longitudinal randomized study "Prevention of Suicide in Primary Care Elderly: Collaborative Trial" to assess the treatment effect on the continuous Hamilton depression score adjusting for the heterogeneity of repeatedly measured binary compliance status. Our motivation for this work comes from a comparison of the effect of two glucose-lowering medications on a clinical cohort of patients with type 2 diabetes. Here, we consider a causal inference problem assessing how well the two medications work relative to one another on two binary endpoint outcomes: cardiovascular disease-related hospitalization and all-cause mortality. Clinically, these glucose-lowering medications can have differential effects on the intermediate outcome, glucose level over time. Ultimately, we want to compare medication effects on the endpoint outcomes among individuals in the same glucose trajectory stratum while accounting for the heterogeneity in baseline covariates (i.e., to obtain 'principal effects' on the endpoint outcomes). The proposed method involves a three-step model estimation procedure. Step 1 identifies principal strata associated with the intermediate variable using hybrid growth mixture modeling analyses. Step 2 obtains the stratum membership using the pseudoclass technique and derives propensity scores for treatment assignment. Step 3 obtains the stratum-specific treatment effect on the endpoint outcome weighted by inverse propensity probabilities derived from Step 2.

Causal inference for bivariate longitudinal quality of life data in presence of death by using global odds ratios

In longitudinal clinical trials, if a subject drops out due to death, certain responses, such as those measuring quality of life (QoL), will not be defined after the time of death. Thus, standard missing data analyses, e.g., under ignorable dropout, are problematic because these approaches implicitly 'impute' values of the response after death. In this paper we define a new survivor average causal effect for a bivariate response in a longitudinal quality of life study that had a high dropout rate with the dropout often due to death (or tumor progression). We show how principal stratification, with a few sensitivity parameters, can be used to draw causal inferences about the joint distribution of these two ordinal quality of life measures.

Rank-based principal stratum sensitivity analyses

We describe rank-based approaches to assess principal stratification treatment effects in studies where the outcome of interest is only well-defined in a subgroup selected after randomization. Our methods are sensitivity analyses, in that estimands are identified by fixing a parameter and then we investigate the sensitivity of results by varying this parameter over a range of plausible values. We present three rank-based test statistics and compare their performance through simulations, and provide recommendations. We also study three different bootstrap approaches for determining levels of significance. Finally, we apply our methods to two studies: an HIV vaccine trial and a prostate cancer prevention trial.

Joint modeling compliance and outcome for causal analysis in longitudinal studies

This article discusses joint modeling of compliance and outcome for longitudinal studies when noncompliance is present. We focus on two-arm randomized longitudinal studies in which subjects are randomized at baseline, treatment is applied repeatedly over time, and compliance behaviors and clinical outcomes are measured and recorded repeatedly over time. In the proposed Markov compliance and outcome model, we use the potential outcome framework to define pre-randomization principal strata from the joint distribution of compliance under treatment and control arms, and estimate the effect of treatment within each principal strata. Besides the causal effect of the treatment, our proposed model can estimate the impact of the causal effect of the treatment at a given time on future compliance. Bayesian methods are used to estimate the parameters. The results are illustrated using a study assessing the effect of cognitive behavior therapy on depression. A simulation study is used to assess the repeated sampling properties of the proposed model.

Intermediate outcomes in randomized clinical trials: an introduction

BACKGROUND

Intermediate outcomes are common and typically on the causal pathway to the final outcome. Some examples include noncompliance, missing data, and truncation by death like pregnancy (e.g. when the trial intervention is given to non-pregnant women and the final outcome is preeclampsia, defined only on pregnant women). The intention-to-treat approach does not account properly for them, and more appropriate alternative approaches like principal stratification are not yet widely known. The purposes of this study are to inform researchers that the intention-to-treat approach unfortunately does not fit all problems we face in experimental research, to introduce the principal stratification approach for dealing with intermediate outcomes, and to illustrate its application to a trial of long term calcium supplementation in women at high risk of preeclampsia.

METHODS

Principal stratification and related concepts are introduced. Two ways for estimating causal effects are discussed and their application is illustrated using the calcium trial, where noncompliance and pregnancy are considered as intermediate outcomes, and preeclampsia is the main final outcome.

RESULTS

The limitations of traditional approaches and methods for dealing with intermediate outcomes are demonstrated. The steps, assumptions and required calculations involved in the application of the principal stratification approach are discussed in detail in the case of our calcium trial.

CONCLUSIONS

The intention-to-treat approach is a very sound one but unfortunately it does not fit all problems we find in randomized clinical trials; this is particularly the case for intermediate outcomes, where alternative approaches like principal stratification should be considered.

Comparing competing risk outcomes within principal strata, with application to studies of mother-to-child transmission of HIV

In randomized trials to prevent breast milk transmission of human immunodeficiency virus (HIV) from mother to infant, investigators are often interested in assessing the effect of a treatment or intervention on the cumulative risk of HIV infection by time (age) t in infants who are alive and uninfected at a certain time point τ(0)  < t. Such comparisons are challenging for two reasons. First, infants are typically randomized at birth (time 0 < τ(0) ) such that comparisons between trial arms among the subset of infants alive and uninfected at τ(0) are subject to selection bias. Second, in most mother-to-child transmission (MTCT) trials competing risks are often present, such as death or cessation of breastfeeding prior to HIV infection. In this paper, we present methods for assessing the causal effect of a treatment on competing risk outcomes within principal strata. In MTCT trials, the causal effect of interest is that of treatment on the risk of HIV infection by time t > τ(0) within the principal stratum of infants who would be alive and uninfected by τ(0) regardless of randomization assignment. We develop large sample nonparametric bounds and a semiparametric sensitivity analysis model for drawing inference about this causal effect. We present a simulation study demonstrating that the proposed methods perform well in finite samples. We apply the proposed methods to a large, recent MTCT trial.

Effective elements of cognitive behaviour therapy for psychosis: results of a novel type of subgroup analysis based on principal stratification

BACKGROUND

Meta-analyses show that cognitive behaviour therapy for psychosis (CBT-P) improves distressing positive symptoms. However, it is a complex intervention involving a range of techniques. No previous study has assessed the delivery of the different elements of treatment and their effect on outcome. Our aim was to assess the differential effect of type of treatment delivered on the effectiveness of CBT-P, using novel statistical methodology.

METHOD

The Psychological Prevention of Relapse in Psychosis (PRP) trial was a multi-centre randomized controlled trial (RCT) that compared CBT-P with treatment as usual (TAU). Therapy was manualized, and detailed evaluations of therapy delivery and client engagement were made. Follow-up assessments were made at 12 and 24 months. In a planned analysis, we applied principal stratification (involving structural equation modelling with finite mixtures) to estimate intention-to-treat (ITT) effects for subgroups of participants, defined by qualitative and quantitative differences in receipt of therapy, while maintaining the constraints of randomization.

RESULTS

Consistent delivery of full therapy, including specific cognitive and behavioural techniques, was associated with clinically and statistically significant increases in months in remission, and decreases in psychotic and affective symptoms. Delivery of partial therapy involving engagement and assessment was not effective.

CONCLUSIONS

Our analyses suggest that CBT-P is of significant benefit on multiple outcomes to patients able to engage in the full range of therapy procedures. The novel statistical methods illustrated in this report have general application to the evaluation of heterogeneity in the effects of treatment.

Assessing the sensitivity of methods for estimating principal causal effects

The framework of principal stratification provides a way to think about treatment effects conditional on post-randomization variables, such as level of compliance. In particular, the complier average causal effect (CACE) - the effect of the treatment for those individuals who would comply with their treatment assignment under either treatment condition - is often of substantive interest. However, estimation of the CACE is not always straightforward, with a variety of estimation procedures and underlying assumptions, but little advice to help researchers select between methods. In this article, we discuss and examine two methods that rely on very different assumptions to estimate the CACE: a maximum likelihood ('joint') method that assumes the 'exclusion restriction,' (ER) and a propensity score-based method that relies on 'principal ignorability.' We detail the assumptions underlying each approach, and assess each methods' sensitivity to both its own assumptions and those of the other method using both simulated data and a motivating example. We find that the ER-based joint approach appears somewhat less sensitive to its assumptions, and that the performance of both methods is significantly improved when there are strong predictors of compliance. Interestingly, we also find that each method performs particularly well when the assumptions of the other approach are violated. These results highlight the importance of carefully selecting an estimation procedure whose assumptions are likely to be satisfied in practice and of having strong predictors of principal stratum membership.

Commentary on "Principal stratification - a goal or a tool?" by Judea Pearl

This commentary takes up Pearl's welcome challenge to clearly articulate the scientific value of principal stratification estimands that we and colleagues have investigated, in the area of randomized placebo-controlled preventive vaccine efficacy trials, especially trials of HIV vaccines. After briefly arguing that certain principal stratification estimands for studying vaccine effects on post-infection outcomes are of genuine scientific interest, the bulk of our commentary argues that the "causal effect predictiveness" (CEP) principal stratification estimand for evaluating immune biomarkers as surrogate endpoints is not of ultimate scientific interest, because it evaluates surrogacy restricted to the setting of a particular vaccine efficacy trial, but is nevertheless useful for guiding the selection of primary immune biomarker endpoints in Phase I/II vaccine trials and for facilitating assessment of transportability/bridging surrogacy.

Principal stratification and attribution prohibition: good ideas taken too far

Pearl's article provides a useful springboard for discussing further the benefits and drawbacks of principal stratification and the associated discomfort with attributing effects to post-treatment variables. The basic insights of the approach are important: pay close attention to modification of treatment effects by variables not observable before treatment decisions are made, and be careful in attributing effects to variables when counterfactuals are ill-defined. These insights have often been taken too far in many areas of application of the approach, including instrumental variables, censoring by death, and surrogate outcomes. A novel finding is that the usual principal stratification estimand in the setting of censoring by death is by itself of little practical value in estimating intervention effects.

Sensitivity Analysis and Bounding of Causal Effects With Alternative Identifying Assumptions

When identification of causal effects relies on untestable assumptions regarding nonidentified parameters, sensitivity of causal effect estimates is often questioned. For proper interpretation of causal effect estimates in this situation, deriving bounds on causal parameters or exploring the sensitivity of estimates to scientifically plausible alternative assumptions can be critical. In this paper, we propose a practical way of bounding and sensitivity analysis, where multiple identifying assumptions are combined to construct tighter common bounds. In particular, we focus on the use of competing identifying assumptions that impose different restrictions on the same non-identified parameter. Since these assumptions are connected through the same parameter, direct translation across them is possible. Based on this cross-translatability, various information in the data, carried by alternative assumptions, can be effectively combined to construct tighter bounds on causal effects. Flexibility of the suggested approach is demonstrated focusing on the estimation of the complier average causal effect (CACE) in a randomized job search intervention trial that suffers from noncompliance and subsequent missing outcomes.

Principal stratification--uses and limitations

Pearl (2011) asked for the causal inference community to clarify the role of the principal stratification framework in the analysis of causal effects. Here, I argue that the notion of principal stratification has shed light on problems of non-compliance, censoring-by-death, and the analysis of post-infection outcomes; that it may be of use in considering problems of surrogacy but further development is needed; that it is of some use in assessing "direct effects"; but that it is not the appropriate tool for assessing "mediation." There is nothing within the principal stratification framework that corresponds to a measure of an "indirect" or "mediated" effect.

Causal assessment of surrogacy in a meta-analysis of colorectal cancer trials

When the true end points (T) are difficult or costly to measure, surrogate markers (S) are often collected in clinical trials to help predict the effect of the treatment (Z). There is great interest in understanding the relationship among S, T, and Z. A principal stratification (PS) framework has been proposed by Frangakis and Rubin (2002) to study their causal associations. In this paper, we extend the framework to a multiple trial setting and propose a Bayesian hierarchical PS model to assess surrogacy. We apply the method to data from a large collection of colon cancer trials in which S and T are binary. We obtain the trial-specific causal measures among S, T, and Z, as well as their overall population-level counterparts that are invariant across trials. The method allows for information sharing across trials and reduces the nonidentifiability problem. We examine the frequentist properties of our model estimates and the impact of the monotonicity assumption using simulations. We also illustrate the challenges in evaluating surrogacy in the counterfactual framework that result from nonidentifiability.

Response to Pearl's comments on principal stratification

Dr. Pearl invites researchers to justify their use of principal stratification. This comment explains how the use of principal stratification simplified a complex mediational problem encountered when evaluating a smoking cessation intervention's effect on reducing smoking withdrawal symptoms.

Clarifying the role of principal stratification in the paired availability design

The paired availability design for historical controls postulated four classes corresponding to the treatment (old or new) a participant would receive if arrival occurred during either of two time periods associated with different availabilities of treatment. These classes were later extended to other settings and called principal strata. Judea Pearl asks if principal stratification is a goal or a tool and lists four interpretations of principal stratification. In the case of the paired availability design, principal stratification is a tool that falls squarely into Pearl's interpretation of principal stratification as "an approximation to research questions concerning population averages." We describe the paired availability design and the important role played by principal stratification in estimating the effect of receipt of treatment in a population using data on changes in availability of treatment. We discuss the assumptions and their plausibility. We also introduce the extrapolated estimate to make the generalizability assumption more plausible. By showing why the assumptions are plausible we show why the paired availability design, which includes principal stratification as a key component, is useful for estimating the effect of receipt of treatment in a population. Thus, for our application, we answer Pearl's challenge to clearly demonstrate the value of principal stratification.

Principal stratification--a goal or a tool?

Principal stratification has recently become a popular tool to address certain causal inference questions, particularly in dealing with post-randomization factors in randomized trials. Here, we analyze the conceptual basis for this framework and invite response to clarify the value of principal stratification in estimating causal effects of interest.