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1. Partial Identifiability in Discrete Data with Measurement Error

   Noam Finkelstein; Roy Adams; Suchi Saria; Ilya Shpitser (July 2021, Proceedings of the Thirty Seventh Conference on Uncertainty in Artificial Intelligence)

   Cassio de Campos; Marloes H. Maathuis (Ed.)

   When data contains measurement errors, it is necessary to make modeling assumptions relating the error-prone measurements to the unobserved true values. Work on measurement error has largely focused on models that fully identify the parameter of interest. As a result, many practically useful models that result in bounds on the target parameter -- known as partial identification -- have been neglected. In this work, we present a method for partial identification in a class of measurement error models involving discrete variables. We focus on models that impose linear constraints on the tar- get parameter, allowing us to compute partial identification bounds using off-the-shelf LP solvers. We show how several common measurement error assumptions can be composed with an extended class of instrumental variable-type models to create such linear constraint sets. We further show how this approach can be used to bound causal parameters, such as the average treatment effect, when treatment or outcome variables are measured with error. Using data from the Oregon Health Insurance Experiment, we apply this method to estimate bounds on the effect Medicaid enrollment has on depression when depression is measured with error. 

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2. Two robust tools for inference about causal effects with invalid instruments

   https://doi.org/10.1111/biom.13415  

   Kang, Hyunseung; Lee, Youjin; Cai, T. Tony; Small, Dylan S. (December 2020, Biometrics)

   Abstract Instrumental variables have been widely used to estimate the causal effect of a treatment on an outcome. Existing confidence intervals for causal effects based on instrumental variables assume that all of the putative instrumental variables are valid; a valid instrumental variable is a variable that affects the outcome only by affecting the treatment and is not related to unmeasured confounders. However, in practice, some of the putative instrumental variables are likely to be invalid. This paper presents two tools to conduct valid inference and tests in the presence of invalid instruments. First, we propose a simple and general approach to construct confidence intervals based on taking unions of well‐known confidence intervals. Second, we propose a novel test for the null causal effect based on a collider bias. Our two proposals outperform traditional instrumental variable confidence intervals when invalid instruments are present and can also be used as a sensitivity analysis when there is concern that instrumental variables assumptions are violated. The new approach is applied to a Mendelian randomization study on the causal effect of low‐density lipoprotein on globulin levels. 

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3. An omitted variable bias framework for sensitivity analysis of instrumental variables

   https://doi.org/10.1093/biomet/asaf004  

   Cinelli, Carlos; Hazlett, Chad (January 2025, Biometrika)

   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. 

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4. An instrumental variable method for point processes: generalized Wald estimation based on deconvolution

   https://doi.org/10.1093/biomet/asad005  

   Jiang, Zhichao; Chen, Shizhe; Ding, Peng (January 2023, Biometrika)

   Summary Point processes are probabilistic tools for modelling event data. While there exists a fast-growing literature on the relationships between point processes, how such relationships connect to causal effects remains unexplored. 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 instrumental variable. We extend the traditional Wald estimation to deal with point process treatment and outcome, showing that it should be performed after a Fourier transform of the intention-to-treat effects on the treatment and outcome, and thus takes the form of deconvolution. We refer to this approach as generalized Wald estimation and propose an estimation strategy based on well-established deconvolution methods. 

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5. Double Machine Learning Density Estimation for Local Treatment Effects with Instruments

   Jung, Y.; Tian, J.; Bareinboim, E. (January 2021, Advances in neural information processing systems)

   It is common to quantify causal effects with mean values, which, however, may fail to capture significant distribution differences of the outcome under different treatments. We study the problem of estimating the density of the causal effect of a binary treatment on a continuous outcome given a binary instrumental variable in the presence of covariates. Specifically, we consider the local treatment effect, which measures the effect of treatment among those who comply with the assignment under the assumption of monotonicity (only the ones who were offered the treatment take it). We develop two families of methods for this task, kernel-smoothing and model-based approximations -- the former smoothes the density by convoluting with a smooth kernel function; the latter projects the density onto a finite-dimensional density class. For both approaches, we derive double/debiased machine learning (DML) based estimators. We study the asymptotic convergence rates of the estimators and show that they are robust to the biases in nuisance function estimation. We illustrate the proposed methods on synthetic data and a real dataset called 401(k). 

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