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We develop an omitted variable bias framework for sensitivity analysis of instrumental variable estimates that naturally handles multiple side effects (violations of the exclusion restriction assumption) and confounders (violations of the ignorability of the instrument assumption) of the instrument, exploits expert knowledge to bound sensitivity parameters and can be easily implemented with standard software. Specifically, we introduce sensitivity statistics for routine reporting, such as (extreme) robustness values for instrumental variables, describing the minimum strength that omitted variables need to have to change the conclusions of a study. Next, we provide visual displays that fully characterize the sensitivity of point estimates and confidence intervals to violations of the standard instrumental variable assumptions. Finally, we offer formal bounds on the worst possible bias under the assumption that the maximum explanatory power of omitted variables is no stronger than a multiple of the explanatory power of observed variables. Conveniently, many pivotal conclusions regarding the sensitivity of the instrumental variable estimate (e.g., tests against the null hypothesis of a zero causal effect) can be reached simply through separate sensitivity analyses of the effect of the instrument on the treatment (the first stage) and the effect of the instrument on the outcome (the reduced form). We apply our methods in a running example that uses proximity to college as an instrumental variable to estimate the returns to schooling. 

We introduce the BREASE framework for the Bayesian analysis of randomized controlled trials with a binary treatment and a binary outcome. Approaching the problem from a causal inference perspective, we propose parameterizing the likelihood in terms of the baseline risk, efficacy, and side effects of the treatment, along with a flexible, yet intuitive and tractable jointly independent beta prior distribution on these parameters, which we show to be a generalization of the Dirichlet prior for the joint distribution of potential outcomes. Our approach has a number of desirable characteristics when compared to current mainstream alternatives: (i) it naturally induces prior dependence between expected outcomes in the treatment and control groups; (ii) as the baseline risk, efficacy and side effects are quantities inherently familiar to clinicians, the hyperparameters of the prior are directly interpretable, thus facilitating the elicitation of prior knowledge and sensitivity analysis; and (iii) it admits analytical formulae for the marginal likelihood, Bayes factor, and other posterior quantities,as well as an exact posterior sampling algorithm and an accurate and fast data-augmented Gibbs sampler in cases where traditional MCMC fails. Empirical examples demonstrate the utility of our methods for estimation, hypothesis testing, and sensitivity analysis of treatment effects. 

This tutorial introduces the package sensemakr for R and Stata, which implements a suite of sensitivity analysis tools for regression models developed in Cinelli and Hazlett (2020, 2022). Given a regression model, sensemakr can compute sensitivity statistics for routine reporting, such as the robustness value , which describes the minimum strength that unobserved confounders need to have to overturn a research conclusion. The package also provides plotting tools that visually demonstrate the sensitivity of point estimates and t-values to hypothetical confounders. Finally, sensemakr implements formal bounds on sensitivity parameters by means of comparison with the explanatory power of observed variables. All these tools are based on the familiar omitted variable bias framework, do not require assumptions regarding the functional form of the treatment assignment mechanism nor the distribution of the unobserved confounders, and naturally handle multiple, non-linear confounders. With sensemakr, users can transparently report the sensitivity of their causal inferences to unobserved confounding, thereby enabling a more precise, quantitative debate as to what can be concluded from imperfect observational studies. 

Abstract Mendelian Randomization (MR) studies are threatened by population stratification, batch effects, and horizontal pleiotropy. Although a variety of methods have been proposed to mitigate those problems, residual biases may still remain, leading to highly statistically significant false positives in large databases. Here we describe a suite of sensitivity analysis tools that enables investigators to quantify the robustness of their findings against such validity threats. Specifically, we propose the routine reporting of sensitivity statistics that reveal the minimal strength of violations necessary to explain away the MR results. We further provide intuitive displays of the robustness of the MR estimate to any degree of violation, and formal bounds on the worst-case bias caused by violations multiple times stronger than observed variables. We demonstrate how these tools can aid researchers in distinguishing robust from fragile findings by examining the effect of body mass index on diastolic blood pressure and Townsend deprivation index. 

Many students of statistics and econometrics express frustration with the way a problem known as “bad control” is treated in the traditional literature. The issue arises when the addition of a variable to a regression equation produces an unintended discrepancy between the regression coefficient and the effect that the coefficient is intended to represent. Avoiding such discrepancies presents a challenge to all analysts in the data intensive sciences. This note describes graphical tools for understanding, visualizing, and resolving the problem through a series of illustrative examples. By making this “crash course” accessible to instructors and practitioners, we hope to avail these tools to a broader community of scientists concerned with the causal interpretation of regression models.