Bias formulas for sensitivity analysis for direct and indirect effects

Abstract

A key question in many studies is how to divide the total effect of an exposure into a component that acts directly on the outcome and a component that acts indirectly, ie, through some intermediate. For example, one might be interested in the extent to which the effect of diet on blood pressure is mediated through sodium intake and the extent to which it operates through other pathways. In the context of such mediation analysis, even if the effect of the exposure on the outcome is unconfounded, estimates of direct and indirect effects will be biased if control is not made for confounders of the mediator-outcome relationship. Often data are not collected on such mediator-outcome confounding variables; the results in this paper allow researchers to assess the sensitivity of their estimates of direct and indirect effects to the biases from such confounding. Specifically, the paper provides formulas for the bias in estimates of direct and indirect effects due to confounding of the exposure-mediator relationship and of the mediator-outcome relationship. Under some simplifying assumptions, the formulas are particularly easy to use in sensitivity analysis. The bias formulas are illustrated by examples in the literature concerning direct and indirect effects in which mediator-outcome confounding may be present.

Figure 1

Causal diagram in which the unmeasured confounder U confounds only the mediator-outcome relationship.

Figure 2

Causal diagram in which the unmeasured confounder U confounds the exposure-outcome, mediator-outcome and exposure-mediator relationship.

Figure 3

Values of δ and γ that lie below the curve would reverse the sign of the direct effect point estimate.