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In many studies, the aim is to learn about the direct exposure effect, that is, the effect not mediated through an intermediate variable. For example, in circulation disease studies it may be of interest to assess whether a suitable level of physical activity can prevent disease, even if it fails to prevent obesity. It is well known that stratification on the intermediate may introduce a so-called posttreatment selection bias. To handle this problem, we use the framework of principal stratification ( Frangakis and Rubin, 2002 , Biometrics58, 21-29) to define a causally relevant estimand--the principal stratum direct effect (PSDE). The PSDE is not identified in our setting. We propose a method of sensitivity analysis that yields a range of plausible values for the causal estimand. We compare our work to similar methods proposed in the literature for handling the related problem of \"truncation by death.\"

In causal mediation analysis, the definitions of the natural direct and indirect effects involve potential outcomes that can never be observed, so-called a priori counterfactuals. This conceptual challenge translates into issues in identification, which requires strong and often unverifiable assumptions, including sequential ignorability. Alternatively, we can deal with post-treatment variables using the principal stratification framework, where causal effects are defined as comparisons of observable potential outcomes. We establish a novel bridge between mediation analysis and principal stratification, which helps to clarify and weaken the commonly used identifying assumptions for natural direct and indirect effects. Using principal stratification, we show how sequential ignorability extrapolates from observable potential outcomes to a priori counterfactuals, and propose alternative weaker principal ignorability-type assumptions. We illustrate the key concepts using a clinical trial.

In causal inference, principal stratification is a framework for dealing with a posttreatment intermediate variable between a treatment and an outcome. In this framework, the principal strata are defined by the joint potential values of the intermediate variable. Because the principal strata are not fully observable, the causal effects within them, also known as the principal causal effects, are not identifiable without additional assumptions. Several previous empirical studies leveraged auxiliary variables to improve the inference of principal causal effects. We establish a general theory for the identification and estimation of principal causal effects with auxiliary variables, which provides a solid foundation for statistical inference and more insights for model building in empirical research. In particular, we consider two commonly used assumptions for principal stratification problems: principal ignorability and the conditional independence between the auxiliary variable and the outcome given principal strata and covariates. Under each assumption, we give nonparametric and semiparametric identification results without modeling the outcome. When neither assumption is plausible, we propose a large class of flexible parametric and semiparametric models for identifying principal causal effects. Our theory not only establishes formal identification results of several models that have been used in previous empirical studies but also generalizes them to allow for different types of outcomes and intermediate variables.

•Many statistical approaches have been applied to vaccine efficacy (VE) trials to evaluate CoPs.•Four distinct statistical frameworks for evaluating CoPs in VE trials are summarized.•Criteria for a ‘good CoP’ according to the four frameworks are suggested.•This article constitutes a reference resource for promoting harmonization of CoP evaluation. A central goal of vaccine research is to characterize and validate immune correlates of protection (CoPs). In addition to helping elucidate immunological mechanisms, a CoP can serve as a valid surrogate endpoint for an infectious disease clinical outcome and thus qualifies as a primary endpoint for vaccine authorization or approval without requiring resource-intensive randomized, controlled phase 3 trials. Yet, it is challenging to persuasively validate a CoP, because a prognostic immune marker can fail as a reliable basis for predicting/inferring the level of vaccine efficacy against a clinical outcome, and because the statistical analysis of phase 3 trials only has limited capacity to disentangle association from cause. Moreover, the multitude of statistical methods garnered for CoP evaluation in phase 3 trials renders the comparison, interpretation, and synthesis of CoP results challenging. Toward promoting broader harmonization and standardization of CoP evaluation, this article summarizes four complementary statistical frameworks for evaluating CoPs in a phase 3 trial, focusing on the frameworks’ distinct scientific objectives as measured and communicated by distinct causal vaccine efficacy parameters. Advantages and disadvantages of the frameworks are considered, dependent on phase 3 trial context, and perspectives are offered on how the frameworks can be applied and their results synthesized.

There is interest in learning about the causal effects of modern contraceptive use on empowerment outcomes. Data on this question often come from family planning (FP) programs that increase access to FP and facilitate contraceptive use among some women, rather than directly assigning use. Women whose contraceptive behavior changes because of these programs (“compliers”) may differ from target populations in ways that alter the consequences of contraceptive use for empowerment outcomes.We propose a two-step approach. First, we use principal stratification and Bayesian Additive Regression Trees (BART) to estimate the effect of modern contraceptive use among compliers in the study population, treating the FP program as an instrument rather than as the treatment of interest. Second, we generalize these complier-specific effects to a broader population by averaging conditional effects over the covariate distribution in the target population, with uncertainty in that distribution quantified via a Bayesian bootstrap applied to external complex survey data.We examine performance in simulation designs previously used to evaluate IV estimators. We then apply the approach to employment among urban women in Nigeria and Senegal, finding strong and heterogeneous effects of contraceptive use. Sensitivity analyses suggest robustness to violations of assumptions for internal and external validity.

In many medical studies, an ultimate failure event, such as death, is likely to be affected by the occurrence and timing of other intermediate clinical events. Both event times are subject to censoring by loss-to-follow-up, but the nonterminal event may be further censored by the occurrence of the primary outcome, but not vice versa. To study the effect of an intervention on both events, the intermediate event may be viewed as a mediator. However, the conventional definitions of direct and indirect effects do not apply, because of the semi-competing risks data structure. We define three principal strata based on whether the potential intermediate event occurs before the potential failure event. This allows us to properly define direct and indirect effects in one stratum, and define total effects for all strata. We discuss the identification conditions for the stratum-specific effects, and propose a semiparametric estimator based on a multivariate logistic stratum membership model and within-stratum proportional hazards models for the event times. By treating the unobserved stratum membership as a latent variable, we propose an expectation-maximization algorithm for the computation. We study the asymptotic properties of the estimators using modern empirical process theory and examine the performance of the estimators in numerical studies.

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 Compilers: 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 Compilers who would otherwise be in home-based care. By contrast, we find no meaningful impact of Head Start on vocabulary for those Compilers 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.

It is common in medical studies that the outcome of interest is truncated by death, meaning that a subject has died before the outcome could be measured. In this case, restricted analysis among survivors may be subject to selection bias. Hence, it is of interest to estimate the survivor average causal effect, defined as the average causal effect among the subgroup consisting of subjects who would survive under either exposure. In this paper, we consider the identification and estimation problems of the survivor average causal effect. We propose to use a substitution variable in place of the latent membership in the always-survivor group. The identification conditions required for a substitution variable are conceptually similar to conditions for a conditional instrumental variable, and may apply to both randomized and observational studies. We show that the survivor average causal effect is identifiable with use of such a substitution variable, and propose novel model parameterizations for estimation of the survivor average causal effect under our identification assumptions. Our approaches are illustrated via simulation studies and a data analysis.

Causal effects are defined as comparisons of potential outcomes under different treatments on a common set of units. Observed values of the potential outcomes are revealed by the assignment mechanism-a probabilistic model for the treatment each unit receives as a function of covariates and potential outcomes. Fisher made tremendous contributions to causal inference through his work on the design of randomized experiments, but the potential outcomes perspective applies to other complex experiments and nonrandomized studies as well. As noted by Kempthorne in his 1976 discussion of Savage's Fisher lecture, Fisher never bridged his work on experimental design and his work on parametric modeling, a bridge that appears nearly automatic with an appropriate view of the potential outcomes framework, where the potential outcomes and covariates are given a Bayesian distribution to complete the model specification. Also, this framework crisply separates scientific inference for causal effects and decisions based on such inference, a distinction evident in Fisher's discussion of tests of significance versus tests in an accept/reject framework. But Fisher never used the potential outcomes framework, originally proposed by Neyman in the context of randomized experiments, and as a result he provided generally flawed advice concerning the use of the analysis of covariance to adjust for posttreatment concomitants in randomized trials.

Despite its monogenic origin, Cerebral Autosomal Dominant Arteriopathy with Subcortical Infarcts and Leukoencephalopathy exhibits marked variability in clinical expression and severity. Variants in the NOTCH3 gene, within epidermal growth factor-like repeat domains 1-6 or 7-34, are known to influence disease onset, but their impact on long-term progression remains unclear. This study assesses mutation location effects on post-first stroke clinical trajectories. Clinical data from a large cohort were analyzed (Patients EGFR 1-6 mutation group n = 210 and 7-34 mutation group n = 116) with target emulated trial framework. To study the impact of mutation location on stroke recurrence, disability (modified Rankin score ≥ 3) and mortality, following a first stroke event. Propensity score matching was used to balance covariates between mutation location groups and principal stratification to consider truncation by death. Events occurrence differences were compared using Restricted Mean Survival Time at 2, 5, 10 and 15 years. At first stroke, patients with mutation in domains 1-6 were younger than those in the 7-34 mutation group (49.51 ± 7.4 vs. 55.00 ± 7.4 years). Ten years after first stroke event, mortality occurred slightly later in the 7-34 group (9.63 [9.33-9.92] vs. 9.11 [8.71-9.52] years, p = 0.04), also at 15 years (14.0 [13.42-14.63] vs. 12.4 [11.62-13.24] years; p = 0.002). Second stroke occurrence did not differ between groups. Time beyond modified Rankin of 3 slightly differed between groups at 5 and 10 years, with a difference of 0.22 [0.01-0.044] and 0.72 [0.14-1.30] year respectively (p = 0.044 and 0.017). Although NOTCH3 variants location influences the delay to the first stroke, it has no or little impact on the recurrence of stroke, risk of disability and death after the first stroke manifestation.