In many causal inference settings, the treatment of interest is not directly observed; instead, one or more error-prone proxy measurements are available, creating a fundamental identification challenge. Building on identification methods for hidden treatments with proxies, we develop a general semiparametric framework for causal effect estimation that accommodates several common estimands. Hidden treatment models differ fundamentally from classical missing-data settings, where semiparametric theory relies on positivity requiring a nonzero probability of observing a complete case for each treatment value. Here this assumption fails by design because the true treatment is never observed, creating new challenges for semiparametric characterization and efficient estimation. To overcome these challenges, we develop a new semiparametric characterization for hidden treatment models by deriving a formal mapping between the orthogonal complement to the nuisance tangent space, which contains all influence functions for a causal functional in the oracle full-data model, and its counterpart in the observed hidden treatment model. This mapping gives closed-form observed-data influence functions and yields the semiparametric efficiency bound. It also leads to semiparametric efficient estimators with a new form of multiple robustness or mixed bias property, enabling inference with nonparametric nuisance estimators. A further challenge is that some nuisance functions depend on the hidden treatment, preventing direct use of standard nonparametric regression methods. We introduce an iterative estimation algorithm and establish its large-sample properties. Simulations demonstrate the finite-sample performance of the proposed estimators, and an application estimates the causal effect of Alzheimer's disease on hippocampal volume using data from the Alzheimer's Disease Neuroimaging Initiative.
We introduce a new method to estimate the Markov equivalence class of a directed acyclic graph (DAG) in the presence of hidden variables, in settings where the underlying DAG among the observed variables is sparse, and there are a few hidden variables that have a direct effect on many of the observed ones. Building on the so-called low rank plus sparse framework, we suggest a two-stage approach which first removes the effect of the hidden variables, and then estimates the Markov equivalence class of the underlying DAG under the assumption that there are no remaining hidden variables. This appr…
Suppose that a sequence of treatments are assigned to influence an outcome of interest that occurs after the last treatment. Between treatments there exist time-dependent covariates that may be posttreatment variables of the earlier treatments and confounders of the subsequent treatments. In this article, we develop a parametric approach to inference of the causal effect of the treatment sequence on the outcome called the sequential causal effect. We construct a point parametrization for the conditional distribution of an outcome given all treatments and time-dependent covariates, in which the…
We generalize the proximal g-formula of Miao, Geng, and Tchetgen Tchetgen (2018) for causal inference under unobserved confounding using proxy variables. Specifically, we show that the formula holds true for all causal models in a certain equivalence class, and this class contains models in which the total number of levels for the set of unobserved confounders can be arbitrarily larger than the number of levels of each proxy variable. Although straightforward to obtain, the result can be significant for applications. Simulations corroborate our formal arguments.
Using Heteroscedasticity for Causal Discovery under Hidden Confounding
School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Center for Applied Statistics and School of Statistics, Renmin University of China（中国人民大学应用统计科学中心及统计学院）；School of Mathematical Sciences and Center for Statistical Science, Peking University（北京大学数学科学学院及统计科学中心）
One of the main tasks of causal inference is estimating well-defined causal parameters. One of the main causal parameters is the average causal effect (ACE) - the expected value of the individual level causal effects in the target population. For binary treatments, the individual level causal effect is defined as contrast between potential outcomes. For continuous outcomes, however, there are many such contrasts in finite samples, thus hampering their use as a useful summary of the causal relationship. Here, we proposed a generalized version of the ACE, where individual level causal effects ar…
The propensity score is a common tool for estimating the causal effect of a binary treatment in observational data. In this setting, matching, subclassification, imputation, or inverse probability weighting on the propensity score can reduce the initial covariate bias between the treatment and control groups. With more than two treatment options, however, estimation of causal effects requires additional assumptions and techniques, the implementations of which have varied across disciplines. This paper reviews current methods, and it identifies and contrasts the treatment effects that each one …
Cause-effect relationships are typically evaluated by comparing outcome responses to binary treatment values, representing two arms of a hypothetical randomized controlled trial. However, in certain applications, treatments of interest are continuous and multidimensional. For example, understanding the causal relationship between severity of radiation therapy, summarized by a multidimensional vector of radiation exposure values and post-treatment side effects is a problem of clinical interest in radiation oncology. An appropriate strategy for making interpretable causal conclusions is to reduc…
Randomized clinical trials (RCTs) are ideal for estimating causal effects, because the distributions of background covariates are similar in expectation across treatment groups. When estimating causal effects using observational data, matching is a commonly used method to replicate the covariate balance achieved in a RCT. Matching algorithms have a rich history dating back to the mid-1900s, but have been used mostly to estimate causal effects between two treatment groups. When there are more than two treatments, estimating causal effects requires additional assumptions and techniques. We propo…
Consider a situation with two treatments, the first of which is randomized but the second is not, and the multifactor version of this. Interest is in treatment effects, defined using standard factorial notation. We define estimators for the treatment effects and explore their properties when there is information about the nonrandomized treatment assignment and when there is no information on the assignment of the nonrandomized treatment. We show when and how hidden treatments can bias estimators and inflate their sampling variances.
We apply causal forests to a dataset derived from the National Study of Learning Mindsets, and consider resulting practical and conceptual challenges. In particular, we discuss how causal forests use estimated propensity scores to be more robust to confounding, and how they handle data with clustered errors.
We develop a Bayesian nonparametric (BNP) approach to evaluate the causal effect of treatment in a randomized trial where a nonterminal event may be censored by a terminal event, but not vice versa (i.e., semi-competing risks). Based on the idea of principal stratification, we define a novel estimand for the causal effect of treatment on the nonterminal event. We introduce identification assumptions, indexed by a sensitivity parameter, and show how to draw inference using our BNP approach. We conduct simulation studies and illustrate our methodology using data from a brain cancer trial.
Johns Hopkins Bloomberg School of Public Health（约翰霍普金斯大学布隆伯格公共卫生学院）；Boston University（波士顿大学）；Johns Hopkins Whiting School of Engineering（约翰霍普金斯大学惠廷工程学院）
In causal estimation problems, the parameter of interest is often only partially identified, implying that the parameter cannot be recovered exactly, even with infinite data. Here, we study Bayesian inference for partially identified treatment effects in multi-treatment causal inference problems with unobserved confounding. In principle, inferring the partially identified treatment effects is natural under the Bayesian paradigm, but the results can be highly sensitive to parameterization and prior specification, often in surprising ways. It is thus essential to understand which aspects of the …
Researchers are often interested in analyzing conditional treatment effects. One variant of this is "causal moderation," which implies that intervention upon a third (moderator) variable would alter the treatment effect. This study considers the conditions under which causal moderation can be identified and presents a generalized framework for estimating causal moderation effects given randomized treatments and non-randomized moderators. As part of the estimation process, it allows researchers to implement their preferred method of covariate adjustment, including parametric and non-parametric …
Causal effects are commonly defined as comparisons of the potential outcomes under treatment and control, but this definition is threatened by the possibility that the treatment or control condition is not well-defined, existing instead in more than one version. A simple, widely applicable analysis is proposed to address the possibility that the treatment or control condition exists in two versions with two different treatment effects. This analysis loses no power in the main comparison of treatment and control, provides additional information about version effects, and controls the family-wis…
Causal inference is known to be very challenging when only observational data are available. Randomized experiments are often costly and impractical and in instrumental variable regression the number of instruments has to exceed the number of causal predictors. It was recently shown in Peters et al. [2016] that causal inference for the full model is possible when data from distinct observational environments are available, exploiting that the conditional distribution of a response variable is invariant under the correct causal model. Two shortcomings of such an approach are the high computatio…
The micro-randomized trial (MRT) is a sequential randomized experimental design to empirically evaluate the effectiveness of mobile health (mHealth) intervention components that may be delivered at hundreds or thousands of decision points. MRTs have motivated a new class of causal estimands, termed "causal excursion effects", for which semiparametric inference can be conducted via a weighted, centered least squares criterion (Boruvka et al., 2018). Existing methods assume between-subject independence and non-interference. Deviations from these assumptions often occur. In this paper, causal exc…
Causal inference in observational studies can be challenging when confounders are subject to missingness. Generally, the identification of causal effects is not guaranteed even under restrictive parametric model assumptions when confounders are missing not at random. To address this, We propose a general framework to establish the identification of causal effects when confounders are subject to treatment-independent missingness, which means that the missing data mechanism is independent of the treatment, given the outcome and possibly missing confounders. We give special consideration to commo…
Matching estimators for average treatment effects are widely used in the binary treatment setting, in which missing potential outcomes are imputed as the average of observed outcomes of all matches for each unit. With more than two treatment groups, however, estimation using matching requires additional techniques. In this paper, we propose a nearest-neighbors matching estimator for use with multiple, nominal treatments, and use simulations to show that this method is precise and has coverage levels that are close to nominal. In addition, we implement the proposed inference methods to examine …
In most nonrandomized observational studies, differences between treatment groups may arise not only due to the treatment but also because of the effect of confounders. Therefore, causal inference regarding the treatment effect is not as straightforward as in a randomized trial. To adjust for confounding due to measured covariates, a variety of methods based on the potential outcomes framework are used to estimate average treatment effects. One of the key assumptions is treatment positivity, which states that the probability of treatment is bounded away from zero and one for any possible combi…
In non-network settings, encouragement designs have been widely used to analyze causal effects of a treatment, policy, or intervention on an outcome of interest when randomizing the treatment was considered impractical or when compliance to treatment cannot be perfectly enforced. Unfortunately, such questions related to treatment compliance have received less attention in network settings and the most well-studied experimental design in networks, the two-stage randomization design, requires perfect compliance with treatment. The paper proposes a new experimental design called peer encouragemen…
Clinical studies sometimes encounter truncation by death, rendering outcomes undefined. Statistical analysis based solely on observed survivors may give biased results because the characteristics of survivors differ between treatment groups. By principal stratification, the survivor average causal effect was proposed as a causal estimand defined in always-survivors. However, this estimand is not identifiable when there is unmeasured confounding between the treatment assignment and survival or outcome process. In this paper, we consider the comparison between an aggressive treatment and a conse…
Estimating causal effects with propensity scores relies upon the availability of treated and untreated units observed at each value of the estimated propensity score. In settings with strong confounding, limited so-called "overlap" in propensity score distributions can undermine the empirical basis for estimating causal effects and yield erratic finite-sample performance of existing estimators. We propose a Bayesian procedure designed to estimate causal effects in settings where there is limited overlap in propensity score distributions. Our method relies on the posterior predictive treatment …
Many spatial phenomena exhibit treatment interference where treatments at one location may affect the response at other locations. Because interference violates the stable unit treatment value assumption, standard methods for causal inference do not apply. We propose a new causal framework to recover direct and spill-over effects in the presence of spatial interference, taking into account that treatments at nearby locations are more influential than treatments at locations further apart. Under the no unmeasured confounding assumption, we show that a generalized propensity score is sufficient …
We present a class of inequality constraints on the set of distributions induced by local interventions on variables governed by a causal Bayesian network, in which some of the variables remain unmeasured. We derive bounds on causal effects that are not directly measured in randomized experiments. We derive instrumental inequality type of constraints on nonexperimental distributions. The results have applications in testing causal models with observational or experimental data.
Assessing the causal effects of interventions on ordinal outcomes is an important objective of many educational and behavioral studies. Under the potential outcomes framework, we can define causal effects as comparisons between the potential outcomes under treatment and control. However, unfortunately, the average causal effect, often the parameter of interest, is difficult to interpret for ordinal outcomes. To address this challenge, we propose to use two causal parameters, which are defined as the probabilities that the treatment is beneficial and strictly beneficial for the experimental uni…
A Versatile Causal Discovery Framework to Allow Causally-Related Hidden Variables
University of California San Diego（加州大学圣地亚哥分校）；Carnegie Mellon University（卡内基梅隆大学）；Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）；KDDI Research（KDDI研究所）
University of South Australia（南澳大学）；Hefei University of Technology（合肥工业大学）
Causal effect estimation from observational data is a crucial but challenging task. Currently, only a limited number of data-driven causal effect estimation methods are available. These methods either provide only a bound estimation of the causal effect of a treatment on the outcome, or generate a unique estimation of the causal effect, but making strong assumptions on data and having low efficiency. In this paper, we identify a practical problem setting and propose an approach to achieving unique and unbiased estimation of causal effects from data with hidden variables. For the approach, we h…
Causal mediation analysis is used to evaluate direct and indirect causal effects of a treatment on an outcome of interest through an intermediate variable or a mediator.It is difficult to identify the direct and indirect causal effects because the mediator cannot be randomly assigned in many real applications. In this article, we consider a causal model including latent confounders between the mediator and the outcome. We present sufficient conditions for identifying the direct and indirect effects and propose an approach for estimating them. The performance of the proposed approach is evaluat…
When drawing causal inferences about the effects of multiple treatments on clustered survival outcomes using observational data, we need to address implications of the multilevel data structure, multiple treatments, censoring and unmeasured confounding for causal analyses. Few off-the-shelf causal inference tools are available to simultaneously tackle these issues. We develop a flexible random-intercept accelerated failure time model, in which we use Bayesian additive regression trees to capture arbitrarily complex relationships between censored survival times and pre-treatment covariates and …
Causal inference is widely used in various fields, such as biology, psychology and economics, etc. In observational studies, we need to balance the covariates before estimating causal effect. This study extends the one-dimensional entropy balancing method to multiple dimensions to balance the covariates. Both parametric and nonparametric methods are proposed to estimate the causal effect of multivariate continuous treatments and theoretical properties of the two estimations are provided. Furthermore, the simulation results show that the proposed method is better than other methods in various c…
Amazon（亚马逊）；Massachusetts Institute of Technology（麻省理工学院）
We extend the definition of the marginal causal effect to the continuous treatment setting and develop a novel characterization of causal bias in the framework of structural causal models. We prove that our derived bias expression is zero if, and only if, the causal effect is identifiable via covariate adjustment. We show that under some restrictions on the structural equations, the causal bias can be estimated efficiently and allows for causal regularization of predictive probabilistic models. We demonstrate the effectiveness of our method for causal bias quantification in various settings wh…
Estimating treatment effects plays a crucial role in causal inference, having many real-world applications like policy analysis and decision making. Nevertheless, estimating treatment effects in the longitudinal setting in the presence of hidden confounders remains an extremely challenging problem. Recently, there is a growing body of work attempting to obtain unbiased ITE estimates from time-dynamic observational data by ignoring the possible existence of hidden confounders. Additionally, many existing works handling hidden confounders are not applicable for continuous-time settings. In this …
