. Author manuscript; available in PMC: 2016 Nov 4.

Published in final edited form as: Eur J Epidemiol. 2015 Nov 4;30(10):1119–1127. doi: 10.1007/s10654-015-0100-z

Table 2.

Definition and empirical analog of total effect and component effects^a

Effect	Counterfactual definition	Empirical analog^b
TE^c	E[Y_x − Y_x^*]^d	∑_z∑_m{E (Y\|x, m, z) P (m\|x, z) − E (Y\|x^, m, z) P (m\|x^, z)}P (z)^e
PDE	E[Y_{xM_x^} − Y_{x^M_x^*}]	∑_z∑_m{E (Y\|x, m, z) − (Y\|x^, m, z)}P (m\|x^, z) P (z)^f
TIE	E[Y_{xM_x} − Y_{xM_x^*}]	∑_z∑_m E (Y\|x, m, z){P (m\|x, z) − P (m\|x^*, z)}P (z)
TDE	E[Y_{xM_x} − Y_{x^*M_x}]	∑_z∑_m{E (Y\|x, m, z) − (Y\|x^, m, z)}P (m\|x, z*) P (z)
PIE	E[Y_{x^M_x} − Y_{x^M_x^*}]	∑_z∑_m E (Y\|x^, m, z){P (m\|x, z) − P (m\|x^, z)}P (z)^f
CDE_M=m^*	E[Y_xm^* − Y_x^m^]	∑_z{E (Y\|x, m^, z) − E (Y\|x^, m^*, z)}P (z)
CDE_sto	E[Y_xM′ − Y_x^*M′]	∑_z∑_m{E (Y\|x, m, z) − E (Y\|x^, m, z*)}P (m′) P (z)
RIE	E[(Y_xm − Y_xm^* − Y_x^m + Y_x^m^)(M_x^)]	∑_z∑_m{E (Y\|x, m, z) − E (Y\|x, m^, z) − E (Y\|x^, m, z) + E (Y\|x^, m^, z)} P (m\|x^*, z) P (z)
MIE	E[(Y_xm − Y_xm^* − Y_x^m + Y_x^m^)(M_x* − M_x^*)]	∑_z∑_m{E (Y\|x, m, z) − E (Y\|x, m^, z) − E (Y\|x^, m, z) + E (Y\|x^, m^, z)} {P (m\|x, z) − P (m\|x^*, z)}P (z)
PAI	E[(Y_xm − Y_xm^* − Y_x^m + Y_x^m^)(M_x*)]	∑_z∑_m{E (Y\|x, m, z) − E (Y\|x, m^, z) − E (Y\|x^, m, z) + E (Y\|x^, m^, z)} P (m\|x, z) P (z)

Y: outcome, X: exposure, M: mediator, Z: covariates; x and m represent the index values whereas x^* and m^* represent the reference values.

Under the stable unit treatment value assumption, consistency, conditional exchangeability, positivity, different types of effect can be identified and estimated using the empirical analogs We use E (Y|x, m, z) as a shorthand for E (Y|X = x, M = m, Z = z), and P (m|x, z) as a shorthand for P (M = m|X = x, Z = z).

E: total effect, PDE: pure direct effect, TIE: total indirect effect, TDE: total direct effect, PIE: pure indirect effect, CDE: controlled direct effect (standard), CDE_sto: stochastic controlled direct effect, RIE: reference interaction effect (referred to as “INT_ref” by VanderWeele), MIE: mediated interaction effect (referred to as “INT_med” by VanderWeele), PAI: portion attributable to interaction.

Effects are defined as risk differences here but other measures of effects are possible (risk ratio, odds ratio etc.).

For continuous M and Z, summations are replaced by integrals and the probability functions by appropriate density functions

These two expressions are known as the mediation formula [47].