. Author manuscript; available in PMC: 2024 Sep 1.

Published in final edited form as: Epidemiology. 2023 Jul 31;34(5):661–672. doi: 10.1097/EDE.0000000000001643

Table 5.

Summary of effect modification by covariates, stratified by combinations of scenarios and link functions.

Model 1: linear mediator model, linear outcome model
Simulation	Coefficient	Inclusion of product terms	Dependence on covariates
Scenario			NDE	NIE
1	β₃ = θ₅ = θ₆ = 0	only A×M	linearly^a	constant
2	θ₅ = θ₆ = 0	adding A×C in mediator model	linearly	linearly
3	θ₆ = 0	adding A×C in outcome model	linearly	linearly
4		adding M×C in outcome model	linearly	non-linearly

Model 2: logistic mediator model, linear outcome model
Simulation	Coefficient	Inclusion of product terms	Dependence on covariates
Scenario			NDE	NIE
1	β₃ = θ₅ = θ₆ = 0	only A×M	non-linearly^a	non-linearly
2	θ₅ = θ₆ = 0	adding A×C in mediator model	non-linearly	non-linearly
3	θ₆ = 0	adding A×C in outcome model	non-linearly	non-linearly
4		adding M×C in outcome model	non-linearly	non-linearly

Model 3: linear mediator model, non-linear outcome model ^b
Simulation	Coefficient	Inclusion of product terms	Dependence of covariates
Scenario			NDE	NIE
1	β₃ = θ₅ = θ₆ = 0	only A×M	non-linearly^a	constant
2	θ₅ = θ₆ = 0	adding A×C in mediator model	non-linearly	non-linearly
3	θ₆ = 0	adding A×C in outcome model	non-linearly	non-linearly
4		adding M×C in outcome model	non-linearly	non-linearly

Model 4: logistic mediator model, non-linear outcome model ^b
Simulation	Coefficient	Inclusion of product terms	Dependence of covariates
Scenario			NDE	NIE
1	β₃ = θ₅ = θ₆ = 0	only A×M	non-linearly^a	non-linearly
2	θ₅ = θ₆ = 0	adding A×C in mediator model	non-linearly	non-linearly
3	θ₆ = 0	adding A×C in outcome model	non-linearly	non-linearly
4		adding M×C in outcome model	non-linearly	non-linearly

: NDE is constant when θ₃ = 0. This is corresponding to the special case shown in Figure 2.

: The non-linear outcome model accommodates logistic, log-linear, Poisson, negative binomial, accelerated failure time model and a Cox proportional hazards model, with appropriate link functions and modeling assumptions. Note that logistic and Cox proportional hazards model require the outcome to be rare. Details are in the Supplement (Section 2.1).

Abbreviations. NDE: natural direct effect; NIE: natural indirect effect.

Coefficients.

θ₃: A×M causal interaction in outcome model

β₃: A×C effect measure modification in mediator model

θ₅: A×C effect measure modification in outcome model

θ₆: M×C effect measure modification in outcome model