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 claim the deconfounder is unbiased when there are no single-cause confounders and $\hat{\mathbf{Z}}$ is "pinpointed." We clarify pinpointing requires each confounder to affect infinitely many treatments. We prove under these assumptions, a naïve semiparametric regression of $\mathbf{Y}$ on $\mathbf{A}$ is asymptotically unbiased. Deconfounder variants nesting this regression are therefore also asymptotically unbiased, but variants using $\hat{\mathbf{Z}}$ and subsets of causes require further untestable assumptions. We replicate every deconfounder analysis with available data and find it fails to consistently outperform naïve regression. In practice, the deconfounder produces implausible estimates in WB's case study to movie earnings: estimates suggest comic author Stan Lee's cameo appearances causally contributed \$15.5 billion, most of Marvel movie revenue. We conclude neither approach is a viable substitute for careful research design in real-world applications.
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…
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 …
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…
Difference-in-differences (diff-in-diff) is a study design that compares outcomes of two groups (treated and comparison) at two time points (pre- and post-treatment) and is widely used in evaluating new policy implementations. For instance, diff-in-diff has been used to estimate the effect that increasing minimum wage has on employment rates and to assess the Affordable Care Act's effect on health outcomes. Although diff-in-diff appears simple, potential pitfalls lurk. In this paper, we discuss one such complication: time-varying confounding. We provide rigorous definitions for confounders in …
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…
Institute for Analytical Sociology（分析社会学研究所）；Linköping University（林雪平大学）；Chalmers University of Technology（查尔姆斯理工大学）
Observational studies of causal effects require adjustment for confounding factors. In the tabular setting, where these factors are well-defined, separate random variables, the effect of confounding is well understood. However, in public policy, ecology, and in medicine, decisions are often made in non-tabular settings, informed by patterns or objects detected in images (e.g., maps, satellite or tomography imagery). Using such imagery for causal inference presents an opportunity because objects in the image may be related to the treatment and outcome of interest. In these cases, we rely on the…
Valid estimation of treatment effects from observational data requires proper control of confounding. If the number of covariates is large relative to the number of observations, then controlling for all available covariates is infeasible. In cases where a sparsity condition holds, variable selection or penalization can reduce the dimension of the covariate space in a manner that allows for valid estimation of treatment effects. In this article, we propose matching on both the estimated propensity score and the estimated prognostic scores when the number of covariates is large relative to the …
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…
We address the problem of inferring the causal effect of an exposure on an outcome across space, using observational data. The data is possibly subject to unmeasured confounding variables which, in a standard approach, must be adjusted for by estimating a nuisance function. Here we develop a method that eliminates the nuisance function, while mitigating the resulting errors-in-variables. The result is a robust and accurate inference method for spatially varying heterogeneous causal effects. The properties of the method are demonstrated on synthetic as well as real data from Germany and the US.
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 (…
Cornell University（康奈尔大学）；Cornell Tech（康奈尔理工学院）；New York University（纽约大学）；Technion（以色列理工学院）
Observational data is increasingly used as a means for making individual-level causal predictions and intervention recommendations. The foremost challenge of causal inference from observational data is hidden confounding, whose presence cannot be tested in data and can invalidate any causal conclusion. Experimental data does not suffer from confounding but is usually limited in both scope and scale. We introduce a novel method of using limited experimental data to correct the hidden confounding in causal effect models trained on larger observational data, even if the observational data does no…
Marginal structural models are a popular tool for investigating the effects of time-varying treatments, but they require an assumption of no unobserved confounders between the treatment and outcome. With observational data, this assumption may be difficult to maintain, and in studies with panel data, many researchers use fixed effects models to purge the data of time-constant unmeasured confounding. Unfortunately, traditional linear fixed effects models are not suitable for estimating the effects of time-varying treatments, since they can only estimate lagged effects under implausible assumpti…
Adjustment for Unmeasured Spatial Confounding in Settings of Continuous Exposure Conditional on the Binary Exposure Status: Conditional Generalized Propensity Score-Based Spatial Matching
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…
We consider estimating the conditional average treatment effect for everyone by eliminating confounding and selection bias. Unfortunately, randomized clinical trials (RCTs) eliminate confounding but impose strict exclusion criteria that prevent sampling of the entire clinical population. Observational datasets are more inclusive but suffer from confounding. We therefore analyze RCT and observational data simultaneously in order to extract the strengths of each. Our solution builds upon Difference in Differences (DD), an algorithm that eliminates confounding from observational data by comparing…
Inferring the causal effect of a non-randomly assigned exposure on an outcome requires adjusting for common causes of the exposure and outcome to avoid biased conclusions. Notwithstanding the efforts investigators routinely make to measure and adjust for such common causes (or confounders), some confounders typically remain unmeasured, raising the prospect of biased inference in observational studies. Therefore, it is crucial that investigators can practically assess their substantive conclusions' relative (in)sensitivity to potential unmeasured confounding. In this article, we propose a sensi…
Columbia University（哥伦比亚大学）；Google Research（谷歌研究院）；The University of Chicago（芝加哥大学）
The defining challenge for causal inference from observational data is the presence of `confounders', covariates that affect both treatment assignment and the outcome. To address this challenge, practitioners collect and adjust for the covariates, hoping that they adequately correct for confounding. However, including every observed covariate in the adjustment runs the risk of including `bad controls', variables that induce bias when they are conditioned on. The problem is that we do not always know which variables in the covariate set are safe to adjust for and which are not. To address this …
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…
The University of Chicago（芝加哥大学）；Google Research（谷歌研究院）
Consider the problem of estimating the causal effect of some attribute of a text document; for example: what effect does writing a polite vs. rude email have on response time? To estimate a causal effect from observational data, we need to adjust for confounding aspects of the text that affect both the treatment and outcome -- e.g., the topic or writing level of the text. These confounding aspects are unknown a priori, so it seems natural to adjust for the entirety of the text (e.g., using a transformer). However, causal identification and estimation procedures rely on the assumption of overla…
Technion IIT（以色列理工学院）；University of California, Los Angeles（加州大学洛杉矶分校）
The paper provides a simple test for deciding, from a given causal diagram, whether two sets of variables have the same bias-reducing potential under adjustment. The test requires that one of the following two conditions holds: either (1) both sets are admissible (i.e., satisfy the back-door criterion) or (2) the Markov boundaries surrounding the manipulated variable(s) are identical in both sets. Applications to covariate selection and model testing are discussed.
We propose a new approach for estimating causal effects when the exposure is measured with error and confounding adjustment is performed via a generalized propensity score (GPS). Using validation data, we propose a regression calibration (RC)-based adjustment for a continuous error-prone exposure combined with GPS to adjust for confounding (RC-GPS). The outcome analysis is conducted after transforming the corrected continuous exposure into a categorical exposure. We consider confounding adjustment in the context of GPS subclassification, inverse probability treatment weighting (IPTW) and match…
Drawing causal inference with observational studies is the central pillar of many disciplines. One sufficient condition for identifying the causal effect is that the treatment-outcome relationship is unconfounded conditional on the observed covariates. It is often believed that the more covariates we condition on, the more plausible this unconfoundedness assumption is. This belief has had a huge impact on practical causal inference, suggesting that we should adjust for all pretreatment covariates. However, when there is unmeasured confounding between the treatment and outcome, estimators adjus…
For settings with a binary treatment and a binary outcome, instrumental variables can be used to construct bounds on a causal treatment effect. With continuous outcomes, meaningful bounds are more difficult to obtain because the domain of the outcome is typically unrestricted. In this paper, we combine an instrumental variable and subjective assumptions in the context of an obser- vational cohort study of HIV-infected women to construct meaningful bounds on the initial-stage causal effect of antiretroviral therapy on CD4 count. The subjective assumptions are encoded in terms of the potential o…
Harvard T.H. Chan School of Public Health（哈佛大学陈曾熙公共卫生学院）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
Estimating the causal effects of a spatially-varying intervention on a spatially-varying outcome may be subject to non-local confounding (NLC), a phenomenon that can bias estimates when the treatments and outcomes of a given unit are dictated in part by the covariates of other nearby units. In particular, NLC is a challenge for evaluating the effects of environmental policies and climate events on health-related outcomes such as air pollution exposure. This paper first formalizes NLC using the potential outcomes framework, providing a comparison with the related phenomenon of causal interferen…
University of Texas at Austin（德克萨斯大学奥斯汀分校）；Chalmers University of Technology（查尔姆斯理工大学）；Linköping University（林雪平大学）
Observational studies require adjustment for confounding factors that are correlated with both the treatment and outcome. In the setting where the observed variables are tabular quantities such as average income in a neighborhood, tools have been developed for addressing such confounding. However, in many parts of the developing world, features about local communities may be scarce. In this context, satellite imagery can play an important role, serving as a proxy for the confounding variables otherwise unobserved. In this paper, we study confounder adjustment in this non-tabular setting, where…
The idea of "stratified medicine" is an important driver of methodological research on the identification of predictive biomarkers. Most methods proposed so far for this purpose have been developed for the use on randomized data only. However, especially for rare cancers, data from clinical registries or observational studies might be the only available data source. For such data, methods for an unbiased estimation of the average treatment effect are well established. Research on confounder adjustment when investigating the heterogeneity of treatment effects and the variables responsible for t…
In this paper, we discuss causal inference on the efficacy of a treatment or medication on a time-to-event outcome with competing risks. Although the treatment group can be randomized, there can be confoundings between the compliance and the outcome. Unmeasured confoundings may exist even after adjustment for measured co- variates. Instrumental variable (IV) methods are commonly used to yield consistent estimations of causal parameters in the presence of unmeasured confoundings. Based on a semi-parametric additive hazard model for the subdistribution hazard, we pro- pose an instrumental variab…
Which Regularized Propensity-Score and Doubly Robust Methods Are Best Calibrated When Exposures or Outcomes Are Rare? A Plasmode Study of Proxy-Based Confounding Adjustment
School of Population and Public Health, University of British Columbia（英属哥伦比亚大学公共卫生与人口健康学院）
In observational studies, identification of ATEs is generally achieved by assuming that the correct set of confounders has been measured and properly included in the relevant models. Because this assumption is both strong and untestable, a sensitivity analysis should be performed. Common approaches include modeling the bias directly or varying the propensity scores to probe the effects of a potential unmeasured confounder. In this paper, we take a novel approach whereby the sensitivity parameter is the "proportion of unmeasured confounding:" the proportion of units for whom the treatment is no…
Estimates of the effects of treatment on cost from observational studies are subject to bias if there are unmeasured confounders. It is therefore advisable in practice to assess the potential magnitude of such biases. We derive a general adjustment formula for loglinear models of mean cost and explore special cases under plausible assumptions about the distribution of the unmeasured confounder. We assess the performance of the adjustment by simulation, in particular, examining robustness to a key assumption of conditional independence between the unmeasured and measured covariates given the tr…
In the absence of a randomized experiment, a key assumption for drawing causal inference about treatment effects is the ignorable treatment assignment. Violations of the ignorability assumption may lead to biased treatment effect estimates. Sensitivity analysis helps gauge how causal conclusions will be altered in response to the potential magnitude of departure from the ignorability assumption. However, sensitivity analysis approaches for unmeasured confounding in the context of multiple treatments and binary outcomes are scarce. We propose a flexible Monte Carlo sensitivity analysis approach…
(This comment has been updated to respond to Wang and Blei's rejoinder [arXiv:1910.07320].) The premise of the deconfounder method proposed in "Blessings of Multiple Causes" by Wang and Blei [arXiv:1805.06826], namely that a variable that renders multiple causes conditionally independent also controls for unmeasured multi-cause confounding, is incorrect. This can be seen by noting that no fact about the observed data alone can be informative about ignorability, since ignorability is compatible with any observed data distribution. Methods to control for unmeasured confounding may be valid with …
Inferring the causal effect of a treatment on an outcome in an observational study requires adjusting for observed baseline confounders to avoid bias. However, adjusting for all observed baseline covariates, when only a subset are confounders of the effect of interest, is known to yield potentially inefficient and unstable estimators of the treatment effect. Furthermore, it raises the risk of finite-sample bias and bias due to model misspecification. For these stated reasons, confounder (or covariate) selection is commonly used to determine a subset of the available covariates that is sufficie…
In causal mediation studies that decompose an average treatment effect into a natural indirect effect (NIE) and a natural direct effect (NDE), examples of post-treatment confounding are abundant. Past research has generally considered it infeasible to adjust for a post-treatment confounder of the mediator-outcome relationship due to incomplete information: it is observed under the actual treatment condition while missing under the counterfactual treatment condition. This study proposes a new sensitivity analysis strategy for handling post-treatment confounding and incorporates it into weightin…
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…
