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

Exchangeability concerning a continuous exposure, X, may be assumed to identify average exposure effects of X, AEE(X). When X is measured with error (Xep), three challenges arise. First, exchangeability regarding Xep does not equal exchangeability regarding X. Second, the non-differential error assumption (NDEA) could be overly stringent in practice. Third, a definition of exchangeability that implies that AEE(Xep) can differ from AEE(X) is lacking. To address them, this article proposes unifying exchangeability and exposure/confounder measurement errors with three novel concepts. The 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, describes when bias due to confounder measurement error arises. PE requires adjustment for E(P)C, which can be performed like confounding adjustment. Under PE, the coefficient of determination (R2) in the regression of Xep against X may sometimes be sufficient to measure the difference between AEE(Xep) and AEE(X) in risk difference and ratio scales. This paper provides comprehensive insight into when AEE(Xep) is a surrogate of AEE(X). Differential errors could be addressed and may not compromise causal inference