Continuous Exposures and Inverse Problems in Causal Inference

Abstract

This dissertation studies challenges that arise in causal inference with continuous exposures, with particular emphasis on the role of ill-posed inverse problems. Common causal estimands in continuous exposure settings are often difficult to interpret, challenging to estimate, or rely on strong and potentially unrealistic identification assumptions. The first project introduces and studies a class of stochastic interventions for continuous exposures that yield scientifically interpretable causal estimands which can be identified from observed data without reliance on the positivity assumption. We establish conditions for identification and propose and study an influence function-based estimator. The estimator’s performance is examined in simulations for both uncensored and right-censored outcomes, and the method is applied to the study of correlates of protection in an HIV vaccine trial. The second project considers a two-sample instrumental variable framework for causal inference when the exposure is observed with error. The causal estimand is formulated as a functional of a solution of an ill-posed integral equation, thus connecting the problem to recent work on statistical inverse problems. An estimating equations-based estimator is proposed, its asymptotic properties are studied, and its finite-sample performance is evaluated through simulations. The method is applied to data from the COVAIL study. The third project scrutinizes common assumptions used in the analysis of statistical inverse problems, which can be difficult to interpret in causal inference settings. These assumptions are explored using tools from microlocal and harmonic analysis, providing further insight into these assumptions and suggesting avenues for future work.