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. 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 3.

Sufficient conditions for constant conditional natural direct effect (NDE) and natural indirect effect (NIE) estimates with respect to covariates.

NDE Linear outcome model Non-linear outcome model*
Linear mediator model θ5 = 0 AND {θ3 = 0 OR β2 = β3 = θ6 = 0}
Logistic mediator model θ5 = 0 AND β2 = β3 = θ6 = 0
NIE Linear outcome model Non-linear outcome model*
Linear mediator model β3 = θ6 = 0
Logistic mediator model β2 = β3 = θ6 = 0

Abbreviations. EMM: effect measure modification; A: exposure; C: covariates; M: mediator.

Coefficients.

θ5 = 0: no EMM of the exposure effect by covariates [A×C] on outcome

θ3 = 0: no A×M interaction

β2 = 0: no confounder in mediator outcome

β3 = 0: no EMM of the exposure effect by covariates [A×C] on mediator

θ6 = 0: no EMM of the mediator effect by covariates [M×C] on outcome

*

: 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).