Toward Causal Inference With Interference

Mediation analysis with intermediate confounding: structural equation modeling viewed through the causal inference lens

The study of mediation has a long tradition in the social sciences and a relatively more recent one in epidemiology. The first school is linked to path analysis and structural equation models (SEMs), while the second is related mostly to methods developed within the potential outcomes approach to causal inference. By giving model-free definitions of direct and indirect effects and clear assumptions for their identification, the latter school has formalized notions intuitively developed in the former and has greatly increased the flexibility of the models involved. However, through its predominant focus on nonparametric identification, the causal inference approach to effect decomposition via natural effects is limited to settings that exclude intermediate confounders. Such confounders are naturally dealt with (albeit with the caveats of informality and modeling inflexibility) in the SEM framework. Therefore, it seems pertinent to revisit SEMs with intermediate confounders, armed with the formal definitions and (parametric) identification assumptions from causal inference. Here we investigate: 1) how identification assumptions affect the specification of SEMs, 2) whether the more restrictive SEM assumptions can be relaxed, and 3) whether existing sensitivity analyses can be extended to this setting. Data from the Avon Longitudinal Study of Parents and Children (1990-2005) are used for illustration.

Point: clarifying policy evidence with potential-outcomes thinking--beyond exposure-response estimation in air pollution epidemiology

The regulatory environment surrounding policies to control air pollution warrants a new type of epidemiologic evidence. Whereas air pollution epidemiology has typically informed policies with estimates of exposure-response relationships between pollution and health outcomes, these estimates alone cannot support current debates surrounding the actual health effects of air quality regulations. We argue that directly evaluating specific control strategies is distinct from estimating exposure-response relationships and that increased emphasis on estimating effects of well-defined regulatory interventions would enhance the evidence that supports policy decisions. Appealing to similar calls for accountability assessment of whether regulatory actions impact health outcomes, we aim to sharpen the analytic distinctions between studies that directly evaluate policies and those that estimate exposure-response relationships, with particular focus on perspectives for causal inference. Our goal is not to review specific methodologies or studies, nor is it to extoll the advantages of "causal" versus "associational" evidence. Rather, we argue that potential-outcomes perspectives can elevate current policy debates with more direct evidence of the extent to which complex regulatory interventions affect health. Augmenting the existing body of exposure-response estimates with rigorous evidence of the causal effects of well-defined actions will ensure that the highest-level epidemiologic evidence continues to support regulatory policies.

Interference and Sensitivity Analysis

Causal inference with interference is a rapidly growing area. The literature has begun to relax the "no-interference" assumption that the treatment received by one individual does not affect the outcomes of other individuals. In this paper we briefly review the literature on causal inference in the presence of interference when treatments have been randomized. We then consider settings in which causal effects in the presence of interference are not identified, either because randomization alone does not suffice for identification, or because treatment is not randomized and there may be unmeasured confounders of the treatment-outcome relationship. We develop sensitivity analysis techniques for these settings. We describe several sensitivity analysis techniques for the infectiousness effect which, in a vaccine trial, captures the effect of the vaccine of one person on protecting a second person from infection even if the first is infected. We also develop two sensitivity analysis techniques for causal effects in the presence of unmeasured confounding which generalize analogous techniques when interference is absent. These two techniques for unmeasured confounding are compared and contrasted.

Hard, harder, hardest: principal stratification, statistical identifiability, and the inherent difficulty of finding surrogate endpoints

In many areas of clinical investigation there is great interest in identifying and validating surrogate endpoints, biomarkers that can be measured a relatively short time after a treatment has been administered and that can reliably predict the effect of treatment on the clinical outcome of interest. However, despite dramatic advances in the ability to measure biomarkers, the recent history of clinical research is littered with failed surrogates. In this paper, we present a statistical perspective on why identifying surrogate endpoints is so difficult. We view the problem from the framework of causal inference, with a particular focus on the technique of principal stratification (PS), an approach which is appealing because the resulting estimands are not biased by unmeasured confounding. In many settings, PS estimands are not statistically identifiable and their degree of non-identifiability can be thought of as representing the statistical difficulty of assessing the surrogate value of a biomarker. In this work, we examine the identifiability issue and present key simplifying assumptions and enhanced study designs that enable the partial or full identification of PS estimands. We also present example situations where these assumptions and designs may or may not be feasible, providing insight into the problem characteristics which make the statistical evaluation of surrogate endpoints so challenging.

Estimating the per-exposure effect of infectious disease interventions

The average effect of an infectious disease intervention (eg, a vaccine) varies across populations with different degrees of exposure to the pathogen. As a result, many investigators favor a per-exposure effect measure that is considered independent of the population level of exposure and that can be used in simulations to estimate the total disease burden averted by an intervention across different populations. However, while per-exposure effects are frequently estimated, the quantity of interest is often poorly defined, and assumptions in its calculation are typically left implicit. In this article, we build upon work by Halloran and Struchiner (Epidemiology. 1995;6:142-151) to develop a formal definition of the per-exposure effect and discuss conditions necessary for its unbiased estimation. With greater care paid to the parameterization of transmission models, their results can be better understood and can thereby be of greater value to decision-makers.

Instrumental variable methods for causal inference

A goal of many health studies is to determine the causal effect of a treatment or intervention on health outcomes. Often, it is not ethically or practically possible to conduct a perfectly randomized experiment, and instead, an observational study must be used. A major challenge to the validity of observational studies is the possibility of unmeasured confounding (i.e., unmeasured ways in which the treatment and control groups differ before treatment administration, which also affect the outcome). Instrumental variables analysis is a method for controlling for unmeasured confounding. This type of analysis requires the measurement of a valid instrumental variable, which is a variable that (i) is independent of the unmeasured confounding; (ii) affects the treatment; and (iii) affects the outcome only indirectly through its effect on the treatment. This tutorial discusses the types of causal effects that can be estimated by instrumental variables analysis; the assumptions needed for instrumental variables analysis to provide valid estimates of causal effects and sensitivity analysis for those assumptions; methods of estimation of causal effects using instrumental variables; and sources of instrumental variables in health studies.

Assessing effects of cholera vaccination in the presence of interference

Interference occurs when the treatment of one person affects the outcome of another. For example, in infectious diseases, whether one individual is vaccinated may affect whether another individual becomes infected or develops disease. Quantifying such indirect (or spillover) effects of vaccination could have important public health or policy implications. In this article we use recently developed inverse-probability weighted (IPW) estimators of treatment effects in the presence of interference to analyze an individually-randomized, placebo-controlled trial of cholera vaccination that targeted 121,982 individuals in Matlab, Bangladesh. Because these IPW estimators have not been employed previously, a simulation study was also conducted to assess the empirical behavior of the estimators in settings similar to the cholera vaccine trial. Simulation study results demonstrate the IPW estimators can yield unbiased estimates of the direct, indirect, total, and overall effects of vaccination when there is interference provided the untestable no unmeasured confounders assumption holds and the group-level propensity score model is correctly specified. Application of the IPW estimators to the cholera vaccine trial indicates the presence of interference. For example, the IPW estimates suggest on average 5.29 fewer cases of cholera per 1000 person-years (95% confidence interval 2.61, 7.96) will occur among unvaccinated individuals within neighborhoods with 60% vaccine coverage compared to neighborhoods with 32% coverage. Our analysis also demonstrates how not accounting for interference can render misleading conclusions about the public health utility of vaccination.

Estimation of treatment efficacy with complier average causal effects (CACE) in a randomized stepped wedge trial

Complier average causal effects (CACE) estimate the impact of an intervention among treatment compliers in randomized trials. Methods used to estimate CACE have been outlined for parallel-arm trials (e.g., using an instrumental variables (IV) estimator) but not for other randomized study designs. Here, we propose a method for estimating CACE in randomized stepped wedge trials, where experimental units cross over from control conditions to intervention conditions in a randomized sequence. We illustrate the approach with a cluster-randomized drinking water trial conducted in rural Mexico from 2009 to 2011. Additionally, we evaluated the plausibility of assumptions required to estimate CACE using the IV approach, which are testable in stepped wedge trials but not in parallel-arm trials. We observed small increases in the magnitude of CACE risk differences compared with intention-to-treat estimates for drinking water contamination (risk difference (RD) = -22% (95% confidence interval (CI): -33, -11) vs. RD = -19% (95% CI: -26, -12)) and diarrhea (RD = -0.8% (95% CI: -2.1, 0.4) vs. RD = -0.1% (95% CI: -1.1, 0.9)). Assumptions required for IV analysis were probably violated. Stepped wedge trials allow investigators to estimate CACE with an approach that avoids the stronger assumptions required for CACE estimation in parallel-arm trials. Inclusion of CACE estimates in stepped wedge trials with imperfect compliance could enhance reporting and interpretation of the results of such trials.

Estimating the effect of cumulative occupational asbestos exposure on time to lung cancer mortality: using structural nested failure-time models to account for healthy-worker survivor bias

BACKGROUND

Previous estimates of the effect of occupational asbestos on lung cancer mortality have been obtained by using methods that are subject to the healthy-worker survivor bias. G-estimation of a structural nested model provides consistent exposure effect estimates under this bias.

METHODS

We estimated the effect of cumulative asbestos exposure on lung cancer mortality in a cohort comprising 2564 textile factory workers who were followed from January 1940 to December 2001.

RESULTS

At entry, median age was 23 years, with 42% of the cohort being women and 20% nonwhite. During the follow-up period, 15% of person-years were classified as occurring while employed and 13% as occupationally exposed to asbestos. For a 100 fiber-year/ml increase in cumulative asbestos, a Weibull model adjusting for sex, race, birth year, baseline exposure, and age at study entry yielded a survival time ratio of 0.88 (95% confidence interval = 0.83 to 0.93). Further adjustment for work status yielded no practical change. The corresponding survival time ratio obtained using g-estimation of a structural nested model was 0.57 (0.33 to 0.96).

CONCLUSIONS

Accounting for the healthy-worker survivor bias resulted in a 35% stronger effect estimate. However, this estimate was considerably less precise. When healthy-worker survivor bias is suspected, methods that account for it should be used.

Causal Inference for a Population of Causally Connected Units

Suppose that we observe a population of causally connected units. On each unit at each time-point on a grid we observe a set of other units the unit is potentially connected with, and a unit-specific longitudinal data structure consisting of baseline and time-dependent covariates, a time-dependent treatment, and a final outcome of interest. The target quantity of interest is defined as the mean outcome for this group of units if the exposures of the units would be probabilistically assigned according to a known specified mechanism, where the latter is called a stochastic intervention. Causal effects of interest are defined as contrasts of the mean of the unit-specific outcomes under different stochastic interventions one wishes to evaluate. This covers a large range of estimation problems from independent units, independent clusters of units, and a single cluster of units in which each unit has a limited number of connections to other units. The allowed dependence includes treatment allocation in response to data on multiple units and so called causal interference as special cases. We present a few motivating classes of examples, propose a structural causal model, define the desired causal quantities, address the identification of these quantities from the observed data, and define maximum likelihood based estimators based on cross-validation. In particular, we present maximum likelihood based super-learning for this network data. Nonetheless, such smoothed/regularized maximum likelihood estimators are not targeted and will thereby be overly bias w.r.t. the target parameter, and, as a consequence, generally not result in asymptotically normally distributed estimators of the statistical target parameter. To formally develop estimation theory, we focus on the simpler case in which the longitudinal data structure is a point-treatment data structure. We formulate a novel targeted maximum likelihood estimator of this estimand and show that the double robustness of the efficient influence curve implies that the bias of the targeted minimum loss-based estimation (TMLE) will be a second-order term involving squared differences of two nuisance parameters. In particular, the TMLE will be consistent if either one of these nuisance parameters is consistently estimated. Due to the causal dependencies between units, the data set may correspond with the realization of a single experiment, so that establishing a (e.g. normal) limit distribution for the targeted maximum likelihood estimators, and corresponding statistical inference, is a challenging topic. We prove two formal theorems establishing the asymptotic normality using advances in weak-convergence theory. We conclude with a discussion and refer to an accompanying technical report for extensions to general longitudinal data structures.

Large sample randomization inference of causal effects in the presence of interference

Recently, increasing attention has focused on making causal inference when interference is possible. In the presence of interference, treatment may have several types of effects. In this paper, we consider inference about such effects when the population consists of groups of individuals where interference is possible within groups but not between groups. A two stage randomization design is assumed where in the first stage groups are randomized to different treatment allocation strategies and in the second stage individuals are randomized to treatment or control conditional on the strategy assigned to their group in the first stage. For this design, the asymptotic distributions of estimators of the causal effects are derived when either the number of individuals per group or the number of groups grows large. Under certain homogeneity assumptions, the asymptotic distributions provide justification for Wald-type confidence intervals (CIs) and tests. Empirical results demonstrate the Wald CIs have good coverage in finite samples and are narrower than CIs based on either the Chebyshev or Hoeffding inequalities provided the number of groups is not too small. The methods are illustrated by two examples which consider the effects of cholera vaccination and an intervention to encourage voting.

Causal inference in occupational epidemiology: accounting for the healthy worker effect by using structural nested models

In a recent issue of the Journal, Kirkeleit et al. (Am J Epidemiol. 2013;177(11):1218-1224) provided empirical evidence for the potential of the healthy worker effect in a large cohort of Norwegian workers across a range of occupations. In this commentary, we provide some historical context, define the healthy worker effect by using causal diagrams, and use simulated data to illustrate how structural nested models can be used to estimate exposure effects while accounting for the healthy worker survivor effect in 4 simple steps. We provide technical details and annotated SAS software (SAS Institute, Inc., Cary, North Carolina) code corresponding to the example analysis in the Web Appendices, available at http://aje.oxfordjournals.org/.

Propensity score weighting with multilevel data

Propensity score methods are being increasingly used as a less parametric alternative to traditional regression to balance observed differences across groups in both descriptive and causal comparisons. Data collected in many disciplines often have analytically relevant multilevel or clustered structure. The propensity score, however, was developed and has been used primarily with unstructured data. We present and compare several propensity-score-weighted estimators for clustered data, including marginal, cluster-weighted, and doubly robust estimators. Using both analytical derivations and Monte Carlo simulations, we illustrate bias arising when the usual assumptions of propensity score analysis do not hold for multilevel data. We show that exploiting the multilevel structure, either parametrically or nonparametrically, in at least one stage of the propensity score analysis can greatly reduce these biases. We applied these methods to a study of racial disparities in breast cancer screening among beneficiaries of Medicare health plans.

Mediation and spillover effects in group-randomized trials: a case study of the 4Rs educational intervention

Peer influence and social interactions can give rise to spillover effects in which the exposure of one individual may affect outcomes of other individuals. Even if the intervention under study occurs at the group or cluster level as in group-randomized trials, spillover effects can occur when the mediator of interest is measured at a lower level than the treatment. Evaluators who choose groups rather than individuals as experimental units in a randomized trial often anticipate that the desirable changes in targeted social behaviors will be reinforced through interference among individuals in a group exposed to the same treatment. In an empirical evaluation of the effect of a school-wide intervention on reducing individual students' depressive symptoms, schools in matched pairs were randomly assigned to the 4Rs intervention or the control condition. Class quality was hypothesized as an important mediator assessed at the classroom level. We reason that the quality of one classroom may affect outcomes of children in another classroom because children interact not simply with their classmates but also with those from other classes in the hallways or on the playground. In investigating the role of class quality as a mediator, failure to account for such spillover effects of one classroom on the outcomes of children in other classrooms can potentially result in bias and problems with interpretation. Using a counterfactual conceptualization of direct, indirect and spillover effects, we provide a framework that can accommodate issues of mediation and spillover effects in group randomized trials. We show that the total effect can be decomposed into a natural direct effect, a within-classroom mediated effect and a spillover mediated effect. We give identification conditions for each of the causal effects of interest and provide results on the consequences of ignoring "interference" or "spillover effects" when they are in fact present. Our modeling approach disentangles these effects. The analysis examines whether the 4Rs intervention has an effect on children's depressive symptoms through changing the quality of other classes as well as through changing the quality of a child's own class.

Estimation of vaccine efficacy and critical vaccination coverage in partially observed outbreaks

Classical approaches to estimate vaccine efficacy are based on the assumption that a person's risk of infection does not depend on the infection status of others. This assumption is untenable for infectious disease data where such dependencies abound. We present a novel approach to estimating vaccine efficacy in a Bayesian framework using disease transmission models. The methodology is applied to outbreaks of mumps in primary schools in the Netherlands. The total study population consisted of 2,493 children in ten primary schools, of which 510 (20%) were known to have been infected, and 832 (33%) had unknown infection status. The apparent vaccination coverage ranged from 12% to 93%, and the apparent infection attack rate varied from 1% to 76%. Our analyses show that vaccination reduces the probability of infection per contact substantially but not perfectly ([Formula: see text] = 0.933; 95CrI: 0.908-0.954). Mumps virus appears to be moderately transmissible in the school setting, with each case yielding an estimated 2.5 secondary cases in an unvaccinated population ([Formula: see text] = 2.49; 95%CrI: 2.36-2.63), resulting in moderate estimates of the critical vaccination coverage (64.2%; 95%CrI: 61.7-66.7%). The indirect benefits of vaccination are highest in populations with vaccination coverage just below the critical vaccination coverage. In these populations, it is estimated that almost two infections can be prevented per vaccination. We discuss the implications for the optimal control of mumps in heterogeneously vaccinated populations.

Causal Inference Under Multiple Versions of Treatment

Abstract: In this article, we discuss causal inference when there are multiple versions of treatment. The potential outcomes framework, as articulated by Rubin, makes an assumption of no multiple versions of treatment, and here we discuss an extension of this potential outcomes framework to accommodate causal inference under violations of this assumption. A variety of examples are discussed in which the assumption may be violated. Identification results are provided for the overall treatment effect and the effect of treatment on the treated when multiple versions of treatment are present and also for the causal effect comparing a version of one treatment to some other version of the same or a different treatment. Further identification and interpretative results are given for cases in which the version precedes the treatment as when an underlying treatment variable is coarsened or dichotomized to create a new treatment variable for which there are effectively “multiple versions”. Results are also given for effects defined by setting the version of treatment to a prespecified distribution. Some of the identification results bear resemblance to identification results in the literature on direct and indirect effects. We describe some settings in which ignoring multiple versions of treatment, even when present, will not lead to incorrect inferences.

Estimating the Effect of a Community-Based Intervention with Two Communities

Due to the need to evaluate the effectiveness of community-based programs in practice, there is substantial interest in methods to estimate the causal effects of community-level treatments or exposures on individual level outcomes. The challenge one is confronted with is that different communities have different environmental factors affecting the individual outcomes, and all individuals in a community share the same environment and intervention. In practice, data are often available from only a small number of communities, making it difficult if not impossible to adjust for these environmental confounders. In this paper we consider an extreme version of this dilemma, in which two communities each receives a different level of the intervention, and covariates and outcomes are measured on a random sample of independent individuals from each of the two populations; the results presented can be straightforwardly generalized to settings in which more than two communities are sampled. We address the question of what conditions are needed to estimate the causal effect of the intervention, defined in terms of an ideal experiment in which the exposed level of the intervention is assigned to both communities and individual outcomes are measured in the combined population, and then the clock is turned back and a control level of the intervention is assigned to both communities and individual outcomes are measured in the combined population. We refer to the difference in the expectation of these outcomes as the marginal (overall) treatment effect. We also discuss conditions needed for estimation of the treatment effect on the treated community. We apply a nonparametric structural equation model to define these causal effects and to establish conditions under which they are identified. These identifiability conditions provide guidance for the design of studies to investigate community level causal effects and for assessing the validity of causal interpretations when data are only available from a few communities. When the identifiability conditions fail to hold, the proposed statistical parameters still provide nonparametric treatment effect measures (albeit non-causal) whose statistical interpretations do not depend on model specifications. In addition, we study the use of a matched cohort sampling design in which the units of different communities are matched on individual factors. Finally, we provide semiparametric efficient and doubly robust targeted MLE estimators of the community level causal effect based on i.i.d. sampling and matched cohort sampling.

Components of the indirect effect in vaccine trials: identification of contagion and infectiousness effects

Vaccination of one person may prevent the infection of another either because the vaccine prevents the first from being infected and from infecting the second, or because, even if the first person is infected, the vaccine may render the infection less infectious. We might refer to the first of these mechanisms as a contagion effect and the second as an infectiousness effect. In the simple setting of a randomized vaccine trial with households of size two, we use counterfactual theory under interference to provide formal definitions of a contagion effect and an unconditional infectiousness effect. Using ideas analogous to mediation analysis, we show that the indirect effect (the effect of one person's vaccine on another's outcome) can be decomposed into a contagion effect and an unconditional infectiousness effect on the risk difference, risk ratio, odds ratio, and vaccine efficacy scales. We provide identification assumptions for such contagion and unconditional infectiousness effects and describe a simple statistical technique to estimate these effects when they are identified. We also give a sensitivity analysis technique to assess how inferences would change under violations of the identification assumptions. The concepts and results of this paper are illustrated with hypothetical vaccine trial data.

Assessing mediation using marginal structural models in the presence of confounding and moderation

This article presents marginal structural models with inverse propensity weighting (IPW) for assessing mediation. Generally, individuals are not randomly assigned to levels of the mediator. Therefore, confounders of the mediator and outcome may exist that limit causal inferences, a goal of mediation analysis. Either regression adjustment or IPW can be used to take confounding into account, but IPW has several advantages. Regression adjustment of even one confounder of the mediator and outcome that has been influenced by treatment results in biased estimates of the direct effect (i.e., the effect of treatment on the outcome that does not go through the mediator). One advantage of IPW is that it can properly adjust for this type of confounding, assuming there are no unmeasured confounders. Further, we illustrate that IPW estimation provides unbiased estimates of all effects when there is a baseline moderator variable that interacts with the treatment, when there is a baseline moderator variable that interacts with the mediator, and when the treatment interacts with the mediator. IPW estimation also provides unbiased estimates of all effects in the presence of nonrandomized treatments. In addition, for testing mediation we propose a test of the null hypothesis of no mediation. Finally, we illustrate this approach with an empirical data set in which the mediator is continuous, as is often the case in psychological research.

The parametric g-formula to estimate the effect of highly active antiretroviral therapy on incident AIDS or death

The parametric g-formula can be used to contrast the distribution of potential outcomes under arbitrary treatment regimes. Like g-estimation of structural nested models and inverse probability weighting of marginal structural models, the parametric g-formula can appropriately adjust for measured time-varying confounders that are affected by prior treatment. However, there have been few implementations of the parametric g-formula to date. Here, we apply the parametric g-formula to assess the impact of highly active antiretroviral therapy on time to acquired immune deficiency syndrome (AIDS) or death in two US-based human immunodeficiency virus cohorts including 1498 participants. These participants contributed approximately 7300 person-years of follow-up (49% exposed to highly active antiretroviral therapy) during which 382 events occurred and 259 participants were censored because of dropout. Using the parametric g-formula, we estimated that antiretroviral therapy substantially reduces the hazard of AIDS or death (hazard ratio = 0.55; 95% confidence limits [CL]: 0.42, 0.71). This estimate was similar to one previously reported using a marginal structural model, 0.54 (95% CL: 0.38, 0.78). The 6.5-year difference in risk of AIDS or death was 13% (95% CL: 8%, 18%). Results were robust to assumptions about temporal ordering, and extent of history modeled, for time-varying covariates. The parametric g-formula is a viable alternative to inverse probability weighting of marginal structural models and g-estimation of structural nested models for the analysis of complex longitudinal data.

The Minicommunity Design to Assess Indirect Effects of Vaccination

We propose the minicommunity design to estimate indirect effects of vaccination. Establishing indirect effects of vaccination in unvaccinated subpopulations could have important implications for global vaccine policies. In the minicommunity design, the household or other small transmission unit serves as the cluster in which to estimate indirect effects of vaccination, similar to studies in larger communities to estimate indirect, total, and overall effects. Examples from the literature include studies in small transmission units to estimate indirect effects of pertussis, pneumococcal, influenza, and cholera vaccines. We characterize the minicommunity design by several methodologic considerations, including the assignment mechanism, ascertainment, the role of transmission outside the transmission unit, and the relation of the size of the transmission unit to number of people vaccinated. The minicommunity study for indirect effects is contrasted with studies to estimate vaccine effects on infectiousness and protective effects under conditions of household exposure within small transmission units. The minicommunity design can be easily implemented in individually randomized studies by enrolling and following-up members of households of the randomized individuals. The methodology for the minicommunity design for estimating indirect effects of vaccination deserves much future research.

Mediation analysis with multiple versions of the mediator

The causal inference literature has provided definitions of direct and indirect effects based on counterfactuals that generalize the approach found in the social science literature. However, these definitions presuppose well-defined hypothetical interventions on the mediator. In many settings, there may be multiple ways to fix the mediator to a particular value, and these various hypothetical interventions may have very different implications for the outcome of interest. In this paper, we consider mediation analysis when multiple versions of the mediator are present. Specifically, we consider the problem of attempting to decompose a total effect of an exposure on an outcome into the portion through the intermediate and the portion through other pathways. We consider the setting in which there are multiple versions of the mediator but the investigator has access only to data on the particular measurement, not information on which version of the mediator may have brought that value about. We show that the quantity that is estimated as a natural indirect effect using only the available data does indeed have an interpretation as a particular type of mediated effect; however, the quantity estimated as a natural direct effect, in fact, captures both a true direct effect and an effect of the exposure on the outcome mediated through the effect of the version of the mediator that is not captured by the mediator measurement. The results are illustrated using 2 examples from the literature, one in which the versions of the mediator are unknown and another in which the mediator itself has been dichotomized.

Extending the sufficient component cause model to describe the Stable Unit Treatment Value Assumption (SUTVA)

Causal inference requires an understanding of the conditions under which association equals causation. The exchangeability or no confounding assumption is well known and well understood as central to this task. More recently the epidemiologic literature has described additional assumptions related to the stability of causal effects. In this paper we extend the Sufficient Component Cause Model to represent one expression of this stability assumption--the Stable Unit Treatment Value Assumption. Approaching SUTVA from an SCC model helps clarify what SUTVA is and reinforces the connections between interaction and SUTVA.

Understanding the impact of career academy attendance: an application of the principal stratification framework for causal effects accounting for partial compliance

BACKGROUND

Results from MDRC's longitudinal, random-assignment evaluation of career-academy high schools reveal that several years after high-school completion, those randomized to receive the academy opportunity realized a $175 (11%) increase in monthly earnings, on average.

OBJECTIVES

In this paper, I investigate the impact of duration of actual academy enrollment, as nearly half of treatment group students either never enrolled or participated for only a portion of high school.

RESEARCH DESIGN

I capitalize on data from this experimental evaluation and utilize a principal stratification framework and Bayesian inference to investigate the causal impact of academy participation.

SUBJECTS

This analysis focuses on a sample of 1,306 students across seven sites in the MDRC evaluation.

MEASURES

Participation is measured by number of years of academy enrollment, and the outcome of interest is average monthly earnings in the period of four to eight years after high school graduation.

RESULTS

I estimate an average causal effect of treatment assignment on subsequent monthly earnings of approximately $588 among males who remained enrolled in an academy throughout high school and more modest impacts among those who participated only partially.

CONCLUSIONS

Different from an instrumental variables approach to treatment non-compliance, which allows for the estimation of linear returns to treatment take-up, the more general framework of principal stratification allows for the consideration of non-linear returns, although at the expense of additional model-based assumptions.

Bounding the infectiousness effect in vaccine trials

In vaccine trials, the vaccination of one person might prevent the infection of another; a distinction can be drawn between the ways such a protective effect might arise. Consider a setting with 2 persons per household in which one of the 2 is vaccinated. Vaccinating the first person may protect the second person by preventing the first from being infected and passing the infection on to the second. Alternatively, vaccinating the first person may protect the second by rendering the infection less contagious even if the first is infected. This latter mechanism is sometimes referred to as an "infectiousness effect" of the vaccine. Crude estimators for the infectiousness effect will be subject to selection bias due to stratification on a postvaccination event, namely the infection status of the first person. We use theory concerning causal inference under interference along with a principal-stratification framework to show that, although the crude estimator is biased, it is, under plausible assumptions, conservative for what one might define as a causal infectiousness effect. This applies to bias from selection due to the persons in the comparison, and also to selection due to pathogen virulence. We illustrate our results with an example from the literature.

A mapping between interactions and interference: implications for vaccine trials

In this paper, we discuss relationships between causal interactions within the counterfactual framework and interference in which the exposure of one person may affect the outcomes of another. We show that the empirical tests for causal interactions can, in fact, all be adapted to empirical tests for particular forms of interference. In the context of interference, by recoding the response as some function of the outcomes of the various persons within a cluster, a wide range of different forms of interference can potentially be detected. The correspondence between causal interactions and forms of interference extends to encompass n-way causal interactions, interference between n persons within a cluster, and multivalued exposures. The theory for causal interactions provides a complete conceptual apparatus for assessing interference as well. The results are illustrated using data from a hypothetical vaccine trial to reason about specific forms of interference and spillover effects that may be present in this vaccine setting. We discuss the implications of this correspondence for our conceptualizations of interaction and for application to vaccine trials and many other settings in which spillover effects may be present.

On causal inference in the presence of interference

Interference is said to be present when the exposure or treatment received by one individual may affect the outcomes of other individuals. Such interference can arise in settings in which the outcomes of the various individuals come about through social interactions. When interference is present, causal inference is rendered considerably more complex, and the literature on causal inference in the presence of interference has just recently begun to develop. In this article we summarise some of the concepts and results from the existing literature and extend that literature in considering new results for finite sample inference, new inverse probability weighting estimators in the presence of interference and new causal estimands of interest.

Causal inference for vaccine effects on infectiousness

If a vaccine does not protect individuals completely against infection, it could still reduce infectiousness of infected vaccinated individuals to others. Typically, vaccine efficacy for infectiousness is estimated based on contrasts between the transmission risk to susceptible individuals from infected vaccinated individuals compared with that from infected unvaccinated individuals. Such estimates are problematic, however, because they are subject to selection bias and do not have a causal interpretation. Here, we develop causal estimands for vaccine efficacy for infectiousness for four different scenarios of populations of transmission units of size two. These causal estimands incorporate both principal stratification, based on the joint potential infection outcomes under vaccine and control, and interference between individuals within transmission units. In the most general scenario, both individuals can be exposed to infection outside the transmission unit and both can be assigned either vaccine or control. The three other scenarios are special cases of the general scenario where only one individual is exposed outside the transmission unit or can be assigned vaccine. The causal estimands for vaccine efficacy for infectiousness are well defined only within certain principal strata and, in general, are identifiable only with strong unverifiable assumptions. Nonetheless, the observed data do provide some information, and we derive large sample bounds on the causal vaccine efficacy for infectiousness estimands. An example of the type of data observed in a study to estimate vaccine efficacy for infectiousness is analyzed in the causal inference framework we developed.

Inference with interference between units in an fMRI experiment of motor inhibition

An experimental unit is an opportunity to randomly apply or withhold a treatment. There is interference between units if the application of the treatment to one unit may also affect other units. In cognitive neuroscience, a common form of experiment presents a sequence of stimuli or requests for cognitive activity at random to each experimental subject and measures biological aspects of brain activity that follow these requests. Each subject is then many experimental units, and interference between units within an experimental subject is likely, in part because the stimuli follow one another quickly and in part because human subjects learn or become experienced or primed or bored as the experiment proceeds. We use a recent fMRI experiment concerned with the inhibition of motor activity to illustrate and further develop recently proposed methodology for inference in the presence of interference. A simulation evaluates the power of competing procedures.

Compound treatments and transportability of causal inference

Ill-defined causal questions present serious problems for observational studies-problems that are largely unappreciated. This paper extends the usual counterfactual framework to consider causal questions about compound treatments for which there are many possible implementations (for example, "prevention of obesity"). We describe the causal effect of compound treatments and their identifiability conditions, with a special emphasis on the consistency condition. We then discuss the challenges of using the estimated effect of a compound treatment in one study population to inform decisions in the same population and in other populations. These challenges arise because the causal effect of compound treatments depends on the distribution of the versions of treatment in the population. Such causal effects can be unpredictable when the versions of treatment are unknown. We discuss how such issues of "transportability" are related to the consistency condition in causal inference. With more carefully framed questions, the results of epidemiologic studies can be of greater value to decision-makers.

Effect partitioning under interference in two-stage randomized vaccine trials

In the presence of interference, the exposure of one individual may affect the outcomes of others. We provide new effect partitioning results under interferences that express the overall effect as a sum of (i) the indirect (or spillover) effect and (ii) a contrast between two direct effects.

Statistical identifiability and the surrogate endpoint problem, with application to vaccine trials

Given a randomized treatment Z, a clinical outcome Y, and a biomarker S measured some fixed time after Z is administered, we may be interested in addressing the surrogate endpoint problem by evaluating whether S can be used to reliably predict the effect of Z on Y. Several recent proposals for the statistical evaluation of surrogate value have been based on the framework of principal stratification. In this article, we consider two principal stratification estimands: joint risks and marginal risks. Joint risks measure causal associations (CAs) of treatment effects on S and Y, providing insight into the surrogate value of the biomarker, but are not statistically identifiable from vaccine trial data. Although marginal risks do not measure CAs of treatment effects, they nevertheless provide guidance for future research, and we describe a data collection scheme and assumptions under which the marginal risks are statistically identifiable. We show how different sets of assumptions affect the identifiability of these estimands; in particular, we depart from previous work by considering the consequences of relaxing the assumption of no individual treatment effects on Y before S is measured. Based on algebraic relationships between joint and marginal risks, we propose a sensitivity analysis approach for assessment of surrogate value, and show that in many cases the surrogate value of a biomarker may be hard to establish, even when the sample size is large.

Direct and indirect effects for neighborhood-based clustered and longitudinal data

Definitions of direct and indirect effects are given for settings in which individuals are clustered in groups or neighborhoods and in which treatments are administered at the group level. A particular intervention may affect individual outcomes both through its effect on the individual and by changing the group or neighborhood itself. Identification conditions are given for controlled direct effects and for natural direct and indirect effects. The interpretation of these identification conditions are discussed within the context of neighborhood research and multilevel modeling. Interventions at a single point in time and time-varying interventions are both considered. The definition of direct and indirect effects requires certain stability or no-interference conditions; some discussion is given as to how these no-interference conditions can be relaxed.

Matching methods for causal inference: A review and a look forward

When estimating causal effects using observational data, it is desirable to replicate a randomized experiment as closely as possible by obtaining treated and control groups with similar covariate distributions. This goal can often be achieved by choosing well-matched samples of the original treated and control groups, thereby reducing bias due to the covariates. Since the 1970's, work on matching methods has examined how to best choose treated and control subjects for comparison. Matching methods are gaining popularity in fields such as economics, epidemiology, medicine, and political science. However, until now the literature and related advice has been scattered across disciplines. Researchers who are interested in using matching methods-or developing methods related to matching-do not have a single place to turn to learn about past and current research. This paper provides a structure for thinking about matching methods and guidance on their use, coalescing the existing research (both old and new) and providing a summary of where the literature on matching methods is now and where it should be headed.