Measurement error modelling with an approximate instrumental variable

Summary. Consider using regression modelling to relate an exposure (predictor) variable to a disease outcome (response) variable. If the exposure variable is measured with error, but this error is ignored in the analysis, then misleading inferences can result. This problem is well known and has spawned a large literature on methods which adjust for measurement error in predictor variables. One theme is that the requisite assumptions about the nature of the measurement error can be stronger than what is actually known in many practical situations. In particular, the assumptions that are required to yield a model which is formally identified from the observable data can be quite strong. The paper deals with one particular strategy for measurement error modelling, namely that of seeking an instrumental variable, i.e. a covariate S which is associated with exposure and conditionally independent of the outcome given exposure. If these two conditions hold exactly, then we call S an exact instrumental variable, and an identified model results. However, the second is not checkable empirically, since the actual exposure is unobserved. In practice then, investigators typically seek a covariate which is plausibly thought to satisfy it. We study inferences which acknowledge the approximate nature of this assumption. In particular, we consider Bayesian inference with a prior distribution that posits that S is probably close to conditionally independent of outcome given exposure. We refer to this as an approximate instrumental variable assumption. Although the approximate instrumental variable assumption is more realistic for most applications, concern arises that a non‐identified model may result. Thus the paper contrasts inferences arising from the approximate instrumental variable assumption with their exact instrumental variable counterparts, with particular emphasis on the benefit of basing inferences on a more realistic model versus the cost of basing inferences on a non‐identified model.