Since the average treatment effect (ATE) measures the change in social welfare, even if positive, there is a risk of negative effect on, say, some 10% of the population. Assessing such risk is difficult, however, because any one individual treatment effect (ITE) is never observed, so the 10% worst-affected cannot be identified, while distributional treatment effects only compare the first deciles within each treatment group, which does not correspond to any 10%-subpopulation. In this paper we consider how to nonetheless assess this important risk measure, formalized as the conditional value at risk (CVaR) of the ITE-distribution. We leverage the availability of pre-treatment covariates and characterize the tightest-possible upper and lower bounds on ITE-CVaR given by the covariate-conditional average treatment effect (CATE) function. We then proceed to study how to estimate these bounds efficiently from data and construct confidence intervals. This is challenging even in randomized experiments as it requires understanding the distribution of the unknown CATE function, which can be very complex if we use rich covariates so as to best control for heterogeneity. We develop a debiasing method that overcomes this and prove it enjoys favorable statistical properties even when CATE and other nuisances are estimated by black-box machine learning or even inconsistently. Studying a hypothetical change to French job-search counseling services, our bounds and inference demonstrate a small social benefit entails a negative impact on a substantial subpopulation.
The Predictive Approaches to Treatment Effect Heterogeneity statement focused on baseline risk as a robust predictor of treatment effect and provided guidance on risk-based assessment of treatment effect heterogeneity in the RCT setting. The aim of this study was to extend this approach to the observational setting using a standardized scalable framework. The proposed framework consists of five steps: 1) definition of the research aim, i.e., the population, the treatment, the comparator and the outcome(s) of interest; 2) identification of relevant databases; 3) development of a prediction mode…
Objective: To compare different risk-based methods for optimal prediction of treatment effects. Methods: We simulated RCT data using diverse assumptions for the average treatment effect, a baseline prognostic index of risk (PI), the shape of its interaction with treatment (none, linear, quadratic or non-monotonic), and the magnitude of treatment-related harms (none or constant independent of the PI). We predicted absolute benefit using: models with a constant relative treatment effect; stratification in quarters of the PI; models including a linear interaction of treatment with the PI; models …
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.
We are interested in the estimation of average treatment effects based on right-censored data of an observational study. We focus on causal inference of differences between t-year absolute event risks in a situation with competing risks. We derive doubly robust estimation equations and implement estimators for the nuisance parameters based on working regression models for the outcome, the censoring and the treatment distribution conditional on auxiliary baseline covariates. We use the functional delta method to show that our estimators are regular asymptotically linear estimators and estimate …
A Guide to Estimating Conditional Average Treatment Effects in Competing Risks Settings
Department of Statistics, TU Dortmund University（图恩-多特蒙德大学统计学系）；Research Center Trustworthy Data Science and Security, University Alliance Ruhr (UA Ruhr)（鲁尔大学联盟可信数据科学与安全研究中心）；Institute for Mathematics, University of Augsburg（艾希施泰特大学数学研究所）
Objectives: Clinical prediction models (CPMs) can inform decision-making concerning treatment initiation. Here, one requires predicted risks assuming that no treatment is given. This is challenging since CPMs are often derived in datasets where patients receive treatment; moreover, treatment can commence post-baseline - treatment drop-ins. This study presents a novel approach of using marginal structural models (MSMs) to adjust for treatment drop-in. Study Design and Setting: We illustrate the use of MSMs in the CPM framework through simulation studies, representing randomised controlled trial…
Treatment effects vary across different patients and estimation of this variability is important for clinical decisions. The aim is to develop a model to estimate the benefit of alternative treatment options for individual patients. Hence, we developed a two-stage prediction model for heterogeneous treatment effects, by combining prognosis research and network meta-analysis methods when individual patient data is available. In a first stage, we develop a prognostic model and we predict the baseline risk of the outcome. In the second stage, we use this baseline risk score from the first stage a…
IBM Research（IBM研究院）；Massachusetts Institute of Technology（麻省理工学院）
Inferring causal individual treatment effect (ITE) from observational data is a challenging problem whose difficulty is exacerbated by the presence of treatment assignment bias. In this work, we propose a new way to estimate the ITE using the domain generalization framework of invariant risk minimization (IRM). IRM uses data from multiple domains, learns predictors that do not exploit spurious domain-dependent factors, and generalizes better to unseen domains. We propose an IRM-based ITE estimator aimed at tackling treatment assignment bias when there is little support overlap between the cont…
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 …
Stanford Graduate School of Business（斯坦福大学商学院）；University of California at San Diego（加州大学圣地亚哥分校）
Understanding the effect of a particular treatment or a policy pertains to many areas of interest, ranging from political economics, marketing to healthcare. In this paper, we develop a non-parametric algorithm for detecting the effects of treatment over time in the context of Synthetic Controls. The method builds on counterfactual predictions from many algorithms without necessarily assuming that the algorithms correctly capture the model. We introduce an inferential procedure for detecting treatment effects and show that the testing procedure is asymptotically valid for stationary, beta mixi…
Many popular methods for building confidence intervals on causal effects under high-dimensional confounding require strong "ultra-sparsity" assumptions that may be difficult to validate in practice. To alleviate this difficulty, we here study a new method for average treatment effect estimation that yields asymptotically exact confidence intervals assuming that either the conditional response surface or the conditional probability of treatment allows for an ultra-sparse representation (but not necessarily both). This guarantee allows us to provide valid inference for average treatment effect i…
I introduce a general, Fisher-style randomization testing framework to conduct nearly exact inference about the lack of effect of a binary treatment in the presence of very few, large clusters when the treatment effect is identified across clusters. The proposed randomization test formalizes and extends the intuitive notion of generating null distributions by assigning placebo treatments to untreated clusters. I show that under simple and easily verifiable conditions, the placebo test leads to asymptotically valid inference in a very large class of empirically relevant models. Examples discuss…
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.
In observational studies, treatments are typically not randomized and therefore estimated treatment effects may be subject to confounding bias. The instrumental variable (IV) design plays the role of a quasi-experimental handle since the IV is associated with the treatment and only affects the outcome through the treatment. In this paper, we present a novel framework for identification and inference using an IV for the marginal average treatment effect amongst the treated (ETT) in the presence of unmeasured confounding. For inference, we propose three different semiparametric approaches: (i) i…
In this supplementary appendix we provide additional results, omitted proofs and extensive simulations that complement the analysis of the main text (arXiv:1201.0224).
To maximize clinical benefit, clinicians routinely tailor treatment to the individual characteristics of each patient, where individualized treatment rules are needed and are of significant research interest to statisticians. In the covariate-adjusted randomization clinical trial with many covariates, we model the treatment effect with an unspecified function of a single index of the covariates and leave the baseline response completely arbitrary. We devise a class of estimators to consistently estimate the treatment effect function and its associated index while bypassing the estimation of th…
In this article we estimate confidence regions of the common measures of (baseline, treatment effect) in observational studies, where the measure of baseline is baseline risk or baseline odds while the measure of treatment effect is odds ratio, risk difference, risk ratio or attributable fraction, and where confounding is controlled in estimation of both baseline and treatment effect. To avoid high complexity of the normal approximation method and the parametric or non-parametric bootstrap method, we obtain confidence regions for measures of (baseline, treatment effect) by generating approxima…
The fundamental problem in treatment effect estimation from observational data is confounder identification and balancing. Most of the previous methods realized confounder balancing by treating all observed pre-treatment variables as confounders, ignoring further identifying confounders and non-confounders. In general, not all the observed pre-treatment variables are confounders that refer to the common causes of the treatment and the outcome, some variables only contribute to the treatment and some only contribute to the outcome. Balancing those non-confounders, including instrumental variabl…
It is important to make robust inference of the conditional average treatment effect from observational data, but this becomes challenging when the confounder is multivariate or high-dimensional. In this article, we propose a double dimension reduction method, which reduces the curse of dimensionality as much as possible while keeping the nonparametric merit. We identify the central mean subspace of the conditional average treatment effect using dimension reduction. A nonparametric regression with prior dimension reduction is also used to impute counterfactual outcomes. This step helps improve…
The practical importance of inference with robustness against large bandwidths for causal effects in regression discontinuity and kink designs is widely recognized. Existing robust methods cover many cases, but do not handle uniform inference for CDF and quantile processes in fuzzy designs, despite its use in the recent literature in empirical microeconomics. In this light, this paper extends the literature by developing a unified framework of inference with robustness against large bandwidths that applies to uniform inference for quantile treatment effects in fuzzy designs, as well as all the…
Regression Not-to-the-Mean: An Oddity of Regression, Illustrated with the Risk of Overdose Deaths
Department of Mathematics and Statistics, Boston University（波士顿大学数学与统计系）；Division of Infectious Diseases and Global Public Health, University of California San Diego（加州大学圣地亚哥分校传染病与全球公共卫生部）
Department of Mathematics（数学系）；King’s College London（伦敦国王学院）；Mathematical Institute（数学研究所）；Leiden University（莱顿大学）
Bayesian approaches have become increasingly popular in causal inference problems due to their conceptual simplicity, excellent performance and in-built uncertainty quantification ('posterior credible sets'). We investigate Bayesian inference for average treatment effects from observational data, which is a challenging problem due to the missing counterfactuals and selection bias. Working in the standard potential outcomes framework, we propose a data-driven modification to an arbitrary (nonparametric) prior based on the propensity score that corrects for the first-order posterior bias, thereb…
There are many settings where researchers are interested in estimating average treatment effects and are willing to rely on the unconfoundedness assumption, which requires that the treatment assignment be as good as random conditional on pre-treatment variables. The unconfoundedness assumption is often more plausible if a large number of pre-treatment variables are included in the analysis, but this can worsen the performance of standard approaches to treatment effect estimation. In this paper, we develop a method for de-biasing penalized regression adjustments to allow sparse regression metho…
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…
In semicompeting risks problems, nonterminal time-to-event outcomes such as time to hospital readmission are subject to truncation by death. These settings are often modeled with illness-death models for the hazards of the terminal and nonterminal events, but evaluating causal treatment effects with hazard models is problematic due to conditioning on survival (a post-treatment outcome) that is embedded in the definition of a hazard. Extending an existing survivor average causal effect (SACE) estimand, we frame the evaluation of treatment effects in the context of semicompeting risks with princ…
Causal inference in a program evaluation setting faces the problem of external validity when the treatment effect in the target population is different from the treatment effect identified from the population of which the sample is representative. This paper focuses on a situation where such discrepancy arises by a stratified sampling design based on the individual treatment status and other characteristics. In such settings, the design probability is known from the sampling design but the target population depends on the underlying population share vector which is often unknown, and except fo…
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…
Covariate-adaptive randomization schemes such as the minimization and stratified permuted blocks are often applied in clinical trials to balance treatment assignments across prognostic factors. The existing theoretical developments on inference after covariate-adaptive randomization are mostly limited to situations where a correct model between the response and covariates can be specified or the randomization method has well-understood properties. Based on stratification with covariate levels utilized in randomization and a further adjusting for covariates not used in randomization, in this ar…
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…
Bias Mitigation in Matched Observational Studies with Continuous Treatments: Calipered Non-Bipartite Matching and Bias-Corrected Estimation and Inference
Treatment effect estimation by comparing observed and predicted outcomes: conditions for valid inference and practical illustration
University Medical Center Utrecht, Utrecht University（乌得勒支大学医学中心，乌得勒支大学）；University Medical Center Groningen（格罗宁根大学医学中心）；MAASTRO Clinic（MAASTRO诊所）；Erasmus MC Cancer Institute（伊拉斯姆斯癌症研究所）
We consider estimation and inference on average treatment effects under unconfoundedness conditional on the realizations of the treatment variable and covariates. Given nonparametric smoothness and/or shape restrictions on the conditional mean of the outcome variable, we derive estimators and confidence intervals (CIs) that are optimal in finite samples when the regression errors are normal with known variance. In contrast to conventional CIs, our CIs use a larger critical value that explicitly takes into account the potential bias of the estimator. When the error distribution is unknown, feas…
In this paper, we study the estimation and inference of the quantile treatment effect under covariate-adaptive randomization. We propose two estimation methods: (1) the simple quantile regression and (2) the inverse propensity score weighted quantile regression. For the two estimators, we derive their asymptotic distributions uniformly over a compact set of quantile indexes, and show that, when the treatment assignment rule does not achieve strong balance, the inverse propensity score weighted estimator has a smaller asymptotic variance than the simple quantile regression estimator. For the in…
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 …
We propose a method for conducting asymptotically valid inference for treatment effects in a multi-valued treatment framework where the number of units in the treatment arms can be small and do not grow with the sample size. We accomplish this by casting the model as a semi-/non-parametric conditional quantile model and using known finite sample results about the law of the indicator function that defines the conditional quantile. Our framework allows for structural functions that are non-additively separable, with flexible functional forms and heteroskedasticy in the residuals, and it also en…
University College London（伦敦大学学院）；University of Oxford（牛津大学）
We address the estimation of conditional average treatment effects (CATEs) for structured treatments (e.g., graphs, images, texts). Given a weak condition on the effect, we propose the generalized Robinson decomposition, which (i) isolates the causal estimand (reducing regularization bias), (ii) allows one to plug in arbitrary models for learning, and (iii) possesses a quasi-oracle convergence guarantee under mild assumptions. In experiments with small-world and molecular graphs we demonstrate that our approach outperforms prior work in CATE estimation.
