Instrumental variables (IV) estimation suffers selection bias when the analysis conditions on the treatment. Judea Pearl's early graphical definition of instrumental variables explicitly prohibited conditioning on the treatment. Nonetheless, the practice remains common. In this paper, we derive exact analytic expressions for IV selection bias across a range of data-generating models, and for various selection-inducing procedures. We present four sets of results for linear models. First, IV selection bias depends on the conditioning procedure (covariate adjustment vs. sample truncation). Second, IV selection bias due to covariate adjustment is the limiting case of IV selection bias due to sample truncation. Third, in certain models, the IV and OLS estimators under selection bound the true causal effect in large samples. Fourth, we characterize situations where IV remains preferred to OLS despite selection on the treatment. These results broaden the notion of IV selection bias beyond sample truncation, replace prior simulation findings with exact analytic formulas, and enable formal sensitivity analyses.
Instrumental variable (IV) methods are widely used to adjust for the bias in estimating treatment effects caused by unmeasured confounders in observational studies. In this manuscript, we provide empirical and theoretical evidence that the IV methods may result in biased treatment effects if applied on a data set in which subjects are preselected based on their received treatments. We frame this as a selection bias problem and propose a procedure that identifies the treatment effect of interest as a function of a vector of sensitivity parameters. We also list assumptions under which analyzing …
Instrumental variables are widely used to deal with unmeasured confounding in observational studies and imperfect randomized controlled trials. In these studies, researchers often target the so-called local average treatment effect as it is identifiable under mild conditions. In this paper, we consider estimation of the local average treatment effect under the binary instrumental variable model. We discuss the challenges for causal estimation with a binary outcome, and show that surprisingly, it can be more difficult than the case with a continuous outcome. We propose novel modeling and estima…
We propose the instrumental variable regime (IVR) method to estimate the causal effects of multiple sequential treatments. This method serves to address the problem of endogenous selections of sequential treatments. An IVR is a sequence of instrumental variables in which each IV instruments for an endogenous treatment variable. Our proposed method generalizes the LATE model in Imbens and Angrist (1994) from a single treatment to many treatments applied sequentially. More precisely, with the IVR this model allows for estimating the local average treatment regime effects (LATRE), possibly condit…
This paper considers the instrumental variable quantile regression model (Chernozhukov and Hansen, 2005, 2013) with a binary endogenous treatment. It offers two identification results when the treatment status is not directly observed. The first result is that, remarkably, the reduced-form quantile regression of the outcome variable on the instrumental variable provides a lower bound on the structural quantile treatment effect under the stochastic monotonicity condition (Small and Tan, 2007; DiNardo and Lee, 2011). This result is relevant, not only when the treatment variable is subject to mis…
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…
Robins 1997 introduced marginal structural models (MSMs), a general class of counterfactual models for the joint effects of time-varying treatment regimes in complex longitudinal studies subject to time-varying confounding. In his work, identification of MSM parameters is established under a sequential randomization assumption (SRA), which rules out unmeasured confounding of treatment assignment over time. We consider sufficient conditions for identification of the parameters of a subclass, Marginal Structural Mean Models (MSMMs), when sequential randomization fails to hold due to unmeasured c…
Estimating the causal effect of a treatment or exposure for a subpopulation is of great interest in many biomedical and economical studies. Expected shortfall, also referred to as the super-quantile, is an attractive effect-size measure that can accommodate data heterogeneity and aggregate local information of effect over a certain region of interest of the outcome distribution. In this article, we propose the ComplieR Expected Shortfall Treatment Effect (CRESTE) model under an instrumental variable framework to quantity the CRESTE for a binary endogenous treatment variable. By utilizing the s…
The two-stage least-squares (2SLS) estimator is known to be biased when its first-stage fit is poor. I show that better first-stage prediction can alleviate this bias. In a two-stage linear regression model with Normal noise, I consider shrinkage in the estimation of the first-stage instrumental variable coefficients. For at least four instrumental variables and a single endogenous regressor, I establish that the standard 2SLS estimator is dominated with respect to bias. The dominating IV estimator applies James-Stein type shrinkage in a first-stage high-dimensional Normal-means problem follow…
Suppose one is interested in estimating causal effects in the presence of potentially unmeasured confounding with the aid of a valid instrumental variable. This paper investigates the problem of making inferences about the average treatment effect when data are fused from two separate sources, one of which contains information on the treatment and the other contains information on the outcome, while values for the instrument and a vector of baseline covariates are recorded in both. We provide a general set of sufficient conditions under which the average treatment effect is nonparametrically i…
Instrumental variable methods can identify causal effects even when the treatment and outcome are confounded. We study the problem of imperfect measurements of the binary instrumental variable, treatment or outcome. We first consider non-differential measurement errors, that is, the mis-measured variable does not depend on other variables given its true value. We show that the measurement error of the instrumental variable does not bias the estimate, the measurement error of the treatment biases the estimate away from zero, and the measurement error of the outcome biases the estimate toward ze…
Unmeasured confounding is a threat to causal inference and individualized decision making. Similar to Cui and Tchetgen Tchetgen (2020); Qiu et al. (2020); Han (2020a), we consider the problem of identification of optimal individualized treatment regimes with a valid instrumental variable. Han (2020a) provided an alternative identifying condition of optimal treatment regimes using the conditional Wald estimand of Cui and Tchetgen Tchetgen (2020); Qiu et al. (2020) when treatment assignment is subject to endogeneity and a valid binary instrumental variable is available. In this note, we provide …
Instrumental variables are commonly used to estimate effects of a treatment afflicted by unmeasured confounding, and in practice instruments are often continuous (e.g., measures of distance, or treatment preference). However, available methods for continuous instruments have important limitations: they either require restrictive parametric assumptions for identification, or else rely on modeling both the outcome and treatment process well (and require modeling effect modification by all adjustment covariates). In this work we develop the first semiparametric doubly robust estimators of the loc…
This note discusses two recent studies on identification of individualized treatment rules using instrumental variables---Cui and Tchetgen Tchetgen (2020) and Qiu et al. (2020). It also proposes identifying assumptions that are alternative to what is used in both studies.
In a randomized controlled trial, treatment switching (also called contamination or crossover) occurs when a patient initially assigned to one treatment arm changes to another arm during the course of follow-up. Overlooking treatment switching might substantially bias the evaluation of treatment efficacy or safety. To account for treatment switching, instrumental variable (IV) methods by leveraging the initial randomized assignment as an IV serve as natural adjustment methods because they allow dependent treatment switching possibly due to underlying prognoses. However, the ``exclusion restric…
Instrumental variables (IVs) are widely used for estimating causal effects in the presence of unmeasured confounding. Under the standard IV model, however, the average treatment effect (ATE) is only partially identifiable. To address this, we propose novel assumptions that allow for identification of the ATE. Our identification assumptions are clearly separated from model assumptions needed for estimation, so that researchers are not required to commit to a specific observed data model in establishing identification. We then construct multiple estimators that are consistent under three differe…
Individualized treatment rules (ITRs) are considered a promising recipe to deliver better policy interventions. One key ingredient in optimal ITR estimation problems is to estimate the average treatment effect conditional on a subject's covariate information, which is often challenging in observational studies due to the universal concern of unmeasured confounding. Instrumental variables (IVs) are widely-used tools to infer the treatment effect when there is unmeasured confounding between the treatment and outcome. In this work, we propose a general framework of approaching the optimal ITR est…
The primary analysis of randomized screening trials for cancer typically adheres to the intention-to-screen principle, measuring cancer-specific mortality reductions between screening and control arms. These mortality reductions result from a combination of the screening regimen, screening technology and the effect of the early, screening-induced, treatment. This motivates addressing these different aspects separately. Here we are interested in the causal effect of early versus delayed treatments on cancer mortality among the screening-detectable subgroup, which under certain assumptions is es…
In randomized trials, researchers are often interested in mediation analysis to understand how a treatment works, in particular how much of a treatment's effect is mediated by an intermediated variable and how much the treatment directly affects the outcome not through the mediator. The standard regression approach to mediation analysis assumes sequential ignorability of the mediator, that is that the mediator is effectively randomly assigned given baseline covariates and the randomized treatment. Since the experiment does not randomize the mediator, sequential ignorability is often not plausi…
Robins (1998) introduced marginal structural models (MSMs), a general class of counterfactual models for the joint effects of time-varying treatment regimes in complex longitudinal studies subject to time-varying confounding. He established identification of MSM parameters under a sequential randomization assumption (SRA), which essentially rules out unmeasured confounding of treatment assignment over time. In this technical report, we consider sufficient conditions for identification of MSM parameters with the aid of a time-varying instrumental variable, when sequential randomization fails to…
Zhejiang University（浙江大学）；Emory University（埃默里大学）；Tsinghua University（清华大学）；Shanghai AI Laboratory（上海人工智能实验室）；Shanghai Institute for Advanced Study of Zhejiang University（浙江大学上海高等研究院）
The advent of the big data era brought new opportunities and challenges to draw treatment effect in data fusion, that is, a mixed dataset collected from multiple sources (each source with an independent treatment assignment mechanism). Due to possibly omitted source labels and unmeasured confounders, traditional methods cannot estimate individual treatment assignment probability and infer treatment effect effectively. Therefore, we propose to reconstruct the source label and model it as a Group Instrumental Variable (GIV) to implement IV-based Regression for treatment effect estimation. In thi…
Instrumental variable methods have been widely used to identify causal effects in the presence of unmeasured confounding. A key identification condition known as the exclusion restriction states that the instrument cannot have a direct effect on the outcome which is not mediated by the exposure in view. In the health and social sciences, such an assumption is often not credible. To address this concern, we consider identification conditions of the population average treatment effect with an invalid instrumental variable which does not satisfy the exclusion restriction, and derive the efficient…
Point processes are probabilistic tools for modeling event data. While there exists a fast-growing literature studying the relationships between point processes, it remains unexplored how such relationships connect to causal effects. In the presence of unmeasured confounders, parameters from point process models do not necessarily have causal interpretations. We propose an instrumental variable method for causal inference with point process treatment and outcome. We define causal quantities based on potential outcomes and establish nonparametric identification results with a binary instrumenta…
There is a fast-growing literature on estimating optimal treatment regimes based on randomized trials or observational studies under a key identifying condition of no unmeasured confounding. Because confounding by unmeasured factors cannot generally be ruled out with certainty in observational studies or randomized trials subject to noncompliance, we propose a general instrumental variable approach to learning optimal treatment regimes under endogeneity. Specifically, we establish identification of both value function $E[Y_{\mathcal{D}(L)}]$ for a given regime $\mathcal{D}$ and optimal regimes…
For linear regression models who are not exactly sparse in the sense that the coefficients of the insignificant variables are not exactly zero, the working models obtained by a variable selection are often biased. Even in sparse cases, after a variable selection, when some significant variables are missing, the working models are biased as well. Thus, under such situations, root-n consistent estimation and accurate prediction could not be expected. In this paper, a novel remodelling method is proposed to produce an unbiased model when quasi-instrumental variables are introduced. The root-n est…
A Doubly Robust Instrumental Variable Approach for Estimating Average Treatment Effects in Time-to-Event Data with Unmeasured Confounding: Application to Real-World Data on ICU Patients with Septic Shock
The instrumental variables (IV) method is a method for making causal inferences about the effect of a treatment based on an observational study in which there are unmeasured confounding variables. The method requires a valid IV, a variable that is independent of the unmeasured confounding variables and is associated with the treatment but which has no effect on the outcome beyond its effect on the treatment. An additional assumption that is often made for the IV method is deterministic monotonicity, which is an assumption that for each subject, the level of the treatment that a subject would t…
In Randomised Controlled Trials (RCT) with treatment non-compliance, instrumental variable approaches are used to estimate complier average causal effects. We extend these approaches to cost-effectiveness analyses, where methods need to recognise the correlation between cost and health outcomes. We propose a Bayesian full likelihood (BFL) approach, which jointly models the effects of random assignment on treatment received and the outcomes, and a three-stage least squares (3sls) method, which acknowledges the correlation between the endpoints, and the endogeneity of the treatment received. Thi…
Estimating dynamic treatment regimes (DTRs) from retrospective observational data is challenging as some degree of unmeasured confounding is often expected. In this work, we develop a framework of estimating properly defined "optimal" DTRs with a time-varying instrumental variable (IV) when unmeasured covariates confound the treatment and outcome, rendering the potential outcome distributions only partially identified. We derive a novel Bellman equation under partial identification, use it to define a generic class of estimands (termed IV-optimal DTRs), and study the associated estimation prob…
This paper considers identification and estimation of the causal effect of the time Z until a subject is treated on a survival outcome T. The treatment is not randomly assigned, T is randomly right censored by a random variable C and the time to treatment Z is right censored by min(T,C). The endogeneity issue is treated using an instrumental variable explaining Z and independent of the error term of the model. We study identification in a fully nonparametric framework. We show that our specification generates an integral equation, of which the regression function of interest is a solution. We …
The instrumental variable method consistently estimates the effect of a treatment when there is unmeasured confounding and a valid instrumental variable. A valid instrumental variable is a variable that is independent of unmeasured confounders and affects the treatment but does not have a direct effect on the outcome beyond its effect on the treatment. Two commonly used estimators for using an instrumental variable to estimate a treatment effect are the two stage least squares estimator and the control function estimator. For linear causal effect models, these two estimators are equivalent, bu…
Clinical trials traditionally employ blinding as a design mechanism to reduce the influence of placebo effects. In practice, however, it can be difficult or impossible to blind study participants and unblinded trials are common in medical research. Here we show how instrumental variables can be used to quantify and disentangle treatment and placebo effects in randomized clinical trials comparing control and active treatments in the presence of confounders. The key idea is to use randomization to separately manipulate treatment assignment and psychological encouragement messages that increase t…
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…
University College London（伦敦大学学院）；University of Oxford（牛津大学）；The Alan Turing Institute（艾伦·图灵研究所）
We propose a kernel-based nonparametric estimator for the causal effect when the cause is corrupted by error. We do so by generalizing estimation in the instrumental variable setting. Despite significant work on regression with measurement error, additionally handling unobserved confounding in the continuous setting is non-trivial: we have seen little prior work. As a by-product of our investigation, we clarify a connection between mean embeddings and characteristic functions, and how learning one simultaneously allows one to learn the other. This opens the way for kernel method research to le…
There is an increasing interest in estimating heterogeneity in causal effects in randomized and observational studies. However, little research has been conducted to understand heterogeneity in an instrumental variables study. In this work, we present a method to estimate heterogeneous causal effects using an instrumental variable approach. The method has two parts. The first part uses subject-matter knowledge and interpretable machine learning techniques, such as classification and regression trees, to discover potential effect modifiers. The second part uses closed testing to test for the st…
It is common in instrumental variable studies for instrument values to be missing, for example when the instrument is a genetic test in Mendelian randomization studies. In this paper we discuss two apparent paradoxes that arise in so-called single consent designs where there is one-sided noncompliance, i.e., where unencouraged units cannot access treatment. The first paradox is that, even under a missing completely at random assumption, a complete-case analysis is biased when knowledge of one-sided noncompliance is taken into account; this is not the case when such information is disregarded. …
Two problems that arise in making causal inferences for non-mortality outcomes such as bronchopulmonary dysplasia (BPD) are unmeasured confounding and censoring by death, i.e., the outcome is only observed when subjects survive. In randomized experiments with noncompliance, instrumental variable methods can be used to control for the unmeasured confounding without censoring by death. But when there is censoring by death, the average causal treatment effect cannot be identified under usual assumptions, but can be studied for a specific subpopulation by using sensitivity analysis with additional…
Instrumental variable methods are widely used in medical and social science research to draw causal conclusions when the treatment and outcome are confounded by unmeasured confounding variables. One important feature of such studies is that the instrumental variable is often applied at the cluster level, e.g., hospitals' or physicians' preference for a certain treatment where each hospital or physician naturally defines a cluster. This paper proposes to embed such observational instrumental variable data into a cluster-randomized encouragement experiment using statistical matching. Potential o…
This paper analyzes the effect of a discrete treatment Z on a duration T. The treatment is not randomly assigned. The confounding issue is treated using a discrete instrumental variable explaining the treatment and independent of the error term of the model. Our framework is nonparametric and allows for random right censoring. This specification generates a nonlinear inverse problem and the average treatment effect is derived from its solution. We provide local and global identification properties that rely on a nonlinear system of equations. We propose an estimation procedure to solve this sy…
Consider the problem of estimating the local average treatment effect with an instrument variable, where the instrument unconfoundedness holds after adjusting for a set of measured covariates. Several unknown functions of the covariates need to be estimated through regression models, such as instrument propensity score and treatment and outcome regression models. We develop a computationally tractable method in high-dimensional settings where the numbers of regression terms are close to or larger than the sample size. Our method exploits regularized calibrated estimation, which involves Lasso …
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…
