On the bias of adjusting for a non-differentially mismeasured discrete confounder

Biological and epidemiological phenomena are often measured with error or imperfectly captured in data. When the true state of this imperfect measure is a confounder of an outcome exposure relationship of interest, it was previously widely believed that adjustment for the mismeasured observed variables provides a less biased estimate of the true average causal effect than not adjusting. However, this is not always the case and depends on both the nature of the measurement and confounding. We describe two sets of conditions under which adjusting for a non-deferentially mismeasured proxy comes closer to the unidentifiable true average causal effect than the unadjusted or crude estimate. The first set of conditions apply when the exposure is discrete or continuous and the confounder is ordinal, and the expectation of the outcome is monotonic in the confounder for both treatment levels contrasted. The second set of conditions apply when the exposure and the confounder are categorical (nominal). In all settings, the mismeasurement must be non-differential, as differential mismeasurement, particularly an unknown pattern, can cause unpredictable results.