The Causal Mediation Formula—A Guide to the Assessment of Pathways and Mechanisms

This can be seen vividly by setting α = γ = 0, implying zero direct and indirect effects; yet, if ϵ ₂ and ϵ ₃ are correlated, the regression coefficient R _{YX ·Z} will not equal zero, but − βcov(ϵ ₂, ϵ ₃)/var(ϵ ₂).

Although Judd and Kenny (1981) recognized the importance of controlling for mediator-output confounders, the point was not mentioned in the influential paper of Baron and Kenny (1986) and, as a result, it has been ignored by most researchers in the social and psychological sciences (Judd and Kenny 2010).

This follows from the fact that the regressional image of Eq. 1, R _YX  − R _{YX ·Z}  = R _ZX R _{YZ ·X} , is a universal identity among regression coefficients of any three variables, and has nothing to do with causation or mediation. It will continue to hold regardless of whether confounders are present, whether the underlying model is linear or nonlinear, or whether the arrows in the model of Fig. 1a point in the right direction. The equality also holds among the OLS estimates of these parameters regardless of sample size (Hahn and Pearl 2011). Note the essential distinction between structural and regressional parameters, often conflated by some writers (Rubin 2010; Sobel 2008); the former convey causal relationships, the latters are purely statistical (Pearl 2011c). Conditions for their equality can be found in p. 150 Pearl (2009).

A complete set of techniques is now available for neutralizing error dependencies, whenever possible, both by covariate adjustment and through the use of instrumental variables (Pearl 2009; Shpitser and Pearl 2008; Tian and Shpitser 2010). These techniques are directly applicable to the analysis of mediations (Pearl 2009, p. 128; Pearl 2011a, d; Shpitser and VanderWeele 2011), but are beyond the scope of this paper.

The general causal expression for CDE(z), which does not assume error-independence is given by: (see Pearl 2009, p. 127) or, using the structural equations of Eq. 2, A necessary and sufficient condition for estimating CDE(z) in observational studies (in the presence of unobserved confounders and any set Z of mediators) can be derived using do-calculus (Pearl 2009, pp. 85–88), and is given in Shpitser and Pearl (2008) and Tian and Shpitser (2010).

Using the structural model of Eq. 2, the formal definition of the natural direct effect reads: Robins and Greenland (1992) called this notion of direct effect “Pure” while Pearl called it “Natural,” to stress the natural, unperturbed distribution of values, Z = F ₂(x, ϵ ₂) at which we “freeze” Z while changing X from X = x to X = x′. For discussions regarding policy implications of NDE versus CDE, see (Albert and Nelson 2011; Hafeman and Schwartz 2009; Joffe et al. 2007; Kaufman 2010; Pearl 2001, 2009, p. 132; Robins 2003; Robins and Richardson 2011).

In the presence of measured and unmeasured confounders, the general conditions under which NDE is estimable from population data are somewhat more stringent than those needed for CDE (footnote 5). For details see Avin et al. (2005), Kaufman (2010), Pearl (2001, 2011d), Petersen et al. (2006), Robins (2003), Robins and Richardson (2011), and Shpitser and VanderWeele (2011) and VanderWeele (2009).

By “explain” we mean “sufficient to sustain even in the absence of direct effect.” By “owed to” we mean “would not occur absent of mediation.” These interpretations follow from the counterfactual definitions formulated in Section “Total, Direct and Indirect Effects”, of which Eqs. 6 and 8 are derived statistical estimands.

The degree of moderation exerted by any variable Z is measured by the difference between the controlled direct effects at two levels of Z, CDE(Z = z ₁) − CDE(Z = z ₀) (see Eq. 3). As in mediation, when confounding is present, an unbiased estimation of moderation requires adjustments for covariates that can be identified by graphical methods (see footnote 5).

In practice, the estimates produced may still suffer from misspecification bias, finite-sample bias, and sample-selection bias (see Bareinboim and Pearl 2011) and one should also address the question of generalizability (or external validity), as treated in Pearl and Bareinboim (2011).