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

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), four challenges arise. First, exchangeability regarding Xep does not equal exchangeability regarding X. Second, a definition of exchangeability that implies that AEE(Xep) can differ from AEE(X) is lacking. Third, there is no formal justification for using AEE(Xep) to estimate AEE(X). Fourth, the non-differential error assumption (NDEA) could be overly stringent in practice. 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 concept, 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. PE requires adjustment for E(P)C. Second, this article reveals how AEE(Xep) can equal AEE(X) even in the presence of differential exposure measurement error and confounder measurement error, which can also be applied to generalized propensity score. This paper provides comprehensive insight into when AEE(Xep) is a surrogate of AEE(X).