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Pathogens usually exist in heterogeneous variants, like subtypes and strains. Quantifying treatment effects on the different variants is important for guiding prevention policies and treatment development. Here we ground analyses of variant-specific effects on a formal framework for causal inference. This allows us to clarify the interpretation of existing methods and define new estimands. Unlike most of the existing literature, we explicitly consider the (realistic) setting with interference in the target population: even if individuals can be sensibly perceived as iid in randomized trial data, there will often be interference in the target population where treatments, like vaccines, are rolled out. Thus, one of our contributions is to derive explicit conditions guaranteeing that commonly reported vaccine efficacy parameters quantify well-defined causal effects, also in the presence of interference. Furthermore, our results give alternative justifications for reporting estimands on the relative, rather than absolute, scale. We illustrate the findings with an analysis of a large HIV1 vaccine trial, where there is interest in distinguishing vaccine effects on viruses with different genome sequences.

The aim of personalized medicine is to tailor treatment decisions to individuals' characteristics. N-of-1 trials are within-person crossover trials that hold the promise of targeting individual-specific effects. While the idea behind N-of-1 trials might seem simple, analyzing and interpreting N-of-1 trials is not straightforward. Here we ground N-of-1 trials in a formal causal inference framework and formalize intuitive claims from the N-of-1 trials literature. We focus on causal inference from a single N-of-1 trial and define a conditional average treatment effect (CATE) that represents a target in this setting, which we call the U-CATE. We discuss assumptions sufficient for identification and estimation of the U-CATE under different causal models where the treatment schedule is assigned at baseline. A simple mean difference is an unbiased, asymptotically normal estimator of the U-CATE in simple settings. We also consider settings where carryover effects, trends over time, time-varying common causes of the outcome, and outcome-outcome effects are present. In these more complex settings, we show that a time-varying g-formula identifies the U-CATE under explicit assumptions. Finally, we analyze data from N-of-1 trials about acne symptoms and show how different assumptions about the data generating process can lead to different analytical strategies.

Knowing whether vaccine protection wanes over time is important for health policy and drug development. However, quantifying waning effects is difficult. A simple contrast of vaccine efficacy at two different times compares different populations of individuals: those who were uninfected at the first time versus those who remain uninfected until the second time. Thus, the contrast of vaccine efficacy at early and late times can not be interpreted as a causal effect. We propose to quantify vaccine waning using the challenge effect, which is a contrast of outcomes under controlled exposures to the infectious agent following vaccination. We identify sharp bounds on the challenge effect under non-parametric assumptions that are broadly applicable in vaccine trials using routinely collected data. We demonstrate that the challenge effect can differ substantially from the conventional vaccine efficacy due to depletion of susceptible individuals from the risk set over time. Finally, we apply the methods to derive bounds on the waning of the BNT162b2 COVID-19 vaccine using data from a placebo-controlled randomized trial. Our estimates of the challenge effect suggest waning protection after 2 months beyond administration of the second vaccine dose.

Point identification of causal effects requires strong assumptions that are unreasonable in many practical settings. However, informative bounds on these effects can often be derived under plausible assumptions. Even when these bounds are wide or cover null effects, they can guide practical decisions based on formal decision theoretic criteria. Here we derive new results on optimal treatment regimes in settings where the effect of interest is bounded. These results are driven by consideration of superoptimal regimes; we define regimes that leverage an individual's natural treatment value, which is typically ignored in the existing literature. We obtain (sharp) bounds for the value function of superoptimal regimes, and provide performance guarantees relative to conventional optimal regimes. As a case study, we consider a commonly studied Marginal Sensitivity Model and illustrate that the superoptimal regime can be identified when conventional optimal regimes are not. We similarly illustrate this property in an instrumental variable setting. Finally, we derive efficient estimators for upper and lower bounds on the superoptimal value in instrumental variable settings, building on recent results on covariate adjusted Balke-Pearl bounds. These estimators are applied to study the effect of prompt ICU admission on survival.

Policy-makers are often faced with the task of distributing a limited supply of resources. To support decision-making in these settings, statisticians are confronted with two challenges: estimands are defined by allocation strategies that are functions of features of all individuals in a cluster; and relatedly the observed data are neither independent nor identically distributed when individuals compete for resources. Existing statistical approaches are inadequate because they ignore at least one of these core features. As a solution, we develop theory for a general policy class of dynamic regimes for clustered data, covering existing results in classical and interference settings as special cases. We cover policy-relevant estimands and articulate realistic conditions compatible with resource-limited observed data. We derive identification and inference results for settings with a finite number of individuals in a cluster, where the observed dataset is viewed as a single draw from a super-population of clusters. We also consider asymptotic estimands when the number of individuals in a cluster is allowed to grow; under explicit conditions, we recover previous results, thereby clarifying when the use of existing methods is permitted. Our general results lay the foundation for future research on dynamic regimes for clustered data, including the longitudinal cluster setting.

In our original article (Sarvet & Stensrud, 2024), we examine twin definitions of "harm" in personalized medicine: one based on predictions of individuals' unmeasurable response types (counterfactual harm), and another based solely on the observations of experiments (interventionist harm). In their commentary, Mueller & Pearl (2024) (MP) read our review as an argument that "counterfactual logic should [...] be purged from consideration of harm and benefit" and "strongly object [...] that a rational decision maker may well apply the interventional perspective to the exclusion of counterfactual considerations." Here we show that this objection is misguided. We analyze MP's examples and derive a general result, showing that determinations of harm through interventionist and counterfactual analyses will always concur. Therefore, individuals who embrace counterfactual formulations and those who object to their use will make equivalent decisions in uncontroversial settings.

We formalize an interpretational error that is common in statistical causal inference, termed identity slippage. This formalism is used to describe historically-recognized fallacies, and analyse a fast-growing literature in statistics and applied fields. We conducted a systematic review of natural language claims in the literature on stochastic mediation parameters, and documented extensive evidence of identity slippage in applications. This framework for error detection is applicable whenever policy decisions depend on the accurate interpretation of statistical results, which is nearly always the case. Therefore, broad awareness of identity slippage will aid statisticians in the successful translation of data into public good.

We present new results on average causal effects in settings with unmeasured exposure-outcome confounding. Our results are motivated by a class of estimands, e.g., frequently of interest in medicine and public health, that are currently not targeted by standard approaches for average causal effects. We recognize these estimands as queries about the average causal effect of an intervening variable. We anchor our introduction of these estimands in an investigation of the role of chronic pain and opioid prescription patterns in the opioid epidemic, and illustrate how conventional approaches will lead unreplicable estimates with ambiguous policy implications. We argue that our altenative effects are replicable and have clear policy implications, and furthermore are non-parametrically identified by the classical frontdoor formula. As an independent contribution, we derive a new semiparametric efficient estimator of the frontdoor formula with a uniform sample boundedness guarantee. This property is unique among previously-described estimators in its class, and we demonstrate superior performance in finite-sample settings. Theoretical results are applied with data from the National Health and Nutrition Examination Survey.

Avoiding harm is an uncontroversial aim of personalized medicine and other epidemiologic initiatives. However, the precise mathematical translation of "harm" is disputable. Here we use a formal causal language to study common, but distinct, definitions of "harm". We clarify that commitment to a definition of harm has important practical and philosophical implications for decision making. We relate our practical and philosophical considerations to ideas from medical ethics and legal practice.

We consider optimal regimes for algorithm-assisted human decision-making. Such regimes are decision functions of measured pre-treatment variables and, by leveraging natural treatment values, enjoy a "superoptimality" property whereby they are guaranteed to outperform conventional optimal regimes. When there is unmeasured confounding, the benefit of using superoptimal regimes can be considerable. When there is no unmeasured confounding, superoptimal regimes are identical to conventional optimal regimes. Furthermore, identification of the expected outcome under superoptimal regimes in non-experimental studies requires the same assumptions as identification of value functions under conventional optimal regimes when the treatment is binary. To illustrate the utility of superoptimal regimes, we derive new identification and estimation results in a common instrumental variable setting. We use these derivations to analyze examples from the optimal regimes literature, including a case study of the effect of prompt intensive care treatment on survival.

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 counterfactual 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 using subject matter knowledge. Furthermore, using results on counting processes, we show how 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. Finally, we propose several estimators and establish their consistency for the various identifying functionals.

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 work, 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 post-vaccine exposure to the virus, using data from a large RCT.

Intercurrent (post-treatment) events occur frequently in randomized trials, and investigators often express interest in treatment effects that suitably take account of these events. A naive conditioning on intercurrent events does 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 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. Certain questions require stronger assumptions for identification. However, we highlight that our proposed estimands require less stringent assumptions than estimands commonly targeted in these settings, including principal stratum effects.

We discuss causal mediation analyses for survival data and propose a new approach based on the additive hazards model. The emphasis is on a dynamic point of view, that is, understanding how the direct and indirect effects develop over time. Hence, importantly, we allow for a time varying mediator. To define direct and indirect effects in such a longitudinal survival setting we take an interventional approach (Didelez (2018)) where treatment is separated into one aspect affecting the mediator and a different aspect affecting survival. In general, this leads to a version of the non-parametric g-formula (Robins (1986)). In the present paper, we demonstrate that combining the g-formula with the additive hazards model and a sequential linear model for the mediator process results in simple and interpretable expressions for direct and indirect effects in terms of relative survival as well as cumulative hazards. Our results generalise and formalise the method of dynamic path analysis (Fosen et al. (2006), Strohmaier et al. (2015)). An application to data from a clinical trial on blood pressure medication is given.

Researchers are often interested in treatment effects on outcomes that are only defined conditional on a post-treatment event status. For example, in a study of the effect of different cancer treatments on quality of life at end of follow-up, the quality of life of individuals who die during the study is undefined. In these settings, a naive contrast of outcomes conditional on the post-treatment variable is not an average causal effect, even in a randomized experiment. Therefore the effect in the principal stratum of those who would have the same value of the post-treatment variable regardless of treatment, such as the always survivors in a truncation by death setting, is often advocated for causal inference. While this principal stratum effect is a well defined causal contrast, it is often hard to justify that it is relevant to scientists, patients or policy makers, and it cannot be identified without relying on unfalsifiable assumptions. Here we formulate alternative estimands, the conditional separable effects, that have a natural causal interpretation under assumptions that can be falsified in a randomized experiment. We provide identification results and introduce different estimators, including a doubly robust estimator derived from the nonparametric influence function. As an illustration, we estimate a conditional separable effect of chemotherapies on quality of life in patients with prostate cancer, using data from a randomized clinical trial.

In competing event settings, a counterfactual contrast of cause-specific cumulative incidences quantifies the total causal effect of a treatment on the event of interest. However, effects of treatment on the competing event may indirectly contribute to this total effect, complicating its interpretation. We previously proposed the separable effects (Stensrud et al, 2019) to define direct and indirect effects of the treatment on the event of interest. This definition presupposes a treatment decomposition into two components acting along two separate causal pathways, one exclusively outside of the competing event and the other exclusively through it. Unlike previous definitions of direct and indirect effects, the separable effects can be subject to empirical scrutiny in a study where separate interventions on the treatment components are available. Here we extend and generalize the notion of the separable effects in several ways, allowing for interpretation, identification and estimation under considerably weaker assumptions. We propose and discuss a definition of separable effects that is applicable to general time-varying structures, where the separable effects can still be meaningfully interpreted, even when they cannot be regarded as direct and indirect effects. We further derive weaker conditions for identification of separable effects in observational studies where decomposed treatments are not yet available; in particular, these conditions allow for time-varying common causes of the event of interest, the competing events and loss to follow-up. For these general settings, we propose semi-parametric weighted estimators that are straightforward to implement. As an illustration, we apply the estimators to study the separable effects of intensive blood pressure therapy on acute kidney injury, using data from a randomized clinical trial.

Investigators often evaluate treatment effects by considering settings in which all individuals are assigned a treatment of interest, assuming that an unlimited number of treatment units are available. However, many real-life treatments are of limited supply and cannot be provided to all individuals in the population. For example, patients on the liver transplant waiting list cannot be assigned a liver transplant immediately at the time they reach highest priority because a suitable organ is not likely to be immediately available. In these cases, investigators may still be interested in the effects of treatment strategies in which a finite number of organs are available at a given time, that is, treatment regimes that satisfy resource constraints. Here, we describe an estimand that can be used to define causal effects of treatment strategies that satisfy resource constraints: proportionally-representative interventions for limited resources. We derive a simple class of inverse probability weighted estimators, and apply one such estimator to evaluate the effect of restricting or expanding utilization of "increased risk" liver organs to treat patients with end-stage liver disease. Our method is designed to evaluate policy-relevant interventions in the setting of finite treatment resources.

In time-to-event settings, the presence of competing events complicates the definition of causal effects. Here we propose the new separable effects to study the causal effect of a treatment on an event of interest. The separable direct effect is the treatment effect on the event of interest not mediated by its effect on the competing event. The separable indirect effect is the treatment effect on the event of interest only through its effect on the competing event. Similar to Robins and Richardson's extended graphical approach for mediation analysis, the separable effects can only be identified under the assumption that the treatment can be decomposed into two distinct components that exert their effects through distinct causal pathways. Unlike existing definitions of causal effects in the presence of competing events, our estimands do not require cross-world contrasts or hypothetical interventions to prevent death. As an illustration, we apply our approach to a randomized clinical trial on estrogen therapy in individuals with prostate cancer.

In failure-time settings, a competing risk 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 through 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.