Assessing whether the effect of exposure on an outcome is completely mediated by a third variable is often done by conditioning on the intermediate variable. However, when an association remains, it is not always clear how this should be interpreted. It may be explained by a causal direct effect of the exposure on the disease, or the adjustment may have been distorted due to various reasons, such as error in the measured mediator or unknown confounding of association between the mediator and the outcome.
In this paper, we study various situations where the conditional relationship between the exposure and the outcome is biased due to different types of measurement error in the mediator. For each of these situations, we quantify the effect on the association parameter. Such formulas can be used as tools for sensitivity analysis or to correct the association parameter for the bias due to measurement error. The performance of the bias formulas is studied by simulation and by applying them to data from a case-control study (Leiden Thrombophilia Study) on risk factors for venous thrombosis. In this study, the question was the extent to which the relationship between blood group and venous thrombosis might be mediated through coagulation factor VIII. We found that measurement error could have strongly biased the estimated direct effect of blood group on thrombosis. The formulas we propose can be a guide for researchers who find a residual association after adjusting for an intermediate variable and who wish to explore other possible explanations before concluding that there is a direct causal effect.