Mediation and mechanism

Abstract

The concepts of mediation and mechanism are contrasted and logical implications holding between theses two concepts are described. The concept of mediation can be formalized using counterfactual definitions of indirect effects; the concept of mechanism can be formalized within the sufficient cause framework. It is shown that both concepts can be illustrated using a single causal diagram. It is also shown that mediation implies mechanism but mechanism need not imply mediation. Discussion is given regarding how the distinction between “statistical causality” and “mechanistic causality” is blurred by recent work in causal inference concerning methods for testing for mediation and mechanism.

Technical appendix

Mediation, controlled direct effects, and natural direct and indirect effects

Note that the total effect Y₁−Y₀ decomposes as the sum of a total direct effect and a pure indirect effect, Y₁−Y₀ = ${\rm{(Y_{1M_{1}}}}\!-\!{\rm{Y_{0M_{1}})}}+{\rm{(Y_{0M_{1}}}}\!-\!{\rm{Y_{0M_{0}})}}$, or as the sum of a total indirect effect and a pure direct effect, Y₁−Y₀ = ${\rm{(Y_{1M_{1}}}}\!-\!{\rm{Y_{1M_{0}})}}+{\rm{(Y_{1M_{0}}}}\!-\!{\rm{Y_{0M_{0}})}}$. If both the pure indirect effect and the total indirect effect are zero, i.e., if both ${\rm{Y_{0M_{1}}}}\!-\!{\rm{Y_{0M_{0}}}=0}$ and ${\rm{Y_{1M_{1}}}}\!-\!{\rm{Y_{0M_{1}}}=0}$, then we will have that Y₁−Y₀ = ${\rm{Y_{1M_{1}}}}\!-\!{\rm{Y_{0M_{1}}}}$ and that Y₁−Y₀ = ${\rm{Y_{1M_{0}}}}\!-\!{\rm{Y_{0M_{0}}}}$ so that the total effect, the total direct effect and the pure direct effect all coincide. We thus say that “M mediates the effect of X on Y” whenever one of the natural indirect effects is non-zero since, if they are both zero then the total effect and the natural direct effects coincide.

Note that it is more difficult to use the controlled direct effect to draw conclusions about mediation. The controlled direct effect takes the form Y_1m−Y_0m. Even if X has no effect on M so that there is no mediation, the controlled direct effect Y_1m−Y_0m may differ from the total effect. For example, suppose that X has no effect on M so that there is no mediation of the effect of X on Y by M but suppose there is interaction between X and M. If there is interaction between the effects of X and M on Y, then Y_1m−Y_0m will differ for different values of m and thus one of the controlled direct effects Y₁₁−Y₀₁ or Y₁₀−Y₀₀ will differ from Y₁−Y₀. Obtaining a controlled direct effect that is different than the total effect is thus not evidence that mediation is present. If there is no interaction between the effects of X and M on Y then Robins [35] has shown that the controlled direct, the total direct effect and the pure direct effect all coincide; furthermore all of these quantities will be equal to the total effect if there are no natural indirect effects; in this case one can compare total effects and controlled direct effects to assess mediation. Thus only under the assumption of no interaction can one use controlled direct effects to assess mediation. See the work of Robins and Greenland [7] and Kaufman et al. [9] for further critique of using controlled direct effects to assess mediation and indirect effects.

Relationship between direct and indirect effects, with a sufficient cause taken as the mediator, and the probabilities of the background component causes

Here we express controlled direct effects, with some sufficient cause S taken as the mediator, in terms of the probabilities of the background components of the sufficient causes in Fig. 1. Note that if S were set to 1 then Y would be 1 since S is a sufficient cause for Y; it thus does not make sense to discuss controlled direct effects of the form Y_x=1,s=1−Y_x=0,s=1 since this controlled direct effect is zero for any sufficient cause and we report only the controlled direct effects of the form Y_x=1,s=0−Y_x=0,s=0. Note that a binary variable Z can be treated as an event and so we will let P(Z) denote P(Z = 1).

For S = BM:

$$ \begin{aligned} {\text{E}}\left[ {{\text{Y}}_{{{\text{x}} =\,1,{\text{s}} =\,0}} - {\text{Y}}_{{{\text{x}} =\,0,{\text{s}} =\,0}} } \right] & =\,{\text{P}}({\text{L or C}}\,{\text{or FM}}_{{{\text{x}} =\,1}} ) - {\text{ P}}({\text{L}}) \\ & =\,{\text{ P}}\left[ {{\text{L}}^{\text{c}} ({\text{C\,or\,FM}}_{{{\text{x}} =\,1}} )} \right] \\ & =\,{\text{ P}}\left[ {{\text{L}}^{\text{c}} {\text{C\,or\,L}}^{\text{c}} {\text{F}}({\text{A\,or\,K}})} \right] \\ \end{aligned} $$

For S = CX:

$$ \begin{aligned} {\text{E}}\left[ {{\text{Y}}_{\text{x = 1,s = 0}} - {\text{Y}}_{{{\text{x}} = 0,{\text{s}} = 0}} } \right] & =\,{\text{P}}({\text{L}}\;{\text{or\;BM}}_{{{\text{x}} = 1}} {\text{ \;or \;FM}}_{{{\text{x}} = 1}} ) - {\text{P}}({\text{L\;or\;BM}}_{{{\text{x}} = 0}} ) \\ & = {\text{ P}}\left[ {{\text{L\;or\;B}}({\text{A\;or\;K}}){\text{ or\;F}}({\text{A\;or\;K}})} \right] - {\text{P}}({\text{L\;or\;BK}}) \\ & = {\text{ P}}\left[ {{\text{L}}^{\text{c}} {\text{K}}^{\text{c}} {\text{BA\;or\;L}}^{\text{c}} {\text{K}}^{\text{c}} {\text{FA}}\,{\text{or\;L}}^{\text{c}} {\text{B}}^{\text{c}} {\text{F}}({\text{A}}\,{\text{or\;K}})} \right] \\ \end{aligned} $$

For S = FXM:

$$ \begin{aligned} {\text{E}}\left[ {{\text{Y}}_{{{\text{x}} = 1,{\text{s}} = 0}} - {\text{Y}}_{{{\text{x}} = 0,{\text{s}} = 0}} } \right] & = {\text{P}}({\text{L}}\,{\text{ or\;BM}}_{{{\text{x}} = 1}} {\text{\;or\;C}}) - {\text{P}}({\text{L\;or\;BM}}_{{{\text{x}} = 0}} ) \\ & = {\text{ P}}\left[ {{\text{L\;or\;B}}({\text{A\;or\;K}}){\text{ or\;C}}} \right] - {\text{P}}({\text{L\;or\;BK}}) \\ & = {\text{ P}}\left[ {{\text{L}}^{\text{c}} {\text{K}}^{\text{c}} {\text{BA\;or\;L}}^{\text{c}} ({\text{B}}^{\text{c}}\;{\text{or \;K}}^{\text{c}} ){\text{C}}} \right] \\ \end{aligned} $$