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 |