Table 1.

Closed-form formulas for NDE and NIE with different mediator and outcome models.

NDE
Linear mediator model, Linear outcome model	$\left(a-a^{\text{*}}\right)\left[{\theta }_1+{\theta }_3\left({\beta }_0+{\beta }_1{a}^{\text{*}}+{\beta }_{2}c+{\beta }_3{a}^{\text{*}}c\right)+{\theta }_{5}c\right]$
Logistic mediator model, Linear outcome model	$\left(a-a^{\text{*}}\right)\left[{\theta }_1+{\theta }_3\cdot \text{expit}\left({\beta }_0+{\beta }_1{a}^{\text{*}}+{\beta }_{2}c+{\beta }_3{a}^{\text{*}}c\right)+{\theta }_{5}c\right]$
Linear mediator model, Non-linear outcome model	$\text{exp}\left(\left[a-a^{\text{*}}\right]\cdot \left[{\theta }_1+{\theta }_{5}c+{\theta }_3\left({\beta }_0+{\beta }_1{a}^{\text{*}}+{\beta }_{2}c+{\beta }_3{a}^{\text{*}}c\right)+\frac{1}{2}{\sigma }^2{\theta }_3\left(2{\theta }_2+{\theta }_{3}a+{\theta }_3{a}^{\text{*}}+2{\theta }_{6}c\right)\right]\right)$
Logistic mediator model, Non-linear outcome model	$\text{exp}\left[\left({\theta }_1+{\theta }_{5}c\right)\left(a-a^{\text{*}}\right)\right]\frac{1+\text{exp}\left({\beta }_0+{\beta }_1{a}^{\text{*}}+{\beta }_{2}c+{\beta }_3{a}^{\text{*}}c+{\theta }_2+{\theta }_{3}a+{\theta }_{6}c\right)}{1+\text{exp}\left({\beta }_0+{\beta }_1{a}^{\text{*}}+{\beta }_{2}c+{\beta }_3{a}^{\text{*}}c+{\theta }_2+{\theta }_3{a}^{\text{*}}+{\theta }_{6}c\right)}$
NIE
Linear mediator model, Linear outcome model	$\left({\theta }_2+{\theta }_{3}a+{\theta }_{6}c\right)\left({\beta }_1+{\beta }_{3}c\right)\left(a-a^{\text{*}}\right)$
Logistic mediator model, Linear outcome model	$\left({\theta }_2+{\theta }_{3}a+{\theta }_{6}c\right)\left[\text{expit}\left({\beta }_0+{\beta }_{1}a+{\beta }_{2}c+{\beta }_{3}ac\right)-\text{expit}\left({\beta }_0+{\beta }_1{a}^{\text{*}}+{\beta }_{2}c+{\beta }_3{a}^{\text{*}}c\right)\right]$
Linear mediator model, Non-linear outcome model	$\text{exp}\left\{\left({\theta }_2+{\theta }_{3}a+{\theta }_{6}c\right)\left({\beta }_1+{\beta }_{3}c\right)\left(a-a^{\text{*}}\right)\right\}$
Logistic mediator model, Non-linear outcome model	$\frac{1+\text{exp}\left({\beta }_0+{\beta }_{1}a+{\beta }_{2}c+{\beta }_{3}ac+{\theta }_2+{\theta }_{3}a+{\theta }_{6}c\right)}{1+\text{exp}\left({\beta }_0+{\beta }_1{a}^{\text{*}}+{\beta }_{2}c+{\beta }_3{a}^{\text{*}}c+{\theta }_2+{\theta }_{3}a+{\theta }_{6}c\right)}\cdot \frac{1+\text{exp}\left({\beta }_0+{\beta }_1{a}^{\text{*}}+{\beta }_{2}c+{\beta }_3{a}^{\text{*}}c\right)}{1+\text{exp}\left({\beta }_0+{\beta }_{1}a+{\beta }_{2}c+{\beta }_{3}ac\right)}$

The formulas for non-linear outcome model accommodate 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). NDE=natural direct effect; NIE=natural indirect effect.