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In a recent issue of this journal, Philip Dawid (2021) proposes a framework for causal inference that is based on statistical decision theory and that is, in many aspects, compatible with the familiar framework of causal graphs (e.g., Directed Acyclic Graphs (DAGs)). This editorial compares the methodological features of the two frameworks as well as their epistemological basis.

We present a method for assessing the sensitivity of the true causal effect to unmeasured confounding. The method requires the analyst to set two intuitive parameters. Otherwise, the method is assumption free. The method returns an interval that contains the true causal effect and whose bounds are arbitrarily sharp, i.e., practically attainable. We show experimentally that our bounds can be tighter than those obtained by the method of Ding and VanderWeele, which, moreover, requires to set one more parameter than our method. Finally, we extend our method to bound the natural direct and indirect effects when there are measured mediators and unmeasured exposure–outcome confounding.

Mediation analysis has been used in many disciplines to explain the mechanism or process that underlies an observed relationship between an exposure variable and an outcome variable via the inclusion of mediators. Decompositions of the total effect (TE) of an exposure variable into effects characterizing mediation pathways and interactions have gained an increasing amount of interest in the last decade. In this work, we develop decompositions for scenarios where two mediators are causally sequential or non-sequential. Current developments in this area have primarily focused on either decompositions without interaction components or with interactions but assuming no causally sequential order between the mediators. We propose a new concept called natural mediated interaction (MI) effect that captures the two-way and three-way interactions for both scenarios and extends the two-way MIs in the literature. We develop a unified approach for decomposing the TE into the effects that are due to mediation only, interaction only, both mediation and interaction, neither mediation nor interaction within the counterfactual framework. Finally, we compare our proposed decomposition to an existing method in a non-sequential two-mediator scenario using simulated data, and illustrate the proposed decomposition for a sequential two-mediator scenario using a real data analysis.

Instrumental variables allow for quantification of cause and effect relationships even in the absence of interventions. To achieve this, a number of causal assumptions must be met, the most important of which is the independence assumption, which states that the instrument and any confounding factor must be independent. However, if this independence condition is not met, can we still work with imperfect instrumental variables? Imperfect instruments can manifest themselves by violations of the instrumental inequalities that constrain the set of correlations in the scenario. In this article, we establish a quantitative relationship between such violations of instrumental inequalities and the minimal amount of measurement dependence required to explain them for the case of discrete observed variables. As a result, we provide adapted inequalities that are valid in the presence of a relaxed measurement dependence assumption in the instrumental scenario. This allows for the adaptation of existing and new lower bounds on the average causal effect for instrumental scenarios with binary outcomes. Finally, we discuss our findings in the context of quantum mechanics.

Recent interest in the external validity of prediction models (i.e., the problem of different train and test distributions, known as dataset shift ) has produced many methods for finding predictive distributions that are invariant to dataset shifts and can be used for prediction in new, unseen environments. However, these methods consider different types of shifts and have been developed under disparate frameworks, making it difficult to theoretically analyze how solutions differ with respect to stability and accuracy. Taking a causal graphical view, we use a flexible graphical representation to express various types of dataset shifts. Given a known graph of the data generating process, we show that all invariant distributions correspond to a causal hierarchy of graphical operators, which disable the edges in the graph that are responsible for the shifts. The hierarchy provides a common theoretical underpinning for understanding when and how stability to shifts can be achieved, and in what ways stable distributions can differ. We use it to establish conditions for minimax optimal performance across environments, and derive new algorithms that find optimal stable distributions. By using this new perspective, we empirically demonstrate that that there is a tradeoff between minimax and average performance.

Adjusting for covariates is a well-established method to estimate the total causal effect of an exposure variable on an outcome of interest. Depending on the causal structure of the mechanism under study, there may be different adjustment sets, equally valid from a theoretical perspective, leading to identical causal effects. However, in practice, with finite data, estimators built on different sets may display different precisions. To investigate the extent of this variability, we consider the simplest non-trivial non-linear model of a v-structure on three nodes for binary data. We explicitly compute and compare the variance of the two possible different causal estimators. Further, by going beyond leading-order asymptotics, we show that there are parameter regimes where the set with the asymptotically optimal variance does depend on the edge coefficients, a result that is not captured by the recent leading-order developments for general causal models. As a practical consequence, the adjustment set selection needs to account for the relative magnitude of the relationships between variables with respect to the sample size and cannot rely on purely graphical criteria.

Applying causal methods to fields such as healthcare, marketing, and economics receives increasing interest. In particular, optimising the individual-treatment-effect – often referred to as uplift modelling – has peaked in areas such as precision medicine and targeted advertising. While existing techniques have proven useful in many settings, they suffer vividly in a dynamic environment. To address this issue, we propose a novel optimisation target that is easily incorporated in bandit algorithms. Incorporating this target creates a causal model which we name an uplifted contextual multi-armed bandit . Experiments on real and simulated data show the proposed method to effectively improve upon the state-of-the-art. All our code is made available online at https://github.com/vub-dl/u-cmab.

In causal inference, a variety of causal effect estimands have been studied, including the sample, uncensored, target, conditional, optimal subpopulation, and optimal weighted average treatment effects. Ad hoc methods have been developed for each estimand based on inverse probability weighting (IPW) and on outcome regression modeling, but these may be sensitive to model misspecification, practical violations of positivity, or both. The contribution of this article is twofold. First, we formulate the generalized average treatment effect (GATE) to unify these causal estimands as well as their IPW estimates. Second, we develop a method based on Kernel optimal matching (KOM) to optimally estimate GATE and to find the GATE most easily estimable by KOM, which we term the Kernel optimal weighted average treatment effect. KOM provides uniform control on the conditional mean squared error of a weighted estimator over a class of models while simultaneously controlling for precision. We study its theoretical properties and evaluate its comparative performance in a simulation study. We illustrate the use of KOM for GATE estimation in two case studies: comparing spine surgical interventions and studying the effect of peer support on people living with HIV.

We study the selection of adjustment sets for estimating the interventional mean under an individualized treatment rule. We assume a non-parametric causal graphical model with, possibly, hidden variables and at least one adjustment set composed of observable variables. Moreover, we assume that observable variables have positive costs associated with them. We define the cost of an observable adjustment set as the sum of the costs of the variables that comprise it. We show that in this setting there exist adjustment sets that are minimum cost optimal, in the sense that they yield non-parametric estimators of the interventional mean with the smallest asymptotic variance among those that control for observable adjustment sets that have minimum cost. Our results are based on the construction of a special flow network associated with the original causal graph. We show that a minimum cost optimal adjustment set can be found by computing a maximum flow on the network, and then finding the set of vertices that are reachable from the source by augmenting paths. The optimaladj Python package implements the algorithms introduced in this article.

This article studies the treatment effect models in which individuals are classified into unobserved groups based on heterogeneous treatment rules. By using a finite mixture approach, we propose a marginal treatment effect (MTE) framework in which the treatment choice and outcome equations can be heterogeneous across groups. Under the availability of instrumental variables specific to each group, we show that the MTE for each group can be separately identified. On the basis of our identification result, we propose a two-step semiparametric procedure for estimating the group-wise MTE. We illustrate the usefulness of the proposed method with an application to economic returns to college education.

Recently, there has been increasing interest in the use of heavily restricted randomization designs which enforce balance on observed covariates in randomized controlled trials. However, when restrictions are strict, there is a risk that the treatment effect estimator will have a very high mean squared error (MSE). In this article, we formalize this risk and propose a novel combinatoric-based approach to describe and address this issue. First, we validate our new approach by re-proving some known properties of complete randomization and restricted randomization. Second, we propose a novel diagnostic measure for restricted designs that only use the information embedded in the combinatorics of the design. Third, we show that the variance of the MSE of the difference-in-means estimator in a randomized experiment is a linear function of this diagnostic measure. Finally, we identify situations in which restricted designs can lead to an increased risk of getting a high MSE and discuss how our diagnostic measure can be used to detect such designs. Our results have implications for any restricted randomization design and can be used to evaluate the trade-off between enforcing balance on observed covariates and avoiding too restrictive designs.

Causal mediation analysis is complicated with multiple effect definitions that require different sets of assumptions for identification. This article provides a systematic explanation of such assumptions. We define five potential outcome types whose means are involved in various effect definitions. We tackle their mean/distribution’s identification, starting with the one that requires the weakest assumptions and gradually building up to the one that requires the strongest assumptions. This presentation shows clearly why an assumption is required for one estimand and not another, and provides a succinct table from which an applied researcher could pick out the assumptions required for identifying the causal effects they target. Using a running example, the article illustrates the assembling and consideration of identifying assumptions for a range of causal contrasts. For several that are commonly encountered in the literature, this exercise clarifies that identification requires weaker assumptions than those often stated in the literature. This attention to the details also draws attention to the differences in the positivity assumption for different estimands, with practical implications. Clarity on the identifying assumptions of these various estimands will help researchers conduct appropriate mediation analyses and interpret the results with appropriate caution given the plausibility of the assumptions.

Analysts often use data-driven approaches to supplement their knowledge when selecting covariates for effect estimation. Multiple variable selection procedures for causal effect estimation have been devised in recent years, but additional developments are still required to adequately address the needs of analysts. We propose a generalized Bayesian causal effect estimation (GBCEE) algorithm to perform variable selection and produce double robust (DR) estimates of causal effects for binary or continuous exposures and outcomes. GBCEE employs a prior distribution that targets the selection of true confounders and predictors of the outcome for the unbiased estimation of causal effects with reduced standard errors. The Bayesian machinery allows GBCEE to directly produce inferences for its estimate. In simulations, GBCEE was observed to perform similarly or to outperform DR alternatives. Its ability to directly produce inferences is also an important advantage from a computational perspective. The method is finally illustrated for the estimation of the effect of meeting physical activity recommendations on the risk of hip or upper-leg fractures among older women in the study of osteoporotic fractures. The 95% confidence interval produced by GBCEE is 61% narrower than that of a DR estimator adjusting for all potential confounders in this illustration.

Residual confounding is a common source of bias in observational studies. In this article, we build upon a series of sensitivity analyses methods for residual confounding developed by Brumback et al. and Chiba whose sensitivity parameters are constructed to quantify deviation from conditional exchangeability, given measured confounders. These sensitivity parameters are combined with the observed data to produce a “bias-corrected” estimate of the causal effect of interest. We provide important generalizations of these sensitivity analyses, by allowing for arbitrary exposures and a wide range of different causal effect measures, through the specification of the target causal effect as a parameter in a generalized linear model with the arbitrary link function. We show how our generalized sensitivity analysis can be easily implemented with standard software, and how its sensitivity parameters can be calibrated against measured confounders. We demonstrate our sensitivity analysis with an application to publicly available data from a cohort study of behavior patterns and coronary heart disease.

Estimation and evaluation of individualized treatment rules have been studied extensively, but real-world treatment resource constraints have received limited attention in existing methods. We investigate a setting in which treatment is intervened upon based on covariates to optimize the mean counterfactual outcome under treatment cost constraints when the treatment cost is random. In a particularly interesting special case, an instrumental variable corresponding to encouragement to treatment is intervened upon with constraints on the proportion receiving treatment. For such settings, we first develop a method to estimate optimal individualized treatment rules. We further construct an asymptotically efficient plug-in estimator of the corresponding average treatment effect relative to a given reference rule.

Statistical power is often a concern for clustered randomized control trials (RCTs) due to variance inflation from design effects and the high cost of adding study clusters (such as hospitals, schools, or communities). While covariate pre-specification can improve power for estimating regression-adjusted average treatment effects (ATEs), further precision gains can be achieved through covariate selection once primary outcomes have been collected. This article uses design-based methods underlying clustered RCTs to develop Lasso methods for the post-hoc selection of covariates for ATE estimation that avoids a lack of transparency and model overfitting. Our focus is on two-stage estimators: in the first stage, Lasso estimation is conducted using data on cluster-level averages or sums, and in the second stage, standard ATE estimators are adjusted for covariates using the first-stage Lasso results. We discuss l 1 {l}_{1} consistency of the estimated Lasso coefficients, asymptotic normality of the ATE estimators, and design-based variance estimation. The nonparametric approach applies to continuous, binary, and discrete outcomes. We present simulation results and demonstrate the method using data from a federally funded clustered RCT testing the effects of school-based programs promoting behavioral health.

It is well-known that dichotomization can cause bias and loss of efficiency in estimation. One can easily construct examples where adjusting for a dichotomized confounder causes bias in causal estimation. There are additional examples in the literature where adjusting for a dichotomized confounder can be more biased than not adjusting at all. The message is clear, do not dichotomize. What is unclear is if there are scenarios where adjusting for the dichotomized confounder always leads to lower bias than not adjusting. We propose several sets of conditions that characterize scenarios where one should always adjust for the dichotomized confounder to reduce bias. We then highlight scenarios where the decision to adjust should be made more cautiously. To our knowledge, this is the first formal presentation of conditions that give information about when one should and potentially should not adjust for a dichotomized confounder.

The study of causal inference has seen recent momentum in machine learning and artificial intelligence (AI), particularly in the domains of transfer learning, reinforcement learning, automated diagnostics, and explainability (among others). Yet, despite its increasing application to address many of the boundaries in modern AI, causal topics remain absent in most AI curricula. This work seeks to bridge this gap by providing classroom-ready introductions that integrate into traditional topics in AI, suggests intuitive graphical tools for the application to both new and traditional lessons in probabilistic and causal reasoning, and presents avenues for instructors to impress the merit of climbing the “causal hierarchy” to address problems at the levels of associational, interventional, and counterfactual inference. Finally, this study shares anecdotal instructor experiences, successes, and challenges integrating these lessons at multiple levels of education.

I thank Ilya Shpitser for his comments on my article, and discuss the use of models with restricted interventions.

I thank Judea Pearl for his discussion of my paper and respond to the points he raises. In particular, his attachment to unaugmented directed acyclic graphs has led to a misapprehension of my own proposals. I also discuss the possibilities for developing a non-manipulative understanding of causality.

Recent studies have indicated that it is possible to protect individuals from HIV infection using passive infusion of monoclonal antibodies. However, in order for monoclonal antibodies to confer robust protection, the antibodies must be capable of neutralizing many possible strains of the virus. This is particularly challenging in the context of a highly diverse pathogen like HIV. It is therefore of great interest to leverage existing observational data sources to discover antibodies that are able to neutralize HIV viruses via residues where existing antibodies show modest protection. Such information feeds directly into the clinical trial pipeline for monoclonal antibody therapies by providing information on (i) whether and to what extent combinations of antibodies can generate superior protection and (ii) strategies for analyzing past clinical trials to identify in vivo evidence of antibody resistance. These observational data include genetic features of many diverse HIV genetic sequences, as well as in vitro measures of antibody resistance. The statistical learning problem we are interested in is developing statistical methodology that can be used to analyze these data to identify important genetic features that are significantly associated with antibody resistance. This is a challenging problem owing to the high-dimensional and strongly correlated nature of the genetic sequence data. To overcome these challenges, we propose an outcome-adaptive, collaborative targeted minimum loss-based estimation approach using random forests. We demonstrate via simulation that the approach enjoys important statistical benefits over existing approaches in terms of bias, mean squared error, and type I error. We apply the approach to the Compile, Analyze, and Tally Nab Panels database to identify AA positions that are potentially causally related to resistance to neutralization by several different antibodies.

Randomized controlled trials (RCTs) sometimes test interventions that aim to improve existing services targeted to a subset of individuals identified after randomization. Accordingly, the treatment could affect the composition of service recipients and the offered services. With such bias, intention-to-treat estimates using data on service recipients and nonrecipients may be difficult to interpret. This article develops causal estimands and inverse probability weighting (IPW) estimators for complier populations in these settings, using a generalized estimating equation approach that adjusts the standard errors for estimation error in the IPW weights. While our focus is on more general clustered RCTs, the methods also apply (reduce) to nonclustered RCTs. Simulations show that the estimators achieve nominal confidence interval coverage under the assumed identification conditions. An empirical application demonstrates the methods using data from a large-scale RCT testing the effects of early childhood services on children’s cognitive development scores. An R program for estimation is available for download.

Randomized controlled trials (RCTs) are often considered the gold standard for estimating causal effect, but they may lack external validity when the population eligible to the RCT is substantially different from the target population. Having at hand a sample of the target population of interest allows us to generalize the causal effect. Identifying the treatment effect in the target population requires covariates to capture all treatment effect modifiers that are shifted between the two sets. Standard estimators then use either weighting (IPSW), outcome modeling (G-formula), or combine the two in doubly robust approaches (AIPSW). However, such covariates are often not available in both sets. In this article, after proving L 1 {L}^{1} -consistency of these three estimators, we compute the expected bias induced by a missing covariate, assuming a Gaussian distribution, a continuous outcome, and a semi-parametric model. Under this setting, we perform a sensitivity analysis for each missing covariate pattern and compute the sign of the expected bias. We also show that there is no gain in linearly imputing a partially unobserved covariate. Finally, we study the substitution of a missing covariate by a proxy. We illustrate all these results on simulations, as well as semi-synthetic benchmarks using data from the Tennessee student/teacher achievement ratio (STAR), and a real-world example from critical care medicine.

In the presence of heterogeneity between the randomized controlled trial (RCT) participants and the target population, evaluating the treatment effect solely based on the RCT often leads to biased quantification of the real-world treatment effect. To address the problem of lack of generalizability for the treatment effect estimated by the RCT sample, we leverage observational studies with large samples that are representative of the target population. This article concerns evaluating treatment effects on survival outcomes for a target population and considers a broad class of estimands that are functionals of treatment-specific survival functions, including differences in survival probability and restricted mean survival times. Motivated by two intuitive but distinct approaches, i.e., imputation based on survival outcome regression and weighting based on inverse probability of sampling, censoring, and treatment assignment, we propose a semiparametric estimator through the guidance of the efficient influence function. The proposed estimator is doubly robust in the sense that it is consistent for the target population estimands if either the survival model or the weighting model is correctly specified and is locally efficient when both are correct. In addition, as an alternative to parametric estimation, we employ the nonparametric method of sieves for flexible and robust estimation of the nuisance functions and show that the resulting estimator retains the root- n n consistency and efficiency, the so-called rate-double robustness. Simulation studies confirm the theoretical properties of the proposed estimator and show that it outperforms competitors. We apply the proposed method to estimate the effect of adjuvant chemotherapy on survival in patients with early-stage resected non-small cell lung cancer.