. 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 4.

Equations used to simulate potential mediators and outcomes in step 2 of the G-computation marginal structural model for the illustrative example^a

	M model and Y model from step 1^b:
	E (M\|x, z; α) = α_M + α_X · x + α_Z · z
	E (M\|x, m, z; β) = β_Y + β_X · x + β_M · m + β_XM · x · m + β_Z · z

Effect	Simulating M^c		Simulating Y^c
TE^d	M_x	= α_M + α_X · x + α_Z · z + ε_M^e	Y_TE	= β_Y + β_X · x + β_M · m_x + β_XM · x · m_x + β_Z · z + ε_Y^e
PDE	M₀	= α_M + α_X · 0 + α_Z · z + ε_M	Y_PDE	= β_Y + β_X · x + β_M · m₀ + β_XM · x · m₀ + β_Z · z + ε_Y
TIE	M_x	= α_M + α_X · x + α_Z · z + ε_M	Y_TIE	= β_Y + β_X · 1 + β_M · m_x + β_XM · 1 · m_x + β_Z · z + ε_Y
TDE	M₁	= α_M + α_X · 1 + α_Z · z + ε_M	Y_TDE	= β_Y + β_X · x + β_M · m₁ + β_XM · x · m₁ + β_Z · z + ε_Y
PIE	M_x	= α_M + α_X · x + α_Z · z + ε_M	Y_PIE	= β_Y + β_X · 0 + β_M · m_x + β_XM · 0 · m_x + β_Z · z + ε_Y
CDE_M=m^*	M^*	= 4.8^f	Y_CDE	= β_Y + β_X · x + β_M · m^* + β_XM · x · m^* + β_Z · z + ε_Y
CDE_sto	M′	= E (M) + ε_M	Y_CDEsto	= β_Y + β_X · x + β_M · m′ + β_XM · x · m′ + β_Z · z + ε_Y
RIE	M₀	= α_M + α_X · 0 + α_Z · z + ε_M	Y_RIE^g	= β_Y + 0 · x + 0 · m_x + β_XM · x · (m₀ − m^) + β_Z · z + ε_Y*
MIE	M_med^h	= α_X · x + ε_M	Y_MIE^g	= β_Y + 0 · x + 0 · m_x + β_XM · x · m_med + β_Z · z + ε_Y
PAI	M₁	= α_M + α_X · 1 + α_Z · z + ε_M	Y_PAI^g	= β_Y + 0 · x + 0 · m_x + β_XM · x · (m₁ − m^) + β_Z · z + ε_Y*

Exposure: smoking (1=yes, 0=no); mediator: body mass index (5-unit increase); outcome: composite health score; covariates: age, gender, education, urbanicity, depression.

Variables used to fit the M model and Y model in step 1 were observed variables.

Lower case “x”, “z”, and “m_x” represented specific values of the random variables intervention “X”, simulated covariate set “Z”, and potential mediator “M_x” respectively and the values can differ for different individuals. Intervention “X” was independent of the simulated covariate set “Z”.

TE: 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.

The root mean square error (RMSE) from the M model and the Y model in step 1 respectively.

The mediator was fixed at 4.8 (BMI=24).

To simulate Y_RIE, Y_MIE, and Y_PAI, we assigned zero for the coefficients for random variables intervention “X” and the potential mediator “M_x” but not the coefficients for the product term between these two variables to mimic “de-activating” the direct and indirect path from X to Y, leaving only a specific type of “interaction” between X and M to transmit the effect of X to Y.

The mediated interaction captures the interaction between X and a version of M that is due to X only. Thus, to simulate M_med, the mediator M responds to no other determinants of M but X.