A subvector of predictor that satisfies the ignorability assumption, whose index set is called a sufficient adjustment set, is crucial for conducting reliable causal inference based on observational data. In this paper, we propose a general family of methods to detect all such sets for the first time in the literature, with no parametric assumptions on the outcome models and with flexible parametric and semiparametric assumptions on the predictor within the treatment groups; the latter induces desired sample-level accuracy. We show that the collection of sufficient adjustment sets can uniquely facilitate multiple types of studies in causal inference, including sharpening the estimation of average causal effect and recovering fundamental connections between the outcome and the treatment hidden in the dependence structure of the predictor. These findings are illustrated by simulation studies and a real data example at the end.
We develop finite-population asymptotic theory for covariate adjustment in randomization-based causal inference for 2K factorial designs. In particular, we confirm that both the unadjusted and covariate-adjusted estimators of the factorial effects are asymptotically normal, and the latter is more precise than the former.
The purpose of this work is to improve the efficiency in estimating the average causal effect (ACE) on the survival scale where right-censoring exists and high-dimensional covariate information is available. We propose new estimators using regularized survival regression and survival random forests (SRF) to make the adjustment for the high dimensional covariates to improve efficiency. We study the behavior of the adjusted estimator under mild assumptions and show theoretical guarantees that the proposed estimators are more efficient than the unadjusted ones asymptotically when using SRF for ad…
We study Bayesian approaches to causal inference via propensity score regression. Much of the Bayesian literature on propensity score methods have relied on approaches that cannot be viewed as fully Bayesian in the context of conventional `likelihood times prior' posterior inference; in addition, most methods rely on parametric and distributional assumptions, and presumed correct specification. We emphasize that causal inference is typically carried out in settings of mis-specification, and develop strategies for fully Bayesian inference that reflect this. We focus on methods based on decision…
Identifying effects of actions (treatments) on outcome variables from observational data and causal assumptions is a fundamental problem in causal inference. This identification is made difficult by the presence of confounders which can be related to both treatment and outcome variables. Confounders are often handled, both in theory and in practice, by adjusting for covariates, in other words considering outcomes conditioned on treatment and covariate values, weighed by probability of observing those covariate values. In this paper, we give a complete graphical criterion for covariate adjustme…
Causal effect estimation from observational data is a challenging problem, especially with high dimensional data and in the presence of unobserved variables. The available data-driven methods for tackling the problem either provide an estimation of the bounds of a causal effect (i.e. nonunique estimation) or have low efficiency. The major hurdle for achieving high efficiency while trying to obtain unique and unbiased causal effect estimation is how to find a proper adjustment set for confounding control in a fast way, given the huge covariate space and considering unobserved variables. In this…
Choosing the Right Approach at the Right Time: A Comparative Analysis of Causal Effect Estimation using Confounder Adjustment and Instrumental Variables
Confounding seriously impairs our ability to learn about causal relations from observational data. Confounding can be defined as a statistical association between two variables due to inputs from a common source (the confounder). For example, if $Z\rightarrow Y$ and $Z\rightarrow X$, then $X$ and $Y$ will be statistically dependent, even if there are no causal connections between the two. There are several approaches available to adjust for confounding, i.e. to remove, or reduce, the association between two variables due to the confounder. Common adjustment techniques include stratifying the a…
Unmeasured confounding is a threat to causal inference in observational studies. In recent years, use of negative controls to mitigate unmeasured confounding has gained increasing recognition and popularity. Negative controls have a longstanding tradition in laboratory sciences and epidemiology to rule out non-causal explanations, although they have been used primarily for bias detection. Recently, Miao et al. (2018) have described sufficient conditions under which a pair of negative control exposure and outcome variables can be used to nonparametrically identify the average treatment effect (…
The Clean Air Act mandates that the National Ambient Air Quality Standards (NAAQS) must be routinely assessed to protect populations based on the latest science. Therefore, researchers should continue to address whether exposure to levels of air pollution below the NAAQS is harmful to human health. The contentious nature surrounding environmental regulations urges us to cast this question within a causal inference framework. Parametric and semi-parametric regression approaches have been used to estimate the exposure-response (ER) curve between ambient air pollution and health outcomes. Most of…
Confounding bias, missing data, and selection bias are three common obstacles to valid causal inference in the data sciences. Covariate adjustment is the most pervasive technique for recovering casual effects from confounding bias. In this paper, we introduce a covariate adjustment formulation for controlling confounding bias in the presence of missing-not-at-random data and develop a necessary and sufficient condition for recovering causal effects using the adjustment. We also introduce an adjustment formulation for controlling both confounding and selection biases in the presence of missing …
Estimating causal effects from observational data is not always possible due to confounding. Identifying a set of appropriate covariates (adjustment set) and adjusting for their influence can remove confounding bias; however, such a set is typically not identifiable from observational data alone. Experimental data do not have confounding bias, but are typically limited in sample size and can therefore yield imprecise estimates. Furthermore, experimental data often include a limited set of covariates, and therefore provide limited insight into the causal structure of the underlying system. In t…
Adjustment Identification Distance: A gadjid for Causal Structure Learning
School of Mathematics and Statistics, University College Dublin（都柏林大学学院数学与统计学院）；Pioneer Centre for AI, University of Copenhagen（哥本哈根大学人工智能先锋中心）；Department of Mathematical Sciences, University of Copenhagen（哥本哈根大学数学科学系）
Covariate adjustment is a commonly used method for total causal effect estimation. In recent years, graphical criteria have been developed to identify all valid adjustment sets, that is, all covariate sets that can be used for this purpose. Different valid adjustment sets typically provide total effect estimates of varying accuracies. Restricting ourselves to causal linear models, we introduce a graphical criterion to compare the asymptotic variances provided by certain valid adjustment sets. We employ this result to develop two further graphical tools. First, we introduce a simple variance re…
We consider estimation of a total causal effect from observational data via covariate adjustment. Ideally, adjustment sets are selected based on a given causal graph, reflecting knowledge of the underlying causal structure. Valid adjustment sets are, however, not unique. Recent research has introduced a graphical criterion for an 'optimal' valid adjustment set (O-set). For a given graph, adjustment by the O-set yields the smallest asymptotic variance compared to other adjustment sets in certain parametric and non-parametric models. In this paper, we provide three new results on the O-set. Firs…
In observational studies, estimation of a causal effect of a treatment on an outcome relies on proper adjustment for confounding. If the number of the potential confounders ($p$) is larger than the number of observations ($n$), then direct control for all potential confounders is infeasible. Existing approaches for dimension reduction and penalization are generally aimed at predicting the outcome, and are less suited for estimation of causal effects. Under standard penalization approaches (e.g. Lasso), if a variable $X_j$ is strongly associated with the treatment $T$ but weakly with the outcom…
Universidad Torcuato Di Tella（托尔夸托·迪·特利亚大学）；CONICET（阿根廷国家科学与技术研究理事会）；Harvard T. H. Chan School of Public Health（哈佛陈曾熙公共卫生学院）
The method of covariate adjustment is often used for estimation of population average treatment effects in observational studies. Graphical rules for determining all valid covariate adjustment sets from an assumed causal graphical model are well known. Restricting attention to causal linear models, a recent article derived two novel graphical criteria: one to compare the asymptotic variance of linear regression treatment effect estimators that control for certain distinct adjustment sets and another to identify the optimal adjustment set that yields the least squares treatment effect estimator…
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…
Covariate adjustment is a widely used approach to estimate total causal effects from observational data. Several graphical criteria have been developed in recent years to identify valid covariates for adjustment from graphical causal models. These criteria can handle multiple causes, latent confounding, or partial knowledge of the causal structure; however, their diversity is confusing and some of them are only sufficient, but not necessary. In this paper, we present a criterion that is necessary and sufficient for four different classes of graphical causal models: directed acyclic graphs (DAG…
A basic principle in the design of observational studies is to approximate the randomized experiment that would have been conducted under controlled circumstances. Now, linear regression models are commonly used to analyze observational data and estimate causal effects. How do linear regression adjustments in observational studies emulate key features of randomized experiments, such as covariate balance, self-weighted sampling, and study representativeness? In this paper, we provide answers to this and related questions by analyzing the implied (individual-level data) weights of linear regress…
In causal inference confounding may be controlled either through regression adjustment in an outcome model, or through propensity score adjustment or inverse probability of treatment weighting, or both. The latter approaches, which are based on modelling of the treatment assignment mechanism and their doubly robust extensions have been difficult to motivate using formal Bayesian arguments, in principle, for likelihood-based inferences, the treatment assignment model can play no part in inferences concerning the expected outcomes if the models are assumed to be correctly specified. On the other…
The Regression Discontinuity (RD) design is a widely used non-experimental method for causal inference and program evaluation. While its canonical formulation only requires a score and an outcome variable, it is common in empirical work to encounter RD analyses where additional variables are used for adjustment. This practice has led to misconceptions about the role of covariate adjustment in RD analysis, from both methodological and empirical perspectives. In this chapter, we review the different roles of covariate adjustment in RD designs, and offer methodological guidance for its correct us…
Propensity score weighting is a tool for causal inference to adjust for measured confounders. Survey data are often collected under complex sampling designs such as multistage cluster sampling, which presents challenges for propensity score modeling and estimation. In addition, for clustered data, there may also be unobserved cluster effects related to both the treatment and the outcome. When such unmeasured confounders exist and are omitted in the propensity score model, the subsequent propensity score adjustment will be biased. We propose a calibrated propensity score weighting adjustment fo…
We consider the estimation of average and counterfactual treatment effects, under two settings: back-door adjustment and front-door adjustment. The goal in both cases is to recover the treatment effect without having an access to a hidden confounder. This objective is attained by first estimating the conditional mean of the desired outcome variable given relevant covariates (the "first stage" regression), and then taking the (conditional) expectation of this function as a "second stage" procedure. We propose to compute these conditional expectations directly using a regression function to the …
University of California, Berkeley（加州大学伯克利分校）；Columbia University（哥伦比亚大学）；University of Michigan（密歇根大学）
Causal inference from observational datasets often relies on measuring and adjusting for covariates. In practice, measurements of the covariates can often be noisy and/or biased, or only measurements of their proxies may be available. Directly adjusting for these imperfect measurements of the covariates can lead to biased causal estimates. Moreover, without additional assumptions, the causal effects are not point-identifiable due to the noise in these measurements. To this end, we study the partial identification of causal effects given noisy covariates, under a user-specified assumption on th…
Propensity score methods are common for estimating a binary treatment effect when treatment assignment is not randomized. When exposure is measured on an ordinal scale (i.e., low - medium - high), however, propensity score inference requires extensions which have received limited attention. Estimands of possible interest with an ordinal exposure are the average treatment effects between each pair of exposure levels. Using these estimands, it is possible to determine an optimal exposure level. Traditional methods, including dichotomization of the exposure or a series of binary propensity score …
Observational studies can play a useful role in assessing the comparative effectiveness of competing treatments. In a clinical trial the randomization of participants to treatment and control groups generally results in well-balanced groups with respect to possible confounders, which makes the analysis straightforward. However, when analysing observational data, the potential for unmeasured confounding makes comparing treatment effects much more challenging. Causal inference methods such as Instrumental Variable and Prior Even Rate Ratio approaches make it possible to circumvent the need to ad…
When investigators seek to estimate causal effects, they often assume that selection into treatment is based only on observed covariates. Under this identification strategy, analysts must adjust for observed confounders. While basic regression models have long been the dominant method of statistical adjustment, more robust methods based on matching or weighting have become more common. Of late, even more flexible methods based on machine learning methods have been developed for statistical adjustment. These machine learning methods are designed to be black box methods with little input from th…
Adjusting for an unmeasured confounder is generally an intractable problem, but in the spatial setting it may be possible under certain conditions. In this paper, we derive necessary conditions on the coherence between the treatment variable of interest and the unmeasured confounder that ensure the causal effect of the treatment is estimable. We specify our model and assumptions in the spectral domain to allow for different degrees of confounding at different spatial resolutions. The key assumption that ensures identifiability is that confounding present at global scales dissipates at local sc…
Having a large number of covariates can have a negative impact on the quality of causal effect estimation since confounding adjustment becomes unreliable when the number of covariates is large relative to the samples available. Propensity score is a common way to deal with a large covariate set, but the accuracy of propensity score estimation (normally done by logistic regression) is also challenged by large number of covariates. In this paper, we prove that a large covariate set can be reduced to a lower dimensional representation which captures the complete information for adjustment in caus…
