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Instrumental variables are widely used in econometrics and epidemiology for identifying and estimating causal effects when an exposure of interest is confounded by unmeasured factors. Despite this popularity, the assumptions invoked to justify the use of instruments differ substantially across the literature. Similarly, statistical approaches for estimating the resulting causal quantities vary considerably, and often rely on strong parametric assumptions. In this work, we compile and organize structural conditions that nonparametrically identify conditional average treatment effects, average treatment effects among the treated, and local average treatment effects, with a focus on identification formulae invoking the conditional Wald estimand. Moreover, we build upon existing work and propose nonparametric efficient estimators of functionals corresponding to marginal and conditional causal contrasts resulting from the various identification paradigms. We illustrate the proposed methods on an observational study examining the effects of operative care on adverse events for cholecystitis patients, and a randomized trial assessing the effects of market participation on political views., Comment: 46 pages, 2 figures

Laparoscopic surgery has been shown through a number of randomized trials to be an effective form of treatment for cholecystitis. Given this evidence, one natural question for clinical practice is: does the effectiveness of laparoscopic surgery vary among patients? It might be the case that, while the overall effect is positive, some patients treated with laparoscopic surgery may respond positively to the intervention while others do not or may be harmed. In our study, we focus on conditional average treatment effects to understand whether treatment effects vary systematically with patient characteristics. Recent methodological work has developed a meta-learner framework for flexible estimation of conditional causal effects. In this framework, nonparametric estimation methods can be used to avoid bias from model misspecification while preserving statistical efficiency. In addition, researchers can flexibly and effectively explore whether treatment effects vary with a large number of possible effect modifiers. However, these methods have certain limitations. For example, conducting inference can be challenging if black-box models are used. Further, interpreting and visualizing the effect estimates can be difficult when there are multi-valued effect modifiers. In this paper, we develop new methods that allow for interpretable results and inference from the meta-learner framework for heterogeneous treatment effects estimation. We also demonstrate methods that allow for an exploratory analysis to identify possible effect modifiers. We apply our methods to a large database for the use of laparoscopic surgery in treating cholecystitis. We also conduct a series of simulation studies to understand the relative performance of the methods we develop. Our study provides key guidelines for the interpretation of conditional causal effects from the meta-learner framework., Comment: 48 pages, 7 figures

Comparative effectiveness research frequently employs the instrumental variable design since randomized trials can be infeasible for many reasons. In this study, we investigate and compare treatments for emergency cholecystitis -- inflammation of the gallbladder. A standard treatment for cholecystitis is surgical removal of the gallbladder, while alternative non-surgical treatments include managed care and pharmaceutical options. As randomized trials are judged to violate the principle of equipoise, we consider an instrument for operative care: the surgeon's tendency to operate. Standard instrumental variable estimation methods, however, often rely on parametric models that are prone to bias from model misspecification. We outline instrumental variable estimation methods based on the doubly robust machine learning framework. These methods enable us to employ various machine learning techniques for nuisance parameter estimation and deliver consistent estimates and fast rates of convergence for valid inference. We use these methods to estimate the primary target causal estimand in an IV design. Additionally, we expand these methods to develop estimators for heterogeneous causal effects, profiling principal strata, and a sensitivity analyses for a key instrumental variable assumption. We conduct a simulation study to demonstrate scenarios where more flexible estimation methods outperform standard methods. Our findings indicate that operative care is generally more effective for cholecystitis patients, although the benefits of surgery can be less pronounced for key patient subgroups.

Quantifying treatment effect heterogeneity is a crucial task in many areas of causal inference, e.g. optimal treatment allocation and estimation of subgroup effects. We study the problem of estimating the level sets of the conditional average treatment effect (CATE), identified under the no-unmeasured-confounders assumption. Given a user-specified threshold, the goal is to estimate the set of all units for whom the treatment effect exceeds that threshold. For example, if the cutoff is zero, the estimand is the set of all units who would benefit from receiving treatment. Assigning treatment just to this set represents the optimal treatment rule that maximises the mean population outcome. Similarly, cutoffs greater than zero represent optimal rules under resource constraints. The level set estimator that we study follows the plug-in principle and consists of simply thresholding a good estimator of the CATE. While many CATE estimators have been recently proposed and analysed, how their properties relate to those of the corresponding level set estimators remains unclear. Our first goal is thus to fill this gap by deriving the asymptotic properties of level set estimators depending on which estimator of the CATE is used. Next, we identify a minimax optimal estimator in a model where the CATE, the propensity score and the outcome model are Holder-smooth of varying orders. We consider data generating processes that satisfy a margin condition governing the probability of observing units for whom the CATE is close to the threshold. We investigate the performance of the estimators in simulations and illustrate our methods on a dataset used to study the effects on mortality of laparoscopic vs open surgery in the treatment of various conditions of the colon., Comment: 38 pages, 4 figures

Experiments studying get-out-the-vote (GOTV) efforts estimate the causal effect of various mobilization efforts on voter turnout. However, there is often substantial noncompliance in these studies. A usual approach is to use an instrumental variable (IV) analysis to estimate impacts for compliers, here being those actually contacted by the investigators. Unfortunately, popular IV estimators can be unstable in studies with a small fraction of compliers. We explore post-stratifying the data (e.g., taking a weighted average of IV estimates within each stratum) using variables that predict complier status (and, potentially, the outcome) to mitigate this. We present the benefits of post-stratification in terms of bias, variance, and improved standard error estimates, and provide a finite-sample asymptotic variance formula. We also compare the performance of different IV approaches and discuss the advantages of our design-based post-stratification approach over incorporating compliance-predictive covariates into the two-stage least squares estimator. In the end, we show that covariates predictive of compliance can increase precision, but only if one is willing to make a bias-variance trade-off by down-weighting or dropping strata with few compliers. By contrast, standard approaches such as two-stage least squares fail to use such information. We finally examine the benefits of our approach in two GOTV applications.

We present a practical guide for the analysis of regression discontinuity (RD) designs in biomedical contexts. We begin by introducing key concepts, assumptions, and estimands within both the continuity-based framework and the local randomization framework. We then discuss modern estimation and inference methods within both frameworks, including approaches for bandwidth or local neighborhood selection, optimal treatment effect point estimation, and robust bias-corrected inference methods for uncertainty quantification. We also overview empirical falsification tests that can be used to support key assumptions. Our discussion focuses on two particular features that are relevant in biomedical research: (i) fuzzy RD designs, which often arise when therapeutic treatments are based on clinical guidelines but patients with scores near the cutoff are treated contrary to the assignment rule; and (ii) RD designs with discrete scores, which are ubiquitous in biomedical applications. We illustrate our discussion with three empirical applications: the effect of CD4 guidelines for anti-retroviral therapy on retention of HIV patients in South Africa, the effect of genetic guidelines for chemotherapy on breast cancer recurrence in the United States, and the effects of age-based patient cost-sharing on healthcare utilization in Taiwan. We provide replication materials employing publicly available statistical software in Python, R and Stata, offering researchers all necessary tools to conduct an RD analysis.

When an exposure of interest is confounded by unmeasured factors, an instrumental variable (IV) can be used to identify and estimate certain causal contrasts. Identification of the marginal average treatment effect (ATE) from IVs relies on strong untestable structural assumptions. When one is unwilling to assert such structure, IVs can nonetheless be used to construct bounds on the ATE. Famously, Balke and Pearl (1997) proved tight bounds on the ATE for a binary outcome, in a randomized trial with noncompliance and no covariate information. We demonstrate how these bounds remain useful in observational settings with baseline confounders of the IV, as well as randomized trials with measured baseline covariates. The resulting bounds on the ATE are non-smooth functionals, and thus standard nonparametric efficiency theory is not immediately applicable. To remedy this, we propose (1) under a novel margin condition, influence function-based estimators of the bounds that can attain parametric convergence rates when the nuisance functions are modeled flexibly, and (2) estimators of smooth approximations of these bounds. We propose extensions to continuous outcomes, explore finite sample properties in simulations, and illustrate the proposed estimators in an observational study targeting the effect of higher education on wages., Comment: 42 pages, 2 figures

In a clustered observational study, a treatment is assigned to groups and all units within the group are exposed to the treatment. We develop a new method for statistical adjustment in clustered observational studies using approximate balancing weights, a generalization of inverse propensity score weights that solve a convex optimization problem to find a set of weights that directly minimize a measure of covariate imbalance, subject to an additional penalty on the variance of the weights. We tailor the approximate balancing weights optimization problem to both adjustment sets by deriving an upper bound on the mean square error for each case and finding weights that minimize this upper bound, linking the level of covariate balance to a bound on the bias. We implement the procedure by specializing the bound to a random cluster-level effects model, leading to a variance penalty that incorporates the signal signal-to-noise ratio and penalizes the weight on individuals and the total weight on groups differently according to the the intra-class correlation.

Inverse probability weights are commonly used in epidemiology to estimate causal effects in observational studies. Researchers can typically focus on either the average treatment effect or the average treatment effect on the treated with inverse probability weighting estimators. However, when overlap between the treated and control groups is poor, this can produce extreme weights that can result in biased estimates and large variances. One alternative to inverse probability weights are overlap weights, which target the population with the most overlap on observed characteristics. While estimates based on overlap weights produce less bias in such contexts, the causal estimand can be difficult to interpret. One alternative to inverse probability weights are balancing weights, which directly target imbalances during the estimation process. Here, we explore whether balancing weights allow analysts to target the average treatment effect on the treated in cases where inverse probability weights are biased due to poor overlap. We conduct three simulation studies and an empirical application. We find that in many cases, balancing weights allow the analyst to still target the average treatment effect on the treated even when overlap is poor. We show that while overlap weights remain a key tool for estimating causal effects, more familiar estimands can be targeted by using balancing weights instead of inverse probability weights.

Research documents that Black patients experience worse general surgery outcomes than white patients in the United States. In this paper, we focus on an important but less-examined category: the surgical treatment of emergency general surgery (EGS) conditions, which refers to medical emergencies where the injury is "endogenous," such as a burst appendix. Our goal is to assess racial disparities for common outcomes after EGS treatment using an administrative database of hospital claims in New York, Florida, and Pennsylvania, and to understand the extent to which differences are attributable to patient-level risk factors versus hospital-level factors. To do so, we use a class of linear weighting estimators that re-weight white patients to have a similar distribution of baseline characteristics as Black patients. This framework nests many common approaches, including matching and linear regression, but offers important advantages over these methods in terms of controlling imbalance between groups, minimizing extrapolation, and reducing computation time. Applying this approach to the claims data, we find that disparities estimates that adjust for the admitting hospital are substantially smaller than estimates that adjust for patient baseline characteristics only, suggesting that hospital-specific factors are important drivers of racial disparities in EGS outcomes.

The clustered observational study (COS) design is the observational study counterpart to the clustered randomized trial. In a COS, a treatment is assigned to intact groups, and all units within the group are exposed to the treatment. However, the treatment is non-randomly assigned. COSs are common in both education and health services research. In education, treatments may be given to all students within some schools but withheld from all students in other schools. In health studies, treatments may be applied to clusters such as hospitals or groups of patients treated by the same physician. In this manuscript, we study the identification of causal effects in clustered observational study designs. We focus on the prospect of differential selection of units to clusters, which occurs when the units' cluster selections depend on the clusters' treatment assignments. Extant work on COSs has made an implicit assumption that rules out the presence of differential selection. We derive the identification results for designs with differential selection and that contexts with differential cluster selection require different adjustment sets than standard designs. We outline estimators for designs with and without differential selection. Using a series of simulations, we outline the magnitude of the bias that can occur with differential selection. We then present two empirical applications focusing on the likelihood of differential selection.

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 use.

Instrumental variable (IV) analyses are becoming common in health services research and epidemiology. IV analyses can be used both to analyze randomized trials with noncompliance and as a form of natural experiment. In these analyses, investigators often adopt a monotonicity assumption, which implies that the relevant effect only applies to a subset of the study population known as compliers. Since the estimated effect is not the average treatment effect of the study population, it is important to compare the characteristics of compliers and non-compliers. Profiling compliers and non-compliers is necessary to understand what subpopulation the researcher is making inferences about, and an important first step in evaluating the external validity (or lack thereof) of the IV estimate for compliers. Here, we discuss the assumptions necessary for profiling, which are weaker than the assumptions necessary for identifying the local average treatment effect if the instrument is randomly assigned. We then outline a simple and general method to characterize compliers and noncompliers using baseline covariates. Next, we extend current methods by deriving standard errors for these estimates. We demonstrate these methods using an IV known as tendency to operate (TTO) from health services research.

We discuss some causal estimands used to study racial discrimination in policing. A central challenge is that not all police-civilian encounters are recorded in administrative datasets and available to researchers. One possible solution is to consider the average causal effect of race conditional on the civilian already being detained by the police. We find that such an estimand can be quite different from the more familiar ones in causal inference and needs to be interpreted with caution. We propose using an estimand new for this context -- the causal risk ratio, which has more transparent interpretation and requires weaker identification assumptions. We demonstrate this through a reanalysis of the NYPD Stop-and-Frisk dataset. Our reanalysis shows that the naive estimator that ignores the post-treatment selection in administrative records may severely underestimate the disparity in police violence between minorities and whites in these and similar data., Comment: Accepted for publication in the American Political Science Review on 14th June, 2021

Comparing outcomes across hospitals, often to identify underperforming hospitals, is a critical task in health services research. However, naive comparisons of average outcomes, such as surgery complication rates, can be misleading because hospital case mixes differ -- a hospital's overall complication rate may be lower due to more effective treatments or simply because the hospital serves a healthier population overall. In this paper, we develop a method of ``direct standardization'' where we re-weight each hospital patient population to be representative of the overall population and then compare the weighted averages across hospitals. Adapting methods from survey sampling and causal inference, we find weights that directly control for imbalance between the hospital patient mix and the target population, even across many patient attributes. Critically, these balancing weights can also be tuned to preserve sample size for more precise estimates. We also derive principled measures of statistical precision, and use outcome modeling and Bayesian shrinkage to increase precision and account for variation in hospital size. We demonstrate these methods using claims data from Pennsylvania, Florida, and New York, estimating standardized hospital complication rates for general surgery patients. We conclude with a discussion of how to detect low performing hospitals.

The method of difference-in-differences (DID) is widely used to study the causal effect of policy interventions in observational studies. DID employs a before and after comparison of the treated and control units to remove bias due to time-invariant unmeasured confounders under the parallel trends assumption. Estimates from DID, however, will be biased if the outcomes for the treated and control units evolve differently in the absence of treatment, namely if the parallel trends assumption is violated. We propose a general identification strategy that leverages two groups of control units whose outcomes relative to the treated units exhibit a negative correlation, and achieves partial identification of the average treatment effect for the treated. The identified set is of a union bounds form that involves the minimum and maximum operators, which makes the canonical bootstrap generally inconsistent and naive methods overly conservative. By utilizing the directional inconsistency of the bootstrap distribution, we develop a novel bootstrap method to construct uniformly valid confidence intervals for the identified set and parameter of interest when the identified set is of a union bounds form, and we establish the method's theoretical properties. We develop a simple falsification test and sensitivity analysis. We apply the proposed strategy for bracketing to study whether minimum wage laws affect employment levels.

Surface mining has become a major method of coal mining in Central Appalachia alongside the traditional underground mining. Concerns have been raised about the health effects of this surface mining, particularly mountaintop removal mining where coal is mined upon steep mountaintops by removing the mountaintop through clearcutting forests and explosives. We have designed a matched observational study to assess the effects of surface mining in Central Appalachia on adverse birth outcomes. This protocol describes for the study the background and motivation, the sample selection and the analysis plan.

Much evidence in comparative effectiveness research is based on observational studies. Researchers who conduct observational studies typically assume that there are no unobservable differences between the treated and control groups. Treatment effects are estimated after adjusting for observed differences between treated and controls. However, treatment effect estimates may be biased due to model misspecification. That is, if the method of treatment effect estimation imposes unduly strong functional form assumptions, treatment effect estimates may be significantly biased. In this study, we compare the performance of a wide variety of treatment effect estimation methods. We do so within the context of the REFLUX study from the UK. In REFLUX, after study qualification, participants were enrolled in either a randomized trial arm or patient preference arm. In the randomized trial, patients were randomly assigned to either surgery or medical management. In the patient preference arm, participants selected to either have surgery or medical management. We attempt to recover the treatment effect estimate from the randomized trial arm using the data from the patient preference arm of the study. We vary the method of treatment effect estimation and record which methods are successful and which are not. We apply over 20 different methods including standard regression models as well as advanced machine learning methods. We find that simple propensity score matching methods perform the worst. We also find significant variation in performance across methods. The wide variation in performance suggests analysts should use multiple methods of estimation as a robustness check.

We investigate the efficacy of surgical versus non-surgical management for two gastrointestinal conditions, colitis and diverticulitis, using observational data. We deploy an instrumental variable design with surgeons' tendencies to operate as an instrument. Assuming instrument validity, we find that non-surgical alternatives can reduce both hospital length of stay and the risk of complications, with estimated effects larger for septic patients than for non-septic patients. The validity of our instrument is plausible but not ironclad, necessitating a sensitivity analysis. Existing sensitivity analyses for IV designs assume effect homogeneity, unlikely to hold here because of patient-specific physiology. We develop a new sensitivity analysis that accommodates arbitrary effect heterogeneity and exploits components explainable by observed features. We find that the results for non-septic patients prove more robust to hidden bias despite having smaller estimated effects. For non-septic patients, two individuals with identical observed characteristics would have to differ in their odds of assignment to a high tendency to operate surgeon by a factor of 2.34 to overturn our finding of a benefit for non-surgical management in reducing length of stay. For septic patients, this value is only 1.64. Simulations illustrate that this phenomenon may be explained by differences in within-group heterogeneity.

Instrumental variable (IV) analyses are becoming common in health services research and epidemiology. Most IV analyses use naturally occurring instruments, such as distance to a hospital. In these analyses, investigators must assume the instrument is as-if randomly assigned. This assumption cannot be tested directly, but it can be falsified. Most falsification tests in the literature compare relative prevalence or bias in observed covariates between the instrument and the exposure. These tests require investigators to make a covariate-by-covariate judgment about the validity of the IV design. Often, only some of the covariates are well-balanced, making it unclear if as-if randomization can be assumed for the instrument across all covariates. We propose an alternative falsification test that compares IV balance or bias to the balance or bias that would have been produced under randomization. A key advantage of our test is that it allows for global balance measures as well as easily interpretable graphical comparisons. Furthermore, our test does not rely on any parametric assumptions and can be used to validly assess if the instrument is significantly closer to being as-if randomized than the exposure. We demonstrate our approach on a recent IV application that uses bed availability in the intensive care unit (ICU) as an instrument for admission to the ICU., Comment: 19 pages

Applied analysts often use the differences-in-differences (DID) method to estimate the causal effect of policy interventions with observational data. The method is widely used, as the required before and after comparison of a treated and control group is commonly encountered in practice. DID removes bias from unobserved time-invariant confounders. While DID removes bias from time-invariant confounders, bias from time-varying confounders may be present. Hence, like any observational comparison, DID studies remain susceptible to bias from hidden confounders. Here, we develop a method of sensitivity analysis that allows investigators to quantify the amount of bias necessary to change a study's conclusions. Our method operates within a matched design that removes bias from observed baseline covariates. We develop methods for both binary and continuous outcomes. We then apply our methods to two different empirical examples from the social sciences. In the first application, we study the effect of changes to disability payments in Germany. In the second, we re-examine whether election day registration increased turnout in Wisconsin.

Cluster randomized trials (CRTs) are popular in public health and in the social sciences to evaluate a new treatment or policy where the new policy is randomly allocated to clusters of units rather than individual units. CRTs often feature both noncompliance, when individuals within a cluster are not exposed to the intervention, and individuals within a cluster may influence each other through treatment spillovers where those who comply with the new policy may affect the outcomes of those who do not. Here, we study the identification of causal effects in CRTs when both noncompliance and treatment spillovers are present. We prove that the standard analysis of CRT data with noncompliance using instrumental variables does not identify the usual complier average causal effect when treatment spillovers are present. We extend this result and show that no analysis of CRT data can unbiasedly estimate local network causal effects. Finally, we develop bounds for these causal effects under the assumption that the treatment is not harmful compared to the control. We demonstrate these results with an empirical study of a deworming intervention in Kenya.

In non-experimental settings, the Regression Discontinuity (RD) design is one of the most credible identification strategies for program evaluation and causal inference. However, RD treatment effect estimands are necessarily local, making statistical methods for the extrapolation of these effects a key area for development. We introduce a new method for extrapolation of RD effects that relies on the presence of multiple cutoffs, and is therefore design-based. Our approach employs an easy-to-interpret identifying assumption that mimics the idea of "common trends" in difference-in-differences designs. We illustrate our methods with data on a subsidized loan program on post-education attendance in Colombia, and offer new evidence on program effects for students with test scores away from the cutoff that determined program eligibility.