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Table 2 Expressions of mediation measures under four different datatypes of the outcome and mediator. Case #1, continuous outcome and continuous mediator; Case #2, continuous outcome and binary mediator; Case #3, binary outcome and continuous mediator; and Case #4, binary outcome and binary mediator

From: Estimating the natural indirect effect and the mediation proportion via the product method

Datatypes

NIE

NDE

Depend on W

Reference(s)

Case #1

β2γ1(x−x∗)

β1(x−x∗)

No

[28]

Case #2

\( \beta _{2}\left \{\frac {\kappa (x,\boldsymbol w)-\kappa (x^{*},\boldsymbol w)}{(1+ \kappa (x,\boldsymbol w))(1+ \kappa (x^{*},\boldsymbol w))}\right \}\)

β1(x−x∗)

Yes

[32]

 

Approx.

β2γ1(x−x∗)

β1(x−x∗)

No

[23]

Case #3

Exact

\(\log \left \{\frac { \int _{m} \exp (\beta _{2} m)\tau (x,x,m,\boldsymbol w) \mathrm {d}m }{ \int _{m} \tau (x,x,m,\boldsymbol w) \mathrm {d}m }\right \} - \log \left \{\frac { \int _{m} \exp (\beta _{2} m)\tau (x^{*},x,m,\boldsymbol w)\mathrm {d}m }{ \int _{m} \tau (x^{*},x,m,\boldsymbol w) \mathrm {d}m }\right \}\)

\(\beta _{1}(x-x^{*})+\log \left \{\frac { \int _{m} \exp (\beta _{2} m)\tau (x^{*},x,m,\boldsymbol w)\mathrm {d}m }{ \int _{m} \tau (x^{*},x,m,\boldsymbol w) \mathrm {d}m }\right \}-\log \left \{\frac { \int _{m} \exp (\beta _{2} m)\tau (x^{*},x^{*},m,\boldsymbol w)\mathrm {d}m }{ \int _{m} \tau (x^{*},x^{*},m,\boldsymbol w) \mathrm {d}m }\right \}\)

Yes

Web Appendix A

 

Probit Approx.

\(\text {logit}\left (\Phi \left \{\frac {s\beta _{0}+s\beta _{1}x+s\beta _{3}^{T} \boldsymbol w + s\beta _{2}(\gamma _{0}+\gamma _{1}x+\gamma _{2}^{T} \boldsymbol w)}{\sqrt {1+s^{2}\beta _{2}^{2}\sigma ^{2}}}\right \}\right) - \text {logit}\left (\Phi \left \{\frac {s\beta _{0}+s\beta _{1}x+s\beta _{3}^{T} \boldsymbol w + s\beta _{2}(\gamma _{0}+\gamma _{1}x^{*}+\gamma _{2}^{T} \boldsymbol w)}{\sqrt {1+s^{2}\beta _{2}^{2}\sigma ^{2}}}\right \}\right)\)

\(\text {logit}\left (\Phi \left \{\frac {s\beta _{0}+s\beta _{1}x+s\beta _{3}^{T} \boldsymbol w + s\beta _{2}(\gamma _{0}+\gamma _{1}x^{*}+\gamma _{2}^{T} \boldsymbol w)}{\sqrt {1+s^{2}\beta _{2}^{2}\sigma ^{2}}}\right \}\right) - \text {logit}\left (\Phi \left \{\frac {s\beta _{0}+s\beta _{1}x^{*}+s\beta _{3}^{T} \boldsymbol w + s\beta _{2}(\gamma _{0}+\gamma _{1}x^{*}+\gamma _{2}^{T} \boldsymbol w)}{\sqrt {1+s^{2}\beta _{2}^{2}\sigma ^{2}}}\right \}\right)\)

Yes

[29]

Case #4

Approx.

\(\beta _{2}\left \{\frac {\kappa (x,\boldsymbol w)-\kappa (x^{*},\boldsymbol w)}{(1+ \kappa (x,\boldsymbol w))(1+ \kappa (x^{*},\boldsymbol w))}\right \}\)

β1(x−x∗)

Yes

[2]

 

Exact

\(\log \left \{\frac {1+e^{\beta _{2}}\eta (x,\boldsymbol w)+\kappa (x^{*},\boldsymbol w)(1+\eta (x,\boldsymbol w))} {1+e^{\beta _{2}}\eta (x,\boldsymbol w)+\kappa (x,\boldsymbol w)(1+\eta (x,\boldsymbol w))}\right \} - \log \left \{\frac {1+e^{\beta _{2}}\eta (x,\boldsymbol w)+e^{\beta _{2}}\kappa (x,\boldsymbol w)(1+\eta (x,\boldsymbol w))}{1+e^{\beta _{2}}\eta (x,\boldsymbol w)+e^{\beta _{2}}\kappa (x^{*},\boldsymbol w)(1+\eta (x,\boldsymbol w))}\right \}\)

\(\beta _{1}(x-x^{*})+\log \left \{\frac {1+e^{\beta _{2}}\eta (x^{*},\boldsymbol w)+\kappa (x^{*},\boldsymbol w)(1+\eta (x^{*},\boldsymbol w))} {1+e^{\beta _{2}}\eta (x,\boldsymbol w)+\kappa (x^{*},\boldsymbol w)(1+\eta (x,\boldsymbol w))}\right \} + \log \left \{\frac {1+e^{\beta _{2}}\eta (x,\boldsymbol w)+e^{\beta _{2}}\kappa (x^{*},\boldsymbol w)(1+\eta (x,\boldsymbol w))}{1+e^{\beta _{2}}\eta (x^{*},\boldsymbol w)+e^{\beta _{2}}\kappa (x^{*},\boldsymbol w)(1+\eta (x^{*},\boldsymbol w))}\right \}\)

Yes

[29] and Web Appendix B

  1. 1Note: NIE and NDE denote the natural indirect effect and natural direct effect, respectively, which are defined for X in change from x∗ to x conditional on W=w, on an identity scale in Cases #1 and #2 and a log odds ratio scale in Cases #3 and #4. Given NIE and NDE, the mediation proportion (MP) can be obtained by \(\frac {\text {NIE}}{\text {NIE}+\text {NDE}}\). In the probit approximation method, s=1/1.6, \(\text {logit}(x)=\log (\frac {x}{1-x})\), and Φ(.) is the cumulative density function for the standard normal distribution