A Unification of Exchangeability and Continuous Exposure and Confounder Measurement Errors: Probabilistic Exchangeability

Measurement errors are pervasive. A deeper understanding of measurement error impacts on research is critical for causal inference. Exchangeability concerning a continuous exposure or treatment, X, may be assumed to identify average exposure/treatment effects of X, AEE(X). When X is measured with error (Xep), exchangeability issues arise, a topic remains largely understudied. First, exchangeability regarding Xep does not equal exchangeability regarding X. Second, there is no formal justification for using AEE(Xep) to estimate AEE(X) under the potential outcomes framework. Third, a definition of exchangeability that implies that AEE(Xep) can differ from AEE(X) is lacking. Fourth, the non-differential error assumption (NDEA) could be overly stringent in practice. Fifth, while confounders or exposure mixtures may be measured with error, raising concerns about residual confounding, methods to correct for measurement errors in both exposures and confounders remain lacking. To address them, first, this article proposes unifying exchangeability and exposure/confounder measurement errors through three concepts. First, Probabilistic Exchangeability (PE) is an exchangeability assumption that allows for the difference between AEE(Xep) and AEE(X). The second, Emergent Pseudo Confounding (EPC), describes the bias introduced by exposure measurement error through mechanisms like confounding mechanisms. The third, Emergent Confounding (EC), describes when bias due to confounder measurement error arises. Second, this article develops correction theories for differential exposure measurement error and confounder measurement error to estimate AEE(X) under PE. This paper provides comprehensive insight into when AEE(Xep) is a surrogate of AEE(X). Differential errors can be addressed, which may not compromise causal inference.