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This study investigated the effect of “dose” and the components of Cognitive Behavioral Therapy (CBT) on treatment effects. It is a secondary analysis of the ACTION (Assessment of Cognitive Therapy Instead of Neuroleptics) trial which investigated CBT for people with schizophrenia spectrum disorders that chose not to take antipsychotic medication. Using instrumental variable methods, we found a “dose-response” such that each CBT session attended, reduced the primary outcome measure (the PANSS total score) by approximately 0.6 points (95% CI −1.20 to −0.06, p = 0.031). This suggests that length of therapy is important for those that receive CBT in the absence of antipsychotic medication. Secondly, using principal stratification we examined the process variables that modified treatment effects. Findings revealed that those who received a longitudinal formulation in the first 4 sessions of CBT had poorer treatment effects than those who did not, however this finding was not statistically significant (95% CI −37.244, 6.677, p = 0.173). However, it is important to note that these findings were evident in an exploratory analysis with a small sample. Future larger scale studies are needed to help understand components of effective treatment.

In a cancer prevention trial, an outcome such as cancer severity cannot be evaluated in individuals who do not develop cancer. In such a situation, the principal stratification approach has been applied. Under this approach, the Principal Strata Effect (PSE) has been considered, which is defined as the effect of treatment on the outcome among the subpopulation in which individuals would have developed cancer under either treatment arm. However, in this study, the author does not apply this approach to the situation. Instead, the author discusses the mediation analysis approach, in which Natural Direct and Indirect Effects (NDE and NIE) are considered. This approach has an advantage as it considers two possible mechanisms of treatment control of cancer severity: The first is that the treatment may prevent an individual from getting cancer, which could be regarded as control of cancer severity; the second is that even if the treatment does not prevent an individual from getting cancer, it may still impair the cancer severity. The former mechanism corresponds to the NIE and the latter corresponds to the NDE, although the PSE can consider only the latter mechanism. Methodologies proposed in the context of vaccine trials are applied to data from a randomized prostate cancer prevention trial.

In respiratory health research, interest often lies in estimating the effect of an exposure on a health outcome. If randomization of the exposure of interest is not possible, estimating its effect is typically complicated by confounding bias. This can often be dealt with by controlling for the variables causing the confounding, if measured, in the statistical analysis. Common statistical methods used to achieve this include multivariable regression models adjusting for selected confounding variables or stratification on those variables. Therefore, a key question is which measured variables need to be controlled for in order to remove confounding. An approach to confounder-selection based on the use of causal diagrams (often called directed acyclic graphs) is discussed. A causal diagram is a visual representation of the causal relationships believed to exist between the variables of interest, including the exposure, outcome and potential confounding variables. After creating a causal diagram for the research question, an intuitive and easy-to-use set of rules can be applied, based on a foundation of rigorous mathematics, to decide which measured variables must be controlled for in the statistical analysis in order to remove confounding, to the extent that is possible using the available data. This approach is illustrated by constructing a causal diagram for the research question: 'Does personal smoking affect the risk of subsequent asthma?'. Using data taken from the Tasmanian Longitudinal Health Study, the statistical analysis suggested by the causal diagram approach was performed.

We show that assumptions that are sufficient for estimating an average treatment effect in randomized trials with non-compliance restrict the subgroup means for always takers, compliers, defiers and never takers to a two-dimensional linear subspace of a four-dimensional space. Implications and special cases are exemplified.

Background Most methodologists recommend intention-to-treat (ITT) analysis in order to minimize bias. Although ITT analysis provides an unbiased estimate for the effect of treatment assignment on the outcome, the estimate is biased for the actual effect of receiving treatment (active treatment) compared to some comparison group (control). Other common analyses include measuring effects in (1) participants who follow their assigned treatment (Per Protocol), (2) participants according to treatment received (As Treated), and (3) those who would comply with recommended treatment (Complier Average Causal Effect (CACE) as estimated by Principal Stratification or Instrumental Variable Analyses). As each of these analyses compares different study subpopulations, they address different research questions. Purpose For each type of analysis, we review and explain (1) the terminology being used, (2) the main underlying concepts, (3) the questions that are answered and whether the method provides valid causal estimates, and (4) the situations when the analysis should be conducted. Methods We first review the major concepts in relation to four nuances of the clinical question, ‘Does treatment improve health?’ After reviewing these concepts, we compare the results of the different analyses using data from two published randomized controlled trials (RCTs). Each analysis has particular underlying assumptions and all require dichotomizing adherence into Yes or No. We apply sensitivity analyses so that intermediate adherence is considered (1) as adherence and (2) as non-adherence. Results The ITT approach provides an unbiased estimate for how active treatment will improve (1) health in the population if a policy or program is enacted or (2) health of patients if a clinician changes treatment practice. The CACE approach generally provides an unbiased estimate of the effect of active treatment on health of patients who would follow the clinician’s advice to take active treatment. Unfortunately, there is no current analysis for clinicians and patients who want to know whether active treatment will improve the patient’s health if taken, which is different from the effect in patients who would follow the clinician’s advice to take active treatment. Sensitivity analysis for the CACE using two published data sets suggests that the underlying assumptions appeared to be violated. Limitations There are several methods within each analytical approach we describe. Our analyses are based on a subset of these approaches. Conclusions Although adherence-based analyses may provide meaningful information, the analytical method should match the clinical question, and investigators should clearly outline why they believe assumptions hold and should provide empirical tests of the assumptions where possible.

We derive nonparametric bounds for local average treatment effects (LATE) without imposing the exclusion restriction assumption or requiring an outcome with bounded support. Instead, we employ assumptions requiring weak monotonicity of mean potential and counterfactual outcomes within or across subpopulations defined by the values of the potential treatment status under each value of the instrument. The key element in our derivation is a result relating LATE to a causal mediation effect, which allows us to exploit partial identification results from the causal mediation analysis literature. The bounds are employed to analyze the effect of attaining a GED, high school, or vocational degree on future labor market outcomes using randomization into a training program as an invalid instrument. The resulting bounds are informative, indicating that the local effect when assigned to training for those whose degree attainment is affected by the instrument is at most 12.7 percentage points on employment and $64.4 o...

In school-based randomized control trials (RCTs), a common design is to follow student cohorts over time. For such designs, education researchers usually focus on the place-based (PB) impact parameter, which is estimated using data collected on all students enrolled in the study schools at each data collection point. A potential problem with this approach, however, is that the PB impact parameter could confound intervention effects on student mobility with more policy-relevant intervention effects on student achievement. Furthermore, the PB parameter pertains to students with different levels of intervention exposure, which complicates the interpretation of the impact findings. To address these issues, this article uses a principal stratification approach to examine the survivor average causal effect (SACE) parameter for original cohort students who would remain in their baseline study schools in either the treatment or control condition. The SACE parameter pertains to those who would receive maximum exposure to the intervention, and thus is a relevant parameter for dosage analyses. A strategy to estimate the SACE parameter is discussed using maximum likelihood (ML) methods for finite mixture models, the expectation-maximization (EM) algorithm, and robust standard errors to adjust for clustering. The estimation approach is demonstrated using data from a recent large-scale, school-based RCT where student mobility was common during the 3-year follow-up period.

Many practical studies in biology, medicine, behavior science and the social sciences seek to establish causal relationship between treatments and outcomes, rather than mere associations. In this paper, we use a graphical model to describe a causal graphical model and study its identification. For an unidentifiable model, we introduce covariates which are always observed into the model so that it becomes identifiable. We then give an identifiable condition of the causal graphical model and prove it mathematically. Finally, we give the algorithm for the identifiable average causal effect of outcomes to the accepted treatment and give an example to illustrate this method and algorithm.

Stratification (among other applications) is a popular technique used in survey practice to improve the accuracy of estimators. Its full potential benefit can be gained by the effective use of auxiliary variables in stratification related to survey variables. This paper focuses on the problem of stratum formation when multiple stratification variables are available. We first review a variance reduction strategy in the case of univariate stratification. We then discuss its use for multivariate situations in convenient and efficient ways using three methods: compromised measures of size, principal components analysis and a K-means clustering algorithm. We also consider three types of compromising factors to data when using these three methods. Finally, we compare their efficiency using data from MU281 Swedish municipality population.

In randomized clinical trials designed to evaluate the effect of a treatment on patients with advanced disease stages, treatment switching is often allowed for ethical reasons. Because the switching is a prognosis-related choice, identification and estimation of the effect of the actual receipt of the treatment becomes problematic. Existing methods in the literature try to reconstruct the ideal situation that would be observed if the switchers had not switched. Rather than focusing on reconstructing the a-priori counterfactual outcome for the switchers, had they not switched, we propose to identify and estimate effects for (latent) subgroups of units according to their switching behaviour. The reference framework of the proposed method is the potential outcome approach. In order to estimate causal effects for sub- groups of units not affected by treatment, we rely on the principal stratification approach (Frangakis and Rubin in Biometrics 58(1): 21–29 2002) [1]. To illustrate the proposed method and evaluate the maintained assumptions, we analyse a dataset from a randomized clinical trial on patients with asymptomatic HIV infection assigned to immediate (the active treatment) or deferred (the control treatment) Zidovudine (ZDV). The results, obtained through a full-Bayesian estimation approach, are promising and emphasize the high heterogeneity of the effects for different latent subgroups defined according to the switching behaviour.

The paper highlights causal inference based on econometric measurement in real-time data environments. Each state has a probability of being realized in real-time. We define state selection bias as arising when real-time environments are ignored. We model indicator variables as measurements that exist partly in all particular theoretically possible states, but show only one configuration on observation. Under real-time randomization within data streams, econometric treatment effects are estimable using controlled and natural experiments motivated by real-time regression analyses. A bias occurs as a result of ignoring concept drift when classical regression statistics are naively applied to real-time experimental data. We present a simple algorithm for difference-in-difference estimation for real-time program evaluations. Finally, a new Problem of Causal Inference is introduced for real-time data environments.

The potential outcomes framework for causal inference has proven valuable in the study of infectious diseases. Conversely, the context of infectious diseases has stimulated many methodological advances in causal inference. In this chapter, we review causal inference problems pertinent to the study of infectious diseases. These include time-varying confounding, interference, and surrogate measures of clinical outcomes. The test-negative study design and the relatively new (to epidemiology) methods of negative controls and regression discontinuity are also covered. Motivation for each topic is given along with examples of application.

Principal stratification and mediation analysis are two ways to conceptualize the mediating role of an intermediate variable in the causal pathways by which a treatment affects an outcome. They are often viewed as competing frameworks, and their role in dealing with issues concerning causal mechanisms has often fired up glowing discussions. However a thoughtful comparative analysis, highlighting the substantive differences between the two frameworks is still lacking. We aim at filling this gap conducting both principal stratification and mediation analysis using, as a motivating example, a prospective, randomized, double-blind study to investigate to which extent the positive overall effect of treatment on postoperative pain control is mediated by postoperative self administration of intra-venous analgesia by patients. Using the Bayesian approach for inference, we estimate both associative and dissociative principal strata effects arising in principal stratification analysis, as well as natural effects and controlled direct effects from mediation analysis. We highlight that principal stratification and mediation analysis focus on different causal estimands, answer different causal questions and involve different sets of identifying assumptions. We discuss these aspects along the results arising from our analyses.

Purpose: We illustrate the effect of survival bias when investigating risk factors for eye disease in elderly populations for whom death is a competing risk. Our investigation focuses on the relationship between smoking and late age-related macular degeneration (AMD) in an observational study impacted by censoring due to death. Methods: Statistical methodology to calculate the survivor average causal effect (SACE) as a sensitivity analysis is described, including example statistical computing code for Stata and R. To demonstrate this method, we examine the causal effect of smoking history at baseline (1990–1994) on the presence of late AMD at the third study wave (2003–2007) using data from the Melbourne Collaborative Cohort Study. Results: Of the 40,506 participants eligible for inclusion, 38,092 (94%) survived until the start of the third study wave, 20,752 (51%) were graded for AMD (60% female, aged 47–85 years, mean 65 ± 8.7 years). Late AMD was detected in 122 participants. Logistic regression showed strong evidence of an increased risk of late AMD for current smokers compared to non-smokers (adjusted naïve odds ratio 2.99, 95% confidence interval, CI, 1.74–5.13). Among participants expected to be alive at the start of follow-up regardless of their smoking status, the estimated SACE odds ratio comparing current smokers to non-smokers was at least 3.42 (95% CI 1.57–5.15). Conclusions: Survival bias can attenuate associations between harmful exposures and diseases of aging. Estimation of the SACE using a sensitivity analysis approach should be considered when conducting epidemiological research within elderly populations.

Summary It has been argued that the use of debit cards may modify cash holding behaviour, as debit card holders may either withdraw cash from automated teller machines or purchase items by using point-of-sale devices at retailers. Within the Rubin causal model, we investigate the causal effects of the use of debit cards on the cash inventories held by households by using data from the Italy Survey of Household Income and Wealth. We adopt the principal stratification approach to incorporate the share of debit card holders who do not use this payment instrument. We use a regression model with the propensity score as the single predictor to adjust for the imbalance in observed covariates. We further develop a sensitivity analysis approach to assess the sensitivity of the proposed model to violation of the key unconfoundedness assumption. Our empirical results suggest statistically significant negative effects of debit cards on the household cash level in Italy.

Estimating treatment effects for subgroups defined by post-treatment behavior (i.e., estimating causal effects in a principal stratification framework) can be technically challenging and heavily reliant on strong assumptions. We investigate an alternative path: using bounds to identify ranges of possible effects that are consistent with the data. This simple approach relies on fewer assumptions and yet can result in policy-relevant findings. As we show, covariates can be used to substantially tighten bounds in a straightforward manner. Via simulation, we demonstrate which types of covariates are maximally beneficial. We conclude with an analysis of a multi-site experimental study of Early College High Schools. When examining the program's impact on students completing the ninth grade "on-track" for college, we find little impact for ECHS students who would otherwise attend a high quality high school, but substantial effects for those who would not. This suggests potential benefit in expanding these programs in areas primarily served by lower quality schools.

In vaccine studies, investigators are often interested in studying effect modifiers of clinical treatment efficacy by biomarker-based principal strata, which is useful for selecting biomarker study endpoints for evaluating treatments in new trials, exploring biological mechanisms of clinical treatment efficacy, and studying mediators of clinical treatment efficacy. However, in trials where participants may enter the study with prior exposure therefore with variable baseline biomarker values, clinical treatment efficacy may depend jointly on a biomarker measured at baseline and measured at a fixed time after vaccination. Therefore, it is of interest to conduct a bivariate effect modification analysis by biomarker-based principal strata and baseline biomarker values. Previous methods allow this assessment if participants who have the biomarker measured at the the fixed time point post randomization would also have the biomarker measured at baseline. However, additional complications in study design could happen in practice. For example, in the Dengue correlates study, baseline biomarker values were only available from a fraction of participants who have biomarkers measured post-randomization. How to conduct the bivariate effect modification analysis in these studies remains an open research question. In this article, we propose an estimated likelihood method to utilize the sub-sampled baseline biomarker in the effect modification analysis and illustrate our method with datasets from two dengue phase 3 vaccine efficacy trials., Comment: Manuscript is planned to submit November, 2017

The vaccination of one person may prevent another from becoming infected, either because the vaccine may prevent the first person from acquiring the infection and thereby reduce the probability of transmission to the second, or because, if the first person is infected, the vaccine may impair the ability of the infectious agent to initiate new infections. The former mechanism is referred as a contagion effect and the latter is referred as an infectiousness effect. By applying a principal stratification approach, the conditional infectiousness effect has been defined, but the contagion effect is not defined using this approach. Recently, new definitions of unconditional infectiousness and contagion effects were provided by applying a mediation analysis approach. In addition, a simple relationship between conditional and unconditional infectiousness effects was found under a number of assumptions. These two infectiousness effects can be assessed by very simple estimation and sensitivity analysis methods under the assumptions. Nevertheless, such simple methods to assess the contagion effect have not been discussed. In this paper, we review the methods of assessing infectiousness effects, and apply them to the inference of the contagion effect. The methods provided here are illustrated with hypothetical vaccine trial data.

Clinical disease registries offer a rich collection of valuable patient information but also pose challenges that require special care and attention in statistical analyses. The goal of this paper is to propose a statistical framework that allows for estimating the effect of surgical insertion of a percutaneous endogastrostomy (PEG) tube for patients living with amyotrophic lateral sclerosis (ALS) using data from a clinical registry. Although all ALS patients are informed about PEG, only some patients agree to the procedure which, leads to the potential for selection bias. Assessing the effect of PEG is further complicated by the aggressively fatal disease, such that time to death competes directly with both the opportunity to receive PEG and clinical outcome measurements. Our proposed methodology handles the “censoring by death” phenomenon through principal stratification and selection bias for PEG treatment through generalized propensity scores. We develop a fully Bayesian modeling approach to estimate the survivor average causal effect (SACE) of PEG on BMI, a surrogate outcome measure of nutrition and quality of life. The use of propensity score methods within the principal stratification framework demonstrates a significant and positive effect of PEG treatment, particularly when time of treatment is included in the treatment definition.

Early childhood education research often compares a group of children who receive the intervention of interest to a group of children who receive care in a range of different care settings. In this paper, we estimate differential impacts of an early childhood intervention by alternative care type, using data from the Head Start Impact Study, a large-scale randomized evaluation. To do so, we utilize a Bayesian principal stratification framework to estimate separate impacts for two types of Compliers: those children who would otherwise be in other center-based care when assigned to control and those who would otherwise be in home-based care. We find strong, positive short-term effects of Head Start on receptive vocabulary for those Compliers who would otherwise be in home-based care. By contrast, we find no meaningful impact of Head Start on vocabulary for those Compliers who would otherwise be in other center-based care. Our findings suggest that alternative care type is a potentially important source of variation in early childhood education interventions.

BackgroundMeta-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.MethodThe 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.ResultsConsistent 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.ConclusionsOur 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.

Graphical models and the corresponding linear structural equation model can be used as a mathematical language for describing a causal-effect relationship. A common problem is argued about which covariates in the diagram should be used to estimate the causal effects. The primary aim of this paper is to show different covariates selection for estimating the causal effects and we evaluated the asymptotic variance of the estimated causal effects with different covariates selection. The comparison results show that the asymptotic variance with controlling C1 and Z is smaller than that with controlling C1 and C2. The simulation results give a promising examination of the proposed method.

Complex causal models, accompanied by causal analyses based on large correlation matrices, are more common in the social sciences than are simple causal models accompanied by a single correlation coefficient. The increased complexity of the former, relative to the latter, seems to carry with it an augmented scientific respectability or credibility. In contrast, the axioms of probability suggest an argument in the opposite direction.

In randomized trials, the most commonly reported method of effect estimation is intention-to-treat (ITT), and to a lesser extent the per-protocol. The ITT is preferred because it is an unbiased estimator of the effect of treatment assignment. However, if there is any non-adherence the ITT is a biased estimate of the treatment effect, defined as the contrast between the potential outcome if treated versus the potential outcome if not treated. The treatment effect is most relevant to patients. Principal stratification is a framework for estimating treatment effects that combines potential outcomes and latent adherence strata. It yields an unbiased estimator of the complier average causal effect (CACE) for a difference in means or proportions, in the setting of all-or-nothing adherence. This paper addresses estimation of the causal hazard ratio for the compliers in a setting of right censoring of a time-to-event. We propose a novel approach to operationalizing principal stratification using weights. We report the results of simulations that vary the amount of adherence and selection bias that show the hazard ratio estimators we propose have minimal bias compared to the ITT, and per-protocol estimators. We demonstrate the approach using a population based randomized controlled trial of colorectal cancer screening subject to a high frequency of nonadherence in the screening arm.

The author presents a nonparametric approach to measuring stratification that highlights the distinction between stratification and inequality. Using pairwise comparison of ranks, the author develops an index of stratification that gauges the overall degree to which population subgroups occupy distinct strata with respect to a hierarchical outcome. This new index possesses a number of desirable properties that are not satisfied by existing measures of stratification. The overall index can be decomposed as a weighted average of pair-specific indices of stratification, which capture the extent of separation between any two particular groups. Besides, this index can be easily extended to measure conditional stratification through control of a third variable. In addition, the author builds a parallel between stratification and inequality in their measurement by developing a general formula of which the index of stratification and the Gini index of inequality can be considered as two special cases. Finally, this new approach is applied to depict the temporal trends of wage stratification by gender, race, and educational attainment over the past three decades in the United States.

Experimental evaluations are increasingly common in the U.S. educational policy-research context. Often, in investigations of multifaceted interventions, researchers and policymakers alike are interested in not only whether a given intervention impacted an outcome but also why. What features of the intervention led to the impacts observed, or what was the causal mechanism or pathway through which treatment assignment resulted in an improved outcome? Quantitative methods for modeling such mediational processes appropriately are an active area of methodological exploration. I contribute to this literature by discussing an approach that relies on the analytic framework of principal stratification. Approaches, such as regression analysis and instrumental variables estimation (a special case of principal stratification), rely on assumptions—sequential ignorability and an exclusion restriction, respectively—that may be too strong to be plausible in practical settings. Principal stratification provides ...

Principal stratification provides an approach to study the effect of an exposure on an outcome within strata defined by the effect of the exposure on some third post-treatment variable (Frangakis and Rubin, 2002). It has been used to give insight into randomized trials with non-compliance (Angrist et al., 1996), to settings where an outcome may be censored by death (Rubin, 2006), and to the study of the effects of a vaccine on post-infection outcomes (Hudgens and Halloran, 2006). There has been more recent interest in using principal stratification to study the extent to which the effect of an exposure on an outcome is mediated by an intermediate variable (e.g. Gallop et al., 2009). In this issue of the Journal of Research on Educational Effectiveness, Page (2012) provides evidence that the Career Academies program had substantial effect on subsequent earnings for those for whom the program would change exposure to the world-of-work, but not for those for whom it would not change exposure to the world-of-work. What I would like to discuss in this commentary is the question of what we learn about mediation from these principal stratification analyses. I believe that principal stratification analyses in the context of an intermediate can be helpful or illuminating. I also think the results of Page (2012) concerning the Career Academies are of substantive interest. However, as I will argue in this commentary, I do not think that the causal effects defined by the principal stratification framework correspond to mediation. Mediation as traditionally understood corresponds to the phenomenon in which a treatment or exposure changes a mediator and that change in the mediator goes on to change the outcome. This is not what the principal stratification framework captures.

Summary The definitive evaluation of treatment to prevent a chronic disease with low incidence in middle age, such as cancer or cardiovascular disease, requires a trial with a large sample size of perhaps 20000 or more. To help to decide whether to implement a large true end point trial, investigators first typically estimate the effect of treatment on a surrogate end point in a trial with a greatly reduced sample size of perhaps 200 subjects. If investigators reject the null hypothesis of no treatment effect in the surrogate end point trial they implicitly assume that they would probably correctly reject the null hypothesis of no treatment effect for the true end point. Surrogate end point trials are generally designed with adequate power to detect an effect of treatment on the surrogate end point. However, we show that a small surrogate end point trial is more likely than a large surrogate end point trial to give a misleading conclusion about the beneficial effect of treatment on the true end point, which can lead to a faulty (and costly) decision about implementing a large true end point prevention trial. If a small surrogate end point trial rejects the null hypothesis of no treatment effect, an intermediate-sized surrogate end point trial could be a useful next step in the decision-making process for launching a large true end point prevention trial.

Purpose We studied the application of the generalized structural mean model (GSMM) of instrumental variable (IV) methods in estimating treatment odds ratios (ORs) for binary outcomes in pharmacoepidemiologic studies and evaluated the bias of GSMM compared to other IV methods. Methods Because of the bias of standard IV methods, including two-stage predictor substitution (2SPS) and two-stage residual inclusion (2SRI) with binary outcomes, we implemented another IV approach based on the GSMM of Vansteelandt and Goetghebeur. We performed simulations under the principal stratification setting and evaluated whether GSMM provides approximately unbiased estimates of the causal OR and compared its bias and mean squared error to that of 2SPS and 2SRI. We then applied different IV methods to a study comparing bezafibrate versus other fibrates on the risk of diabetes. Results Our simulations showed that unlike the standard logistic, 2SPS, and 2SRI procedures, our implementation of GSMM provides an approximately unbiased estimate of the causal OR even under unmeasured confounding. However, for the effect of bezafibrate versus other fibrates on the risk of diabetes, the GSMM and two-stage approaches yielded similarly attenuated and statistically non-significant OR estimates. The attenuation of the OR by the two-stage and GSMM IV approaches suggests unmeasured confounding, although violations of the IV assumptions or differences in the parameters estimated could be playing a role. Conclusion The GSMM IV approach provides approximately unbiased adjustment for unmeasured confounding on binary outcomes when a valid IV is available. Copyright © 2012 John Wiley & Sons, Ltd.

Typically, in the practice of causal inference from observational studies, a parametric model is assumed for the joint population density of potential outcomes and treatment assignments, and possibly this is accompanied by the assumption of no hidden bias. However, both assumptions are questionable for real data, the accuracy of causal inference is compromised when the data violates either assumption, and the parametric assumption precludes capturing a more general range of density shapes (e.g., heavier tail behavior and possible multi-modalities). We introduce a flexible, Bayesian nonparametric causal model to provide more accurate causal inferences. The model makes use of a stick-breaking prior, which has the flexibility to capture any multi-modalities, skewness and heavier tail behavior in this joint population density, while accounting for hidden bias. We prove the asymptotic consistency of the posterior distribution of the model, and illustrate our causal model through the analysis of small and large observational data sets.

It is frequently of interest to estimate the intervention effect that adjusts for post-randomization variables in clinical trials. In the recently completed HPTN 035 trial, there is differential condom use between the three microbicide gel arms and the No Gel control arm, so that intention to treat (ITT) analyses only assess the net treatment effect that includes the indirect treatment effect mediated through differential condom use. Various statistical methods in causal inference have been developed to adjust for post-randomization variables. We extend the principal stratification framework to time-varying behavioral variables in HIV prevention trials with a time-to-event endpoint, using a partially hidden Markov model (pHMM). We formulate the causal estimand of interest, establish assumptions that enable identifiability of the causal parameters, and develop maximum likelihood methods for estimation. Application of our model on the HPTN 035 trial reveals an interesting pattern of prevention effectiveness among different condom-use principal strata.

A major goal of vaccine development is the identification of immune responses that are responsible for vaccine efficacy. In theory, modest vaccines could be successfully improved by increasing such immune responses. And for a vaccine with a great benefit in one population, inducing such immune response in a different population could help one conclude the vaccine would have great benefit there. Such identification is tricky because the immune response to vaccination can only be measured in the vaccine group and thus immune responses might only be identifying individuals with a constitutional ability to remain uninfected, rather than being causal. Define the vaccine induced immune response as X(1). The value X(1) is a potential outcome; it is measured directly in vaccinees but unobserved in the placebo group. Our goal is to regress outcome on X(1) separately in the vaccine and placebo groups and to see if the vaccine effect varies with X(1). Put another way, our goal is to see if there is a vaccine by X(1) interaction. Regression of outcome on X(1) is easy to do in the vaccine group, but difficult in the placebo group as X(1) is not observed. In this paper we derive bounds on the regression curve in the placebo group. For a continuous endpoint these bounds can be unhelpful, or can help modestly temper our enthusiasm for a role of X(1) on the vaccine effect. For binary outcomes with 100% placebo infection the bound is very tight but unhelpful as 100% infection precludes identification of any covariate with a differential effect on placebo infection. We apply these methods to experiments of anthrax vaccine in rabbits with survival to challenge as the outcome and demonstrate how to extrapolate the model to humans.

In medical studies, there are many situations where the final outcomes are truncated by death, in which patients die before outcomes of interest are measured. In this article we consider identifiability and estimation of causal effects by principal stratification when some outcomes are truncated by death. Previous studies mostly focused on large sample bounds, Bayesian analysis, sensitivity analysis. In this article, we propose a new method for identifying the causal parameter of interest under a nonparametric and semiparametric model. We show that the causal parameter of interest is identifiable under some regularity assumptions and the assumption that there exists a pretreatment covariate whose conditional distributions among two principal strata are not the same, but our approach does not need the assumption of a mixture normal distribution for outcomes as required by Zhang, Rubin, and Mealli (2009). Hence, the proposed method is applicable not only to a continuous outcome but also to a binary outcome....

Principal strata are defined by the potential values of a post-treatment variable, and a principal effect is a causal effect within a principal stratum. Identifying the principal effect within every principal stratum is quite challenging. In this paper, we propose an approach for identifying principal effects on a binary outcome via a pre-treatment covariate. We prove the identifiability with single post-treatment intervention under the monotonicity assumption. Furthermore, we discuss the local identifiability with multicomponent intervention. Simulations are performed to evaluate our approach. We also apply it to a real data set from the Improving Mood-Promoting Access to Collaborate Treatment program.

In their initial work, Judd and Kenny (1981) focused on a restricted case: (a) a randomized experiment, with (b) linear relationships between the treatment condition and the measured intervening variable (mediator) and between the mediator and the measured response variable of interest, (c) temporal precedence of the mediator before the outcome, and (d) complete transmission of the effect of the treatment on the outcome through the mediator. Baron and Kenny (1986) greatly generalized the analysis by considering cases in which all four of these restrictions were relaxed. Indeed, the antecedent variable was no longer stipulated to be a treatment but could also be a nonmanipulated variable. This generalization broadened the range of questions that could be addressed, but often came at a cost of weakening the basis for causal inference of meditational effects. Judd and Kenny and Baron and Kenny were careful to specify the assumptions of meditational analysis: the meditational model and the functional form of the relationship between variables are correctly specified, there is no measurement error in the mediator, and there is no reciprocal causation between the mediator and outcome. Judd and Kenny also included disclaimers about causal inference: “We should recognize a meditational analysis for what it really is: A correlational analysis” (p. 208). Rosenbaum (1984) described further complexities that arise from attempting to adjust estimates of effects for the influence of variables such as mediators that are measured after treatment. Baron and Kenny's (1986) implicit weakening of the conditions that are needed for testing mediation has been followed by a proliferation of meditational analyses in cross-sectional studies with nonmanipulated antecedent variables. This development has raised the question of what researchers can learn about mediation from cross-sectional versus stronger longitudinal designs. Cole and Maxwell (Cole & Maxwell, 2003; Maxwell & Cole, 2007) began addressing this question in a series of important papers. Focusing primarily on commonly used panel designs with equally spaced measurement waves and autoregressive longitudinal models, they showed that the conditions under which longitudinal and cross-sectional estimates of meditational effects will be identical are limited to rare special cases. The target article by Maxwell, Cole, and Mitchell (2011) extended their previous work to the more common case in which direct and indirect (meditational) effects of antecedent variables may be present on consequent variables. They found that estimates of meditational effects from cross-sectional designs may be seriously biased. Cases exist in which the cross-sectional design erroneously detects a meditational effect when it does not in fact exist, and conversely also fails to detect a meditational effect when it does exist. Such results raise grave questions about the routine use of cross-sectional meditational analyses. Three different perspectives on Maxwell et al.'s (2011) findings are raised in the commentaries. Reichardt (2011) builds on earlier work on causal lags (e.g., Gollob & Reichardt, 1987), showing that estimates of meditational effects can easily be incorrect in longitudinal autoregressive models, even when all of the assumptions of autoregressive models are met. Three waves of data may not be sufficient to control for mediation that occurs at intermediate points not represented in the measurement design. Shrout (2011) applies insights from his recent work on causal inference in psychopathology (Shrout, Keyes, & Ornstein, 2011). Among these are examination of the assumptions of autoregressive models, consideration of the possibility that alternative longitudinal models may map more closely onto many processes hypothesized by researchers, and consideration of differences between within- and between-individual models that may lead to different effects. Careful consideration of the science, the hypothesized nature of the processes, coupled with the inclusion of unique design elements, can limit the possible set of potential influences and strengthen causal inferences, even potentially in cross-sectional designs. Finally, Imai, Jo, and Stuart (2011) build on their work applying Rubin's (2005) potential outcomes model to causal inference problems (e.g., Imai, Keele, & Tingley, 2010). They show conditions necessary to identify indirect (meditational) and direct effects for the special case of a binary antecedent variable producing changes in a binary mediator, which in turn produces changes in a binary outcome. Their analysis highlights the strong and untestable assumptions needed in this framework to clearly identify unambiguous causal meditational effects, even in the simplified case of binary variables. They provide an analysis of Maxwell et al.'s (2011) longitudinal approach and offer alternative design and analysis approaches using crossover designs and principal stratification using propensity scores. This collection of articles highlights that researchers need to go beyond simply proposing and testing meditational models in a routine manner. As with any causal inference problem, careful attention to theory, previous research, research design, and measurement is needed. The application of Rubin's (2005) potential outcomes model can help clarify the assumptions and the sets of conditions needed for clear causal inference. Appropriate conditioning strategies can greatly strengthen the causal inferences that can be reached. Although not explicitly developed in this set of papers, the application of Campbell's perspective (Shadish, Cook, & Campbell, 2002) can help identify and prioritize plausible threats to internal validity; design elements can be added to the basic design the help rule out those threats and strengthen causal inference (see West & Thoemmes, 2010). The simple meditational model presents challenges both to Rubin's potential outcomes model and Campbell's perspective. These challenges only increase as more complex mediational models that researchers would like to consider are proposed. The four articles in this section provide some important steps toward meeting those challenges.

Summary Many studies involving causal questions are often concerned with understanding the causal pathways by which a treatment affects an outcome. Thus, the concept of ‘direct’versus‘indirect’ effects comes into play. We tackle the problem of disentangling direct and indirect effects by investigating new augmented experimental designs, where the treatment is randomized, and the mediating variable is not forced, but only randomly encouraged. There are two key features of our framework: we adopt a principal stratification approach, and we mainly focus on principal strata effects, avoiding involving a priori counterfactual outcomes. Using non-parametric identification strategies, we provide a set of assumptions, which allow us to identify partially the causal estimands of interest: the principal strata direct effects. Some examples are shown to illustrate our design and causal estimands of interest. Large sample bounds for the principal strata average direct effects are provided, and a simple hypothetical example is used to show how our augmented design can be implemented and how the bounds can be calculated. Finally our augmented design is compared and contrasted with a standard randomized design.

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 article, the authors propose a practical way of bounding and sensitivity analysis, where multiple identifying assumptions are combined to construct tighter common bounds. In particular, the authors focus on the use of competing identifying assumptions that impose different restrictions on the same nonidentified 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.

In many clinical trials, compliance with assigned treatment could be measured on a continuous scale (e.g., the proportion of assigned treatment actually taken). In general, inference about principal causal effects can be challenging, particularly when there are two active treatments; the problem is exacerbated for continuous measures of compliance. We address this issue by first proposing a structural model for the principal effects. We then specify compliance models within each arm of the study. These marginal models are identifiable. The joint distribution of the observed and counterfactual compliance variables is assumed to follow a Gaussian copula model, which links the two marginal models and includes a dependence parameter that cannot be identified. This dependence parameter can be varied as part of a sensitivity analysis. We illustrate the methodology with an analysis of data from a smoking cessation trial. As part of the analysis, we estimate causal effects at particular levels of the compliance variables and within subpopulations that have similar compliance behavior.

Within the principal stratification framework for causal inference, modeling partial compliance is challenging because the continuous nature of the principal strata raises subtle specification issues. In this context, we propose an approach based on the assumption that the joint distribution of the degree of compliance to the treatment and the degree of compliance to the control follows a Plackett copula, so that their association is modeled in a flexible way through a single parameter. Moreover, given the two compliances, the distribution of the outcomes is parameterized in a flexible way through a regression model which may include interaction and quadratic terms and may also be heteroscedastic. In order to estimate the parameters of the resulting model, and then the causal effect of the treatment, we adopt a maximum likelihood approach via the EM algorithm. In applying this approach, the marginal distributions of the two compliances are estimated by their empirical distribution functions, so that no co...

Recently a new definition of surrogate endpoint, the "principal surrogate," was proposed based on causal associations between treatment effects on the biomarker and on the clinical endpoint. Despite its appealing interpretation, limited research has been conducted to evaluate principal surrogates, and existing methods focus on risk models that consider a single biomarker. How to compare principal surrogate value of biomarkers or general risk models that consider multiple biomarkers remains an open research question. We propose to characterize a marker or risk model's principal surrogate value based on the distribution of risk difference between interventions. In addition, we propose a novel summary measure (the standardized total gain) that can be used to compare markers and to assess the incremental value of a new marker. We develop a semiparametric estimated-likelihood method to estimate the joint surrogate value of multiple biomarkers. This method accommodates two-phase sampling of biomarkers and is more widely applicable than existing nonparametric methods by incorporating continuous baseline covariates to predict the biomarker(s), and is more robust than existing parametric methods by leaving the error distribution of markers unspecified. The methodology is illustrated using a simulated example set and a real data set in the context of HIV vaccine trials.

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.