Consider a study with binary exposure, outcome, and confounder, where the confounder is nondifferentially misclassified. Epidemiologists have long accepted the unproven but oft-cited result that, if the confounder is binary, then odds ratios, risk ratios, and risk differences that control for the mismeasured confounder will lie between the crude and the true measures. In this paper, we provide an analytic proof of the result in the absence of a qualitative interaction between treatment and confounder, and we demonstrate via counterexample that the result need not hold when there is such a qualitative interaction. We also present an analytic proof of the result for the effect of treatment among the treated and describe extensions to measures conditional on or standardized over other covariates.
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