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

Exchangeability concerning a continuous exposure, X, implies no confounding bias when identifying average exposure effects of X, AEE(X). When X is measured with error (Xep), two challenges arise in identifying AEE(X). Firstly, exchangeability regarding Xep does not equal exchangeability regarding X. Secondly, the necessity of the non-differential error assumption (NDEA), overly stringent in practice, remains uncertain. To address them, this article proposes unifying exchangeability and exposure and confounder measurement errors with three novel concepts. The first, Probabilistic Exchangeability (PE), states that the outcomes of those with Xep=e are probabilistically exchangeable with the outcomes of those truly exposed to X=eT. The relationship between AEE(Xep) and AEE(X) in risk difference and ratio scales is mathematically expressed as a probabilistic certainty, termed exchangeability probability (Pe). Squared Pe (Pe.sq) quantifies the extent to which AEE(Xep) differs from AEE(X) due to exposure measurement error through mechanisms not akin to confounding mechanisms. The coefficient of determination (R.sq) in the regression of X against Xep may sometimes be sufficient to measure Pe.sq. The second concept, Emergent Pseudo Confounding (EPC), describes the bias introduced by exposure measurement error through mechanisms akin to confounding mechanisms. PE can hold when EPC is controlled for, which is weaker than NDEA. The third, Emergent Confounding, describes when bias due to confounder measurement error arises. Adjustment for E(P)C can be performed like confounding adjustment to ensure PE. This paper provides formal justifications for using AEE(Xep) and maximum insight into potential divergence of AEE(Xep) from AEE(X) and how to measure it.