Principal stratification and attribution prohibition: good ideas taken too far

Marshall Joffe's Contributions to Causal Inference, Biostatistics, and Epidemiology

We pay tribute to Marshall Joffe, PhD, and his substantial contributions to the field of causal inference with focus in biostatistics and epidemiology. By compiling narratives written by us, his colleagues, we not only present highlights of Marshall's research and their significance for causal inference but also offer a portrayal of Marshall's personal accomplishments and character. Our discussion of Marshall's research notably includes (but is not limited to) handling of posttreatment variables such as noncompliance, employing G-estimation for treatment effects on failure-time outcomes, estimating effects of time-varying exposures subject to time-dependent confounding, and developing a causal framework for case-control studies. We also provide a description of some of Marshall's unpublished work, which is accompanied by a bonus anecdote. We discuss future research directions related to Marshall's research. While Marshall's impact in causal inference and the world outside of it cannot be wholly captured by our words, we hope nonetheless to present some of what he has done for our field and what he has meant to us and to his loved ones.

Causal inference with recurrent and competing events

Many research questions concern treatment effects on outcomes that can recur several times in the same individual. For example, medical researchers are interested in treatment effects on hospitalizations in heart failure patients and sports injuries in athletes. Competing events, such as death, complicate causal inference in studies of recurrent events because once a competing event occurs, an individual cannot have more recurrent events. Several statistical estimands have been studied in recurrent event settings, with and without competing events. However, the causal interpretations of these estimands, and the conditions that are required to identify these estimands from observed data, have yet to be formalized. Here we use a formal framework for causal inference to formulate several causal estimands in recurrent event settings, with and without competing events. When competing events exist, we clarify when commonly used classical statistical estimands can be interpreted as causal quantities from the causal mediation literature, such as (controlled) direct effects and total effects. Furthermore, we show that recent results on interventionist mediation estimands allow us to define new causal estimands with recurrent and competing events that may be of particular clinical relevance in many subject matter settings. We use causal directed acyclic graphs and single world intervention graphs to illustrate how to reason about identification conditions for the various causal estimands based on subject matter knowledge. Furthermore, using results on counting processes, we show that our causal estimands and their identification conditions, which are articulated in discrete time, converge to classical continuous time counterparts in the limit of fine discretizations of time. We propose estimators and establish their consistency for the various identifying functionals. Finally, we use the proposed estimators to compute the effect of blood pressure lowering treatment on the recurrence of acute kidney injury using data from the Systolic Blood Pressure Intervention Trial.

Identification of Vaccine Effects When Exposure Status Is Unknown

Results from randomized controlled trials (RCTs) help determine vaccination strategies and related public health policies. However, defining and identifying estimands that can guide policies in infectious disease settings is difficult, even in an RCT. The effects of vaccination critically depend on characteristics of the population of interest, such as the prevalence of infection, the number of vaccinated, and social behaviors. To mitigate the dependence on such characteristics, estimands, and study designs, that require conditioning or intervening on exposure to the infectious agent have been advocated. But a fundamental problem for both RCTs and observational studies is that exposure status is often unavailable or difficult to measure, which has made it impossible to apply existing methodology to study vaccine effects that account for exposure status. In this study, we present new results on this type of vaccine effects. Under plausible conditions, we show that point identification of certain relative effects is possible even when the exposure status is unknown. Furthermore, we derive sharp bounds on the corresponding absolute effects. We apply these results to estimate the effects of the ChAdOx1 nCoV-19 vaccine on SARS-CoV-2 disease (COVID-19) conditional on postvaccine exposure to the virus, using data from a large RCT.

Translating questions to estimands in randomized clinical trials with intercurrent events

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

A causal framework for classical statistical estimands in failure-time settings with competing events

In failure-time settings, a competing event is any event that makes it impossible for the event of interest to occur. For example, cardiovascular disease death is a competing event for prostate cancer death because an individual cannot die of prostate cancer once he has died of cardiovascular disease. Various statistical estimands have been defined as possible targets of inference in the classical competing risks literature. Many reviews have described these statistical estimands and their estimating procedures with recommendations about their use. However, this previous work has not used a formal framework for characterizing causal effects and their identifying conditions, which makes it difficult to interpret effect estimates and assess recommendations regarding analytic choices. Here we use a counterfactual framework to explicitly define each of these classical estimands. We clarify that, depending on whether competing events are defined as censoring events, contrasts of risks can define a total effect of the treatment on the event of interest or a direct effect of the treatment on the event of interest not mediated by the competing event. In contrast, regardless of whether competing events are defined as censoring events, counterfactual hazard contrasts cannot generally be interpreted as causal effects. We illustrate how identifying assumptions for all of these counterfactual estimands can be represented in causal diagrams, in which competing events are depicted as time-varying covariates. We present an application of these ideas to data from a randomized trial designed to estimate the effect of estrogen therapy on prostate cancer mortality.

Dealing with death when studying disease or physiological marker: the stochastic system approach to causality

The stochastic system approach to causality is applied to situations where the risk of death is not negligible. This approach grounds causality on physical laws, distinguishes system and observation and represents the system by multivariate stochastic processes. The particular role of death is highlighted, and it is shown that local influences must be defined on the random horizon of time of death. We particularly study the problem of estimating the effect of a factor V on a process of interest Y, taking death into account. We unify the cases where Y is a counting process (describing an event) and the case where Y is quantitative; we examine the case of observations in continuous and discrete time and we study the issue of whether the mechanism leading to incomplete data can be ignored. Finally, we give an example of a situation where we are interested in estimating the effect of a factor (blood pressure) on cognitive ability in elderly.

Epidemiology, genetic epidemiology and Mendelian randomisation: more need than ever to attend to detail

In the current era, with increasing availability of results from genetic association studies, finding genetic instruments for inferring causality in observational epidemiology has become apparently simple. Mendelian randomisation (MR) analyses are hence growing in popularity and, in particular, methods that can incorporate multiple instruments are being rapidly developed for these applications. Such analyses have enormous potential, but they all rely on strong, different, and inherently untestable assumptions. These have to be clearly stated and carefully justified for every application in order to avoid conclusions that cannot be replicated. In this article, we review the instrumental variable assumptions and discuss the popular linear additive structural model. We advocate the use of tests for the null hypothesis of ‘no causal effect’ and calculation of the bounds for a causal effect, whenever possible, as these do not rely on parametric modelling assumptions. We clarify the difference between a randomised trial and an MR study and we comment on the importance of validating instruments, especially when considering them for joint use in an analysis. We urge researchers to stand by their convictions, if satisfied that the relevant assumptions hold, and to interpret their results causally since that is the only reason for performing an MR analysis in the first place.

The challenging interpretation of instrumental variable estimates under monotonicity

Background

Instrumental variable (IV) methods are often used to identify 'local' causal effects in a subgroup of the population of interest. Such 'local' effects may not be ideal for informing clinical or policy decision making. When the instrument is non-causal, additional difficulties arise for interpreting 'local' effects. Little attention has been paid to these difficulties, even though commonly proposed instruments in epidemiology are non-causal (e.g. proxies for physician's preference; genetic variants in some Mendelian randomization studies).

Methods

For IV estimates obtained from both causal and non-causal instruments under monotonicity, we present results to help investigators pose four questions about the local effect estimates obtained in their studies. (1) To what subgroup of the population does the effect pertain? Can we (2) estimate the size of or (3) describe the characteristics of this subgroup relative to the study population? (4) Can the sensitivity of the effect estimate to deviations from monotonicity be quantified?

Results

We show that the common interpretations and approaches for answering these four questions are generally only appropriate in the case of causal instruments.

Conclusions

Appropriate interpretation of an IV estimate under monotonicity as a 'local' effect critically depends on whether the proposed instrument is causal or non-causal. The results and formal proofs presented here can help in the transparent reporting of IV results and in enhancing the use of IV estimates in informing decision-making efforts.

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

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

A "placement of death" approach for studies of treatment effects on ICU length of stay

Length of stay in the intensive care unit (ICU) is a common outcome measure in randomized trials of ICU interventions. Because many patients die in the ICU, it is difficult to disentangle treatment effects on length of stay from effects on mortality; conventional analyses depend on assumptions that are often unstated and hard to interpret or check. We adapt a proposal from Rosenbaum that addresses concerns about selection bias and makes its assumptions explicit. A composite outcome is constructed that equals ICU length of stay if the patient was discharged alive and indicates death otherwise. Given any preference ordering that compares death with possible lengths of stay, we can estimate the intervention's effects on the composite outcome distribution. Sensitivity analyses can show results for different preference orderings. We discuss methods for constructing approximate confidence intervals for treatment effects on quantiles of the outcome distribution or on proportions of patients with outcomes preferable to various cutoffs. Strengths and weaknesses of possible primary significance tests (including the Wilcoxon-Mann-Whitney rank sum test and a heteroskedasticity-robust variant due to Brunner and Munzel) are reviewed. An illustrative example reanalyzes a randomized trial of an ICU staffing intervention.

Surrogate Endpoint Evaluation: Principal Stratification Criteria and the Prentice Definition

A common problem of interest within a randomized clinical trial is the evaluation of an inexpensive response endpoint as a valid surrogate endpoint for a clinical endpoint, where a chief purpose of a valid surrogate is to provide a way to make correct inferences on clinical treatment effects in future studies without needing to collect the clinical endpoint data. Within the principal stratification framework for addressing this problem based on data from a single randomized clinical efficacy trial, a variety of definitions and criteria for a good surrogate endpoint have been proposed, all based on or closely related to the "principal effects" or "causal effect predictiveness (CEP)" surface. We discuss CEP-based criteria for a useful surrogate endpoint, including (1) the meaning and relative importance of proposed criteria including average causal necessity (ACN), average causal sufficiency (ACS), and large clinical effect modification; (2) the relationship between these criteria and the Prentice definition of a valid surrogate endpoint; and (3) the relationship between these criteria and the consistency criterion (i.e., assurance against the "surrogate paradox"). This includes the result that ACN plus a strong version of ACS generally do not imply the Prentice definition nor the consistency criterion, but they do have these implications in special cases. Moreover, the converse does not hold except in a special case with a binary candidate surrogate. The results highlight that assumptions about the treatment effect on the clinical endpoint before the candidate surrogate is measured are influential for the ability to draw conclusions about the Prentice definition or consistency. In addition, we emphasize that in some scenarios that occur commonly in practice, the principal strata sub-populations for inference are identifiable from the observable data, in which cases the principal stratification framework has relatively high utility for the purpose of effect modification analysis, and is closely connected to the treatment marker selection problem. The results are illustrated with application to a vaccine efficacy trial, where ACN and ACS for an antibody marker are found to be consistent with the data and hence support the Prentice definition and consistency.

Invited commentary: Estimating population impact in the presence of competing events

The formal approach in the field of causal inference has enabled epidemiologists to clarify several complications that arise when estimating the effect of an intervention on a health outcome of interest. When the outcome is a failure time or longitudinal process, researchers must often deal with competing events. In this issue of the Journal, Picciotto et al. (Am J Epidemiol. 2015;181(8):563-570) use structural nested failure time models to assess potential population effects of hypothetical interventions and censor competing events. In the present commentary, we discuss 2 interpretations that result from treating competing events as censored observations and how they relate to measures of public health impact. We also comment on 2 alternative approaches for handling competing events: an inverse probability weighting estimator of the survivor average causal effect and the parametric g-formula, which can be used to estimate a functional of the subdistribution of the event of interest. We argue that careful consideration of the tradeoff between the interpretation of the parameters from each approach and the assumptions required to estimate these parameters should guide researchers on the various ways to handle competing events in epidemiologic research.

Doubly robust estimation and causal inference in longitudinal studies with dropout and truncation by death

Motivated by aging research, we propose an estimator of the effect of a time-varying exposure on an outcome in longitudinal studies with dropout and truncation by death. We use an inverse-probability weighted (IPW) estimator to derive a doubly robust augmented inverse-probability weighted (AIPW) estimator. IPW estimation involves weights for the exposure mechanism, dropout, and mortality; AIPW estimation additionally involves estimating data-generating models via regression. We demonstrate that the estimators identify a causal contrast that is a function of principal strata effects under a set of assumptions. Simulations show that AIPW estimation is unbiased when weights or outcome regressions are correct, and that AIPW estimation is more efficient than IPW estimation when all models are correct. We apply the method to a study of vitamin D and gait speed among older adults.

Nonparametric Bounds and Sensitivity Analysis of Treatment Effects

This paper considers conducting inference about the effect of a treatment (or exposure) on an outcome of interest. In the ideal setting where treatment is assigned randomly, under certain assumptions the treatment effect is identifiable from the observable data and inference is straightforward. However, in other settings such as observational studies or randomized trials with noncompliance, the treatment effect is no longer identifiable without relying on untestable assumptions. Nonetheless, the observable data often do provide some information about the effect of treatment, that is, the parameter of interest is partially identifiable. Two approaches are often employed in this setting: (i) bounds are derived for the treatment effect under minimal assumptions, or (ii) additional untestable assumptions are invoked that render the treatment effect identifiable and then sensitivity analysis is conducted to assess how inference about the treatment effect changes as the untestable assumptions are varied. Approaches (i) and (ii) are considered in various settings, including assessing principal strata effects, direct and indirect effects and effects of time-varying exposures. Methods for drawing formal inference about partially identified parameters are also discussed.

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.

Sharpening bounds on principal effects with covariates

Estimation of treatment effects in randomized studies is often hampered by possible selection bias induced by conditioning on or adjusting for a variable measured post-randomization. One approach to obviate such selection bias is to consider inference about treatment effects within principal strata, that is, principal effects. A challenge with this approach is that without strong assumptions principal effects are not identifiable from the observable data. In settings where such assumptions are dubious, identifiable large sample bounds may be the preferred target of inference. In practice these bounds may be wide and not particularly informative. In this work we consider whether bounds on principal effects can be improved by adjusting for a categorical baseline covariate. Adjusted bounds are considered which are shown to never be wider than the unadjusted bounds. Necessary and sufficient conditions are given for which the adjusted bounds will be sharper (i.e., narrower) than the unadjusted bounds. The methods are illustrated using data from a recent, large study of interventions to prevent mother-to-child transmission of HIV through breastfeeding. Using a baseline covariate indicating low birth weight, the estimated adjusted bounds for the principal effect of interest are 63% narrower than the estimated unadjusted bounds.

Rank-based principal stratum sensitivity analyses

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

Designs combining instrumental variables with case-control: estimating principal strata causal effects

The instrumental variables framework is commonly used for the estimation of causal effects from cohort samples. However, the combination of instrumental variables with more efficient designs such as case-control sampling requires new methodological consideration. For example, as the use of Mendelian randomization studies is increasing and the cost of genotyping and gene expression data can be high, the analysis of data gathered from more cost-effective sampling designs is of prime interest. We show that the standard instrumental variables analysis does not appropriately estimate the causal effects of interest when the instrumental variables design is combined with the case-control design. We also propose a method that can estimate the causal effects in such combined designs. We illustrate the method with a study in oncology.