It is a truth universally acknowledged that an observed association without known mechanism must be in want of a causal estimate. However, causal estimation from observational data often relies on the (untestable) assumption of `no unobserved confounding'. Violations of this assumption can induce bias in effect estimates. In principle, such bias could invalidate or reverse the conclusions of a study. However, in some cases, we might hope that the influence of unobserved confounders is weak relative to a `large' estimated effect, so the qualitative conclusions are robust to bias from unobserved confounding. The purpose of this paper is to develop \emph{Austen plots}, a sensitivity analysis tool to aid such judgments by making it easier to reason about potential bias induced by unobserved confounding. We formalize confounding strength in terms of how strongly the confounder influences treatment assignment and outcome. For a target level of bias, an Austen plot shows the minimum values of treatment and outcome influence required to induce that level of bias. Domain experts can then make subjective judgments about whether such strong confounders are plausible. To aid this judgment, the Austen plot additionally displays the estimated influence strength of (groups of) the observed covariates. Austen plots generalize the classic sensitivity analysis approach of Imbens [Imb03]. Critically, Austen plots allow any approach for modeling the observed data and producing the initial estimate. We illustrate the tool by assessing biases for several real causal inference problems, using a variety of machine learning approaches for the initial data analysis. Code is available at https://github.com/anishazaveri/austen_plots
The Graduate University for Advanced Studies（综合研究大学院大学）；The Institute of Statistical Mathematics（统计数学研究所）
NOTE: This preprint has a flawed theoretical formulation. Please avoid it and refer to the ICLR22 publication https://openreview.net/forum?id=q7n2RngwOM. Also, arXiv:2109.15062 contains some new ideas on unobserved Confounding. As an important problem of causal inference, we discuss the identification and estimation of treatment effects under unobserved confounding. Representing the confounder as a latent variable, we propose Intact-VAE, a new variant of variational autoencoder (VAE), motivated by the prognostic score that is sufficient for identifying treatment effects. We theoretically show …
The estimation of the treatment effect is often biased in the presence of unobserved confounding variables which are commonly referred to as hidden variables. Although a few methods have been recently proposed to handle the effect of hidden variables, these methods often overlook the possibility of any interaction between the observed treatment variable and the unobserved covariates. In this work, we address this shortcoming by studying a multivariate response regression problem with both unobserved and heterogeneous confounding variables of the form $Y=A^T X+ B^T Z+ \sum_{j=1}^{p} C^T_j X_j Z…
University of Cambridge（剑桥大学）；The Alan Turing Institute（艾伦·图灵研究所）；University of California Los Angeles（加州大学洛杉矶分校）
Unobserved confounding is one of the greatest challenges for causal discovery. The case in which unobserved variables have a widespread effect on many of the observed ones is particularly difficult because most pairs of variables are conditionally dependent given any other subset, rendering the causal effect unidentifiable. In this paper we show that beyond conditional independencies, under the principle of independent mechanisms, unobserved confounding in this setting leaves a statistical footprint in the observed data distribution that allows for disentangling spurious and causal effects. Us…
Unobserved confounding is a central barrier to drawing causal inferences from observational data. Several authors have recently proposed that this barrier can be overcome in the case where one attempts to infer the effects of several variables simultaneously. In this paper, we present two simple, analytical counterexamples that challenge the general claims that are central to these approaches. In addition, we show that nonparametric identification is impossible in this setting. We discuss practical implications, and suggest alternatives to the methods that have been proposed so far in this lin…
We consider causal inference in the presence of unobserved confounding. We study the case where a proxy is available for the unobserved confounding in the form of a network connecting the units. For example, the link structure of a social network carries information about its members. We show how to effectively use the proxy to do causal inference. The main idea is to reduce the causal estimation problem to a semi-supervised prediction of both the treatments and outcomes. Networks admit high-quality embedding models that can be used for this semi-supervised prediction. We show that the method …
We study the problem of learning conditional average treatment effects (CATE) from observational data with unobserved confounders. The CATE function maps baseline covariates to individual causal effect predictions and is key for personalized assessments. Recent work has focused on how to learn CATE under unconfoundedness, i.e., when there are no unobserved confounders. Since CATE may not be identified when unconfoundedness is violated, we develop a functional interval estimator that predicts bounds on the individual causal effects under realistic violations of unconfoundedness. Our estimator t…
We study the problem of learning personalized decision policies from observational data while accounting for possible unobserved confounding. Previous approaches, which assume unconfoundedness, i.e., that no unobserved confounders affect both the treatment assignment as well as outcome, can lead to policies that introduce harm rather than benefit when some unobserved confounding is present, as is generally the case with observational data. Instead, since policy value and regret may not be point-identifiable, we study a method that minimizes the worst-case estimated regret of a candidate policy…
In this work, we propose an approach for assessing sensitivity to unobserved confounding in studies with multiple outcomes. We demonstrate how prior knowledge unique to the multi-outcome setting can be leveraged to strengthen causal conclusions beyond what can be achieved from analyzing individual outcomes in isolation. We argue that it is often reasonable to make a shared confounding assumption, under which residual dependence amongst outcomes can be used to simplify and sharpen sensitivity analyses. We focus on a class of factor models for which we can bound the causal effects for all outcom…
Causal Discovery of Linear Non-Gaussian Causal Models with Unobserved Confounding
TUM School of Computation, Information and Technology（慕尼黑工业大学计算、信息与技术学院）；Technical University of Munich（慕尼黑工业大学）；University of British Columbia（不列颠哥伦比亚大学）；Munich Center for Machine Learning（慕尼黑机器学习中心）
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 …
Munich Center for Machine Learning（慕尼黑机器学习中心）；LMU Munich（慕尼黑大学）
Estimating conditional average treatment effects (CATEs) from observational data is relevant in many fields such as personalized medicine. However, in practice, the treatment assignment is usually confounded by unobserved variables and thus introduces bias. A remedy to remove the bias is the use of instrumental variables (IVs). Such settings are widespread in medicine (e.g., trials where the treatment assignment is used as binary IV). In this paper, we propose a novel, multiply robust machine learning framework, called MRIV, for estimating CATEs using binary IVs and thus yield an unbiased CATE…
University of Oxford（牛津大学）；Technion – Israel Institute of Technology（以色列理工学院）
We study the problem of learning conditional average treatment effects (CATE) from high-dimensional, observational data with unobserved confounders. Unobserved confounders introduce ignorance -- a level of unidentifiability -- about an individual's response to treatment by inducing bias in CATE estimates. We present a new parametric interval estimator suited for high-dimensional data, that estimates a range of possible CATE values when given a predefined bound on the level of hidden confounding. Further, previous interval estimators do not account for ignorance about the CATE associated with s…
For observational studies, we study the sensitivity of causal inference when treatment assignments may depend on unobserved confounders. We develop a loss minimization approach for estimating bounds on the conditional average treatment effect (CATE) when unobserved confounders have a bounded effect on the odds ratio of treatment selection. Our approach is scalable and allows flexible use of model classes in estimation, including nonparametric and black-box machine learning methods. Based on these bounds for the CATE, we propose a sensitivity analysis for the average treatment effect (ATE). Our…
Confounding by unmeasured spatial variables has received some attention in the spatial statistics and causal inference literatures, but concepts and approaches have remained largely separated. In this paper, we aim to bridge these distinct strands of statistics by considering unmeasured spatial confounding within a causal inference framework, and estimating effects using outcome regression tools popular within the spatial literature. First, we show how using spatially correlated random effects in the outcome model, an approach common among spatial statisticians, does not necessarily mitigate b…
Understanding and quantifying cause and effect is an important problem in many domains. The generally-agreed solution to this problem is to perform a randomised controlled trial. However, even when randomised controlled trials can be performed, they usually have relatively short duration's due to cost considerations. This makes learning long-term causal effects a very challenging task in practice, since the long-term outcome is only observed after a long delay. In this paper, we study the identification and estimation of long-term treatment effects when both experimental and observational data…
Causal inference is difficult in the presence of unobserved confounders. We introduce the instrumented common confounding (ICC) approach to (nonparametrically) identify causal effects with instruments, which are exogenous only conditional on some unobserved common confounders. The ICC approach is most useful in rich observational data with multiple sources of unobserved confounding, where instruments are at most exogenous conditional on some unobserved common confounders. Suitable examples of this setting are various identification problems in the social sciences, nonlinear dynamic panels, and…
We consider recovering causal structure from multivariate observational data. We assume the data arise from a linear structural equation model (SEM) in which the idiosyncratic errors are allowed to be dependent in order to capture possible latent confounding. Each SEM can be represented by a graph where vertices represent observed variables, directed edges represent direct causal effects, and bidirected edges represent dependence among error terms. Specifically, we assume that the true model corresponds to a bow-free acyclic path diagram; i.e., a graph that has at most one edge between any pai…
Unobserved confounding is a major hurdle for causal inference from observational data. Confounders---the variables that affect both the causes and the outcome---induce spurious non-causal correlations between the two. Wang & Blei (2018) lower this hurdle with "the blessings of multiple causes," where the correlation structure of multiple causes provides indirect evidence for unobserved confounding. They leverage these blessings with an algorithm, called the deconfounder, that uses probabilistic factor models to correct for the confounders. In this paper, we take a causal graphical view of the …
Unobserved spatial confounding variables are prevalent in environmental and ecological applications where the system under study is complex and the data are often observational. Instrumental variables (IVs) are a common way to address unobserved confounding; however, the efficacy of using IVs on spatial confounding is largely unknown. This paper explores the effectiveness of IVs in this situation -- with particular attention paid to the spatial scale of the instrument. We show that, in case of spatially-dependent treatments, IVs are most effective when they vary at a finer spatial resolution t…
Studying the relationship between neuroanatomy and cognitive decline due to Alzheimer's has been a major research focus in the last decade. However, to infer cause-effect relationships rather than simple associations from observational data, we need to (i) express the causal relationships leading to cognitive decline in a graphical model, and (ii) ensure the causal effect of interest is identifiable from the collected data. We derive a causal graph from the current clinical knowledge on cause and effect in the Alzheimer's disease continuum, and show that identifiability of the causal effect re…
Observing the unobserved confounding through its effects: toward randomized trial-like estimates from real-world survival data
Operations Research Center, Massachusetts Institute of Technology（麻省理工学院运营研究中心）；Sloan School of Management, Massachusetts Institute of Technology（麻省理工学院斯隆管理学院）；Department of Surgery, Memorial Sloan Kettering Cancer Center（纪念斯隆凯特琳癌症中心外科部）；Charité-Universitätsmedizin Berlin, Corporate Member of Freie Universität Berlin and Humboldt-Universität zu Berlin（柏林夏里特医学院，柏林自由大学和洪堡大学联合成员）
New York University（纽约大学）；Columbia University（哥伦比亚大学）
Causal inference from observational data requires assumptions. These assumptions range from measuring confounders to identifying instruments. Traditionally, causal inference assumptions have focused on estimation of effects for a single treatment. In this work, we construct techniques for estimation with multiple treatments in the presence of unobserved confounding. We develop two assumptions based on shared confounding between treatments and independence of treatments given the confounder. Together, these assumptions lead to a confounder estimator regularized by mutual information. For this e…
To estimate direct and indirect effects of an exposure on an outcome from observed data strong assumptions about unconfoundedness are required. Since these assumptions cannot be tested using the observed data, a mediation analysis should always be accompanied by a sensitivity analysis of the resulting estimates. In this article we propose a sensitivity analysis method for parametric estimation of direct and indirect effects when the exposure, mediator and outcome are all binary. The sensitivity parameters consist of the correlation between the error terms of the mediator and outcome models, th…
Questions concerning mediated causal effects are of great interest in psychology, cognitive science, medicine, social science, public health, and many other disciplines. For instance, about 60% of recent papers published in leading journals in social psychology contain at least one mediation test (Rucker, Preacher, Tormala, & Petty, 2011). Standard parametric approaches to mediation analysis employ regression models, and either the "difference method" (Judd & Kenny, 1981), more common in epidemiology, or the "product method" (Baron & Kenny, 1986), more common in the social sciences. In this pa…
Participant noncompliance, in which participants do not follow their assigned treatment protocol, often obscures the causal relationship between treatment and treatment effect in randomized trials. In the longitudinal setting, the G-computation algorithm can adjust for confounding to estimate causal effects. Typically, G-computation assumes that both 1) compliance is observed; and 2) the densities of the confounders can be correctly specified. We aim to develop a G-computation estimator in the setting where both assumptions are violated. For 1), in place of unobserved compliance, we substitute…
We examine interval estimation of the effect of a treatment T on an outcome Y given the existence of an unobserved confounder U. Using Hölder's inequality, we derive a set of bounds on the confounding bias |E[Y|T=t]-E[Y|do(T=t)]| based on the degree of unmeasured confounding (i.e., the strength of the connection U->T, and the strength of U->Y). These bounds are tight either when U is independent of T or when U is independent of Y given T (when there is no unobserved confounding). We focus on a special case of this bound depending on the total variation distance between the distributions p(U) a…
Propensity score matching is a common tool for adjusting for observed confounding in observational studies, but is known to have limitations in the presence of unmeasured confounding. In many settings, researchers are confronted with spatially-indexed data where the relative locations of the observational units may serve as a useful proxy for unmeasured confounding that varies according to a spatial pattern. We develop a new method, termed Distance Adjusted Propensity Score Matching (DAPSm) that incorporates information on units' spatial proximity into a propensity score matching procedure. We…
MPI for Intelligent Systems（马克斯·普朗克智能系统研究所）；University of Cambridge（剑桥大学）；Microsoft Research（微软研究院）；University of Oxford（牛津大学）；The Alan Turing Institute（艾伦·图灵研究所）；University College London（伦敦大学学院）
Causal approaches to fairness have seen substantial recent interest, both from the machine learning community and from wider parties interested in ethical prediction algorithms. In no small part, this has been due to the fact that causal models allow one to simultaneously leverage data and expert knowledge to remove discriminatory effects from predictions. However, one of the primary assumptions in causal modeling is that you know the causal graph. This introduces a new opportunity for bias, caused by misspecifying the causal model. One common way for misspecification to occur is via unmeasure…
Quantile-Optimal Policy Learning under Unmeasured Confounding
Department of Statistics and Data Science, Yale University（统计与数据科学系，耶鲁大学）；Department of Decision Sciences, George Washington University（决策科学系，乔治华盛顿大学）；Cowles Foundation for Research in Economics, Yale University（经济学研究基金会，耶鲁大学）
Stanford University（斯坦福大学）；Hoover Institution（胡佛研究所）；Wharton School, University of Pennsylvania（宾夕法尼亚大学沃顿商学院）；Princeton University（普林斯顿大学）
The empirical practice of using factor models to adjust for shared, unobserved confounders, $\mathbf{Z}$, in observational settings with multiple treatments, $\mathbf{A}$, is widespread in fields including genetics, networks, medicine, and politics. Wang and Blei (2019, WB) formalizes these procedures and develops the "deconfounder," a causal inference method using factor models of $\mathbf{A}$ to estimate "substitute confounders," $\hat{\mathbf{Z}}$, then estimating treatment effects by regressing the outcome, $\mathbf{Y}$, on part of $\mathbf{A}$ while adjusting for $\hat{\mathbf{Z}}$. WB cl…
Instrumental and Proximal Causal Inference with Gaussian Processes
Department of Statistics and Actuarial Science, University of Hong Kong（香港大学统计与精算科学系）；Rational Intelligence Lab, CISPA Helmholtz Center for Information Security（CISPA霍普夫信息安全中心理性智能实验室）；Australian Institute for Machine Learning, Adelaide University（阿德莱德大学澳大利亚机器学习研究所）；College of Computing & Data Science, Nanyang Technological University（南洋理工大学计算与数据科学学院）
Mediation analysis is a strategy for understanding the mechanisms by which treatments or interventions affect later outcomes. Mediation analysis is frequently applied in randomized trial settings, but typically assumes: a) that randomized assignment is the exposure of interest as opposed to actual take-up of the intervention, and b) no unobserved confounding of the mediator-outcome relationship. In contrast to the rich literature on instrumental variable (IV) methods to estimate a total effect of a non-randomized exposure, there has been almost no research into using IV as an identification st…
