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Graphical explanation of causal mediation analysis

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Kazuki Yoshida
Kazuki Yoshida

The notion of the total effect decomposition into the natural direct effect and the natural indirect effect can be facilitated by intuitively understanding the nested counterfactual Y(1,M(0)).

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/43 Counterfactual definitions How is mediation formalized in the counterfactual framework? 5
/43 Counterfactual outcome Yi(a,m) What would happen to you if you were forced to receive treatment value a and mediator value m? 6
/43 Nested (composite) counterfactual outcome Yi(a,Mi(a*)) What would happen to you if you were forced to receive treatment value a and mediator value M(a*)? M(a*) is your natural value of the mediator under treatment value a*. When a = a* the counterfactual reduces to Yi(a) (causal consistency assumption). When a ¡Ù a* the counterfactual is a cross-world counterfactual. 7
/43 Natural effect decomposition Yi(1,Mi(1)) ? Yi(0,Mi(0)) Yi(1,Mi(1)) ? Yi(1,Mi(0)) + Yi(1,Mi(0)) ? Yi(0,Mi(0)) Yi(1) ? Yi(0) Total Effect Total Effect Natural Indirect Effect Natural Direct Effect Total Effect (as nested counterfactuals) = = 8
/43 Interpretation Pearl. Psychol Methods 2014;19:459 Naimi et al. Int J Epidemiol 2014;43:1656 Natural Direct Effect: Yi(1,Mi(0)) ? Yi(0,Mi(0)) Natural Indirect Effect: Yi(1,Mi(1)) ? Yi(1,Mi(0)) - change in the outcome when the mediator changes as though the exposure had (but when, in actuality, the exposure hadn¡¯t) - change in the outcome if we were to ¡®freeze¡¯ the mediator value for each person at the level it would have taken had the person¡¯s exposure been some referent level (but when, in actuality, the person¡¯s exposure status changes) 9 Characterizing Yi(a,Mi(a*)) more concretely may help¡­
/43 Treatment Assigned to drug 21 3 4 5 76 8 9 10 Yoshida & Desai A&R 2019 (modified) Treatment Assigned to placebo 21 3 4 5 76 8 9 10 Treated world Untreated world
/43 Treatment Assigned to drug Mediator Assigned to biomarker as if drug 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Yoshida & Desai A&R 2019 (modified) Treatment Assigned to placebo Mediator Assigned to biomarker as if placebo 21 3 4 5 76 8 9 10 11 Mediator under no treatment Mediator under treatment
/43 Treatment Assigned to drug Mediator Assigned to biomarker as if drug Outcome Gout flares Scenarios 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Yoshida & Desai A&R 2019 (modified) 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Treatment Assigned to placebo Mediator Assigned to biomarker as if placebo Outcome Gout flares 21 3 4 5 76 8 9 10 E[Y(0,M(0))] = E[Y(0)] E[Y(1,M(1))] = E[Y(1)] 12 Outcome under no treatment Outcome under treatment
/43 Treatment Assigned to drug Mediator Assigned to biomarker as if drug Outcome Gout flares Scenarios Total Effect Residual Risk 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Yoshida & Desai A&R 2019 (modified) 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Treatment Assigned to placebo Mediator Assigned to biomarker as if placebo Outcome Gout flares 21 3 4 5 76 8 9 10 E[Y(0,M(0))] = E[Y(0)] E[Y(1,M(1))] = E[Y(1)] 13
/43 Scenarios Total Effect Residual Risk 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Yoshida & Desai A&R 2019 (modified) Identical 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Treatment Assigned to drug Mediator Assigned to biomarker as if drug Outcome Gout flares Treatment Assigned to placebo Mediator Assigned to biomarker as if placebo Outcome Gout flares Exposure Assigned to drug E[Y(0,M(0))] E[Y(1,M(1))] 14 Take treatment from treated world
/43 Scenarios Total Effect Residual Risk 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Yoshida & Desai A&R 2019 (modified) Identical Identical 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Exposure Assigned to placebo Exposure Assigned to drug Mediator Assigned to biomarker as if placebo Mediator Assigned to biomarker as if placebo Exposure Assigned to drug Mediator Assigned to biomarker as if drug Outcome Gout flares Outcome Gout flares E[Y(0,M(0))] E[Y(1,M(1))] 15 Take mediators from untreated world
/43 Scenarios Total Effect Residual Risk Identical Identical 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Yoshida & Desai A&R 2019 (modified) 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Exposure Assigned to placebo Exposure Assigned to drug Mediator Assigned to biomarker as if placebo Mediator Assigned to biomarker as if placebo Exposure Assigned to drug Mediator Assigned to biomarker as if drug Outcome Gout flares Outcome Gout flares Outcome Gout flares E[Y(0,M(0))] E[Y(1,M(0))] E[Y(1,M(1))] 16 Hopefully, we get some intermediate
/43 Scenarios Natural Direct Effect Natural Indirect Effect Total Effect Residual Risk Identical Identical 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Yoshida & Desai A&R 2019 (modified) 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Exposure Assigned to placebo Exposure Assigned to drug Mediator Assigned to biomarker as if placebo Mediator Assigned to biomarker as if placebo Exposure Assigned to drug Mediator Assigned to biomarker as if drug Outcome Gout flares Outcome Gout flares Outcome Gout flares E[Y(0,M(0))] E[Y(1,M(0))] E[Y(1,M(1))] 17 TE partitioned nicely!

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Graphical explanation of causal mediation analysis

  • 1. /43 Counterfactual definitions How is mediation formalized in the counterfactual framework? 5
  • 2. /43 Counterfactual outcome Yi(a,m) What would happen to you if you were forced to receive treatment value a and mediator value m? 6
  • 3. /43 Nested (composite) counterfactual outcome Yi(a,Mi(a*)) What would happen to you if you were forced to receive treatment value a and mediator value M(a*)? M(a*) is your natural value of the mediator under treatment value a*. When a = a* the counterfactual reduces to Yi(a) (causal consistency assumption). When a ¡Ù a* the counterfactual is a cross-world counterfactual. 7
  • 4. /43 Natural effect decomposition Yi(1,Mi(1)) ? Yi(0,Mi(0)) Yi(1,Mi(1)) ? Yi(1,Mi(0)) + Yi(1,Mi(0)) ? Yi(0,Mi(0)) Yi(1) ? Yi(0) Total Effect Total Effect Natural Indirect Effect Natural Direct Effect Total Effect (as nested counterfactuals) = = 8
  • 5. /43 Interpretation Pearl. Psychol Methods 2014;19:459 Naimi et al. Int J Epidemiol 2014;43:1656 Natural Direct Effect: Yi(1,Mi(0)) ? Yi(0,Mi(0)) Natural Indirect Effect: Yi(1,Mi(1)) ? Yi(1,Mi(0)) - change in the outcome when the mediator changes as though the exposure had (but when, in actuality, the exposure hadn¡¯t) - change in the outcome if we were to ¡®freeze¡¯ the mediator value for each person at the level it would have taken had the person¡¯s exposure been some referent level (but when, in actuality, the person¡¯s exposure status changes) 9 Characterizing Yi(a,Mi(a*)) more concretely may help¡­
  • 6. /43 Treatment Assigned to drug 21 3 4 5 76 8 9 10 Yoshida & Desai A&R 2019 (modified) Treatment Assigned to placebo 21 3 4 5 76 8 9 10 Treated world Untreated world
  • 7. /43 Treatment Assigned to drug Mediator Assigned to biomarker as if drug 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Yoshida & Desai A&R 2019 (modified) Treatment Assigned to placebo Mediator Assigned to biomarker as if placebo 21 3 4 5 76 8 9 10 11 Mediator under no treatment Mediator under treatment
  • 8. /43 Treatment Assigned to drug Mediator Assigned to biomarker as if drug Outcome Gout flares Scenarios 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Yoshida & Desai A&R 2019 (modified) 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Treatment Assigned to placebo Mediator Assigned to biomarker as if placebo Outcome Gout flares 21 3 4 5 76 8 9 10 E[Y(0,M(0))] = E[Y(0)] E[Y(1,M(1))] = E[Y(1)] 12 Outcome under no treatment Outcome under treatment
  • 9. /43 Treatment Assigned to drug Mediator Assigned to biomarker as if drug Outcome Gout flares Scenarios Total Effect Residual Risk 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Yoshida & Desai A&R 2019 (modified) 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Treatment Assigned to placebo Mediator Assigned to biomarker as if placebo Outcome Gout flares 21 3 4 5 76 8 9 10 E[Y(0,M(0))] = E[Y(0)] E[Y(1,M(1))] = E[Y(1)] 13
  • 10. /43 Scenarios Total Effect Residual Risk 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Yoshida & Desai A&R 2019 (modified) Identical 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Treatment Assigned to drug Mediator Assigned to biomarker as if drug Outcome Gout flares Treatment Assigned to placebo Mediator Assigned to biomarker as if placebo Outcome Gout flares Exposure Assigned to drug E[Y(0,M(0))] E[Y(1,M(1))] 14 Take treatment from treated world
  • 11. /43 Scenarios Total Effect Residual Risk 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Yoshida & Desai A&R 2019 (modified) Identical Identical 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Exposure Assigned to placebo Exposure Assigned to drug Mediator Assigned to biomarker as if placebo Mediator Assigned to biomarker as if placebo Exposure Assigned to drug Mediator Assigned to biomarker as if drug Outcome Gout flares Outcome Gout flares E[Y(0,M(0))] E[Y(1,M(1))] 15 Take mediators from untreated world
  • 12. /43 Scenarios Total Effect Residual Risk Identical Identical 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Yoshida & Desai A&R 2019 (modified) 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Exposure Assigned to placebo Exposure Assigned to drug Mediator Assigned to biomarker as if placebo Mediator Assigned to biomarker as if placebo Exposure Assigned to drug Mediator Assigned to biomarker as if drug Outcome Gout flares Outcome Gout flares Outcome Gout flares E[Y(0,M(0))] E[Y(1,M(0))] E[Y(1,M(1))] 16 Hopefully, we get some intermediate
  • 13. /43 Scenarios Natural Direct Effect Natural Indirect Effect Total Effect Residual Risk Identical Identical 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Yoshida & Desai A&R 2019 (modified) 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 21 3 4 5 76 8 9 10 Exposure Assigned to placebo Exposure Assigned to drug Mediator Assigned to biomarker as if placebo Mediator Assigned to biomarker as if placebo Exposure Assigned to drug Mediator Assigned to biomarker as if drug Outcome Gout flares Outcome Gout flares Outcome Gout flares E[Y(0,M(0))] E[Y(1,M(0))] E[Y(1,M(1))] 17 TE partitioned nicely!