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Counterfactual-based mediation analysis Workshop 2 Rhian Daniel London School of Hygiene and Tropical Medicine CIMPOD 28th February, 2017 Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 1/55 Setting the scene Case study

  1. Setting the scene Case study Q&A Wrapping up References Causal diagram M 1 C M 2 X Y • We want to separate the effect of SES on survival into an effect via screening and an effect via treatment, and an effect via neither. • This is complicated by the fact that M 1 can affect M 2 . • In fact, we don’t have data on screening, but we’ll use age and stage at diagnosis as a proxy for screening. • So our M 1 is in fact a vector. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 9/55

  2. Setting the scene Case study Q&A Wrapping up References Outline Setting the scene 1 Quick summary of yesterday Today’s case study Mediation analysis with multiple mediators Sequential mediation analysis Interventional effects for multiple mediators Case study 2 Q&A 3 References 4 Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 10/55

  3. Setting the scene Case study Q&A Wrapping up References Counterfactuals and estimands for multiple mediators — With one mediator, we needed: M ( x ) , Y ( x , m ) , Y ( x , M ( x ′ )) Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 11/55

  4. Setting the scene Case study Q&A Wrapping up References Counterfactuals and estimands for multiple mediators — With one mediator, we needed: M ( x ) , Y ( x , m ) , Y ( x , M ( x ′ )) — With two, we need: M 1 ( x ) , M 2 ( x , m 1 ) , Y ( x , m 1 , m 2 ) Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 11/55

  5. Setting the scene Case study Q&A Wrapping up References Counterfactuals and estimands for multiple mediators — With one mediator, we needed: M ( x ) , Y ( x , m ) , Y ( x , M ( x ′ )) — With two, we need: M 1 ( x ) , M 2 ( x , m 1 ) , Y ( x , m 1 , m 2 ) and M 2 ( x , M 1 ( x ′ )) Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 11/55

  6. Setting the scene Case study Q&A Wrapping up References Counterfactuals and estimands for multiple mediators — With one mediator, we needed: M ( x ) , Y ( x , m ) , Y ( x , M ( x ′ )) — With two, we need: M 1 ( x ) , M 2 ( x , m 1 ) , Y ( x , m 1 , m 2 ) and M 2 ( x , M 1 ( x ′ )) and Y ( x , M 1 ( x ′ ) , M 2 ( x ′′ , M 1 ( x ′′′ ))) Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 11/55

  7. Setting the scene Case study Q&A Wrapping up References Counterfactuals and estimands for multiple mediators — With one mediator, we needed: M ( x ) , Y ( x , m ) , Y ( x , M ( x ′ )) — With two, we need: M 1 ( x ) , M 2 ( x , m 1 ) , Y ( x , m 1 , m 2 ) and M 2 ( x , M 1 ( x ′ )) and Y ( x , M 1 ( x ′ ) , M 2 ( x ′′ , M 1 ( x ′′′ ))) — Natural path-specific effects are defined as contrasts between these for carefully chosen values of x , x ′ , x ′′ , x ′′′ . Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 11/55

  8. Setting the scene Case study Q&A Wrapping up References Counterfactuals and estimands for multiple mediators — With one mediator, we needed: M ( x ) , Y ( x , m ) , Y ( x , M ( x ′ )) — With two, we need: M 1 ( x ) , M 2 ( x , m 1 ) , Y ( x , m 1 , m 2 ) and M 2 ( x , M 1 ( x ′ )) and Y ( x , M 1 ( x ′ ) , M 2 ( x ′′ , M 1 ( x ′′′ ))) — Natural path-specific effects are defined as contrasts between these for carefully chosen values of x , x ′ , x ′′ , x ′′′ . Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 11/55

  9. Setting the scene Case study Q&A Wrapping up References Counterfactuals and estimands for multiple mediators — With one mediator, we needed: M ( x ) , Y ( x , m ) , Y ( x , M ( x ′ )) — With two, we need: M 1 ( x ) , M 2 ( x , m 1 ) , Y ( x , m 1 , m 2 ) and M 2 ( x , M 1 ( x ′ )) and Y ( x , M 1 ( x ′ ) , M 2 ( x ′′ , M 1 ( x ′′′ ))) — Natural path-specific effects are defined as contrasts between these for carefully chosen values of x , x ′ , x ′′ , x ′′′ . Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 11/55

  10. Setting the scene Case study Q&A Wrapping up References Direct effect — A natural direct effect (through neither M 1 nor M 2 ) is of the form: Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 12/55

  11. Setting the scene Case study Q&A Wrapping up References Direct effect — A natural direct effect (through neither M 1 nor M 2 ) is of the form: E { Y ( 1 , M 1 ( x ′ ) , M 2 ( x ′′ , M 1 ( x ′′′ ))) − Y ( 0 , M 1 ( x ′ ) , M 2 ( x ′′ , M 1 ( x ′′′ ))) } Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 12/55

  12. Setting the scene Case study Q&A Wrapping up References Direct effect — A natural direct effect (through neither M 1 nor M 2 ) is of the form: E { Y ( 1 , M 1 ( x ′ ) , M 2 ( x ′′ , M 1 ( x ′′′ ))) − Y ( 0 , M 1 ( x ′ ) , M 2 ( x ′′ , M 1 ( x ′′′ ))) } — The first argument changes and all other arguments stay the same, making it a direct effect. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 12/55

  13. Setting the scene Case study Q&A Wrapping up References Direct effect — A natural direct effect (through neither M 1 nor M 2 ) is of the form: E { Y ( 1 , M 1 ( x ′ ) , M 2 ( x ′′ , M 1 ( x ′′′ ))) − Y ( 0 , M 1 ( x ′ ) , M 2 ( x ′′ , M 1 ( x ′′′ ))) } — The first argument changes and all other arguments stay the same, making it a direct effect. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 12/55

  14. Setting the scene Case study Q&A Wrapping up References Direct effect — A natural direct effect (through neither M 1 nor M 2 ) is of the form: E { Y ( 1 , M 1 ( x ′ ) , M 2 ( x ′′ , M 1 ( x ′′′ ))) − Y ( 0 , M 1 ( x ′ ) , M 2 ( x ′′ , M 1 ( x ′′′ ))) } — The first argument changes and all other arguments stay the same, making it a direct effect. — There are 8 choices for how to fix x ′ , x ′′ , x ′′′ . Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 12/55

  15. Setting the scene Case study Q&A Wrapping up References Direct effect — A natural direct effect (through neither M 1 nor M 2 ) is of the form: E { Y ( 1 , M 1 ( 0 ) , M 2 ( 0 , M 1 ( 0 ))) − Y ( 0 , M 1 ( 0 ) , M 2 ( 0 , M 1 ( 0 ))) } — The first argument changes and all other arguments stay the same, making it a direct effect. — There are 8 choices for how to fix x ′ , x ′′ , x ′′′ . — We can choose ( x ′ , x ′′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NDE-000. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 12/55

  16. Setting the scene Case study Q&A Wrapping up References Direct effect — A natural direct effect (through neither M 1 nor M 2 ) is of the form: E { Y ( 1 , M 1 ( 0 ) , M 2 ( 0 , M 1 ( 1 ))) − Y ( 0 , M 1 ( 0 ) , M 2 ( 0 , M 1 ( 1 ))) } — The first argument changes and all other arguments stay the same, making it a direct effect. — There are 8 choices for how to fix x ′ , x ′′ , x ′′′ . — We can choose ( x ′ , x ′′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NDE-000. — Similarly, can choose ( x ′ , x ′′ , x ′′′ ) = ( 0 , 0 , 1 ) . We call this NDE-001. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 12/55

  17. Setting the scene Case study Q&A Wrapping up References Direct effect — A natural direct effect (through neither M 1 nor M 2 ) is of the form: E { Y ( 1 , M 1 ( 0 ) , M 2 ( 1 , M 1 ( 0 ))) − Y ( 0 , M 1 ( 0 ) , M 2 ( 1 , M 1 ( 0 ))) } — The first argument changes and all other arguments stay the same, making it a direct effect. — There are 8 choices for how to fix x ′ , x ′′ , x ′′′ . — We can choose ( x ′ , x ′′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NDE-000. — Similarly, can choose ( x ′ , x ′′ , x ′′′ ) = ( 0 , 1 , 0 ) . We call this NDE-010. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 12/55

  18. Setting the scene Case study Q&A Wrapping up References Direct effect — A natural direct effect (through neither M 1 nor M 2 ) is of the form: E { Y ( 1 , M 1 ( 0 ) , M 2 ( 1 , M 1 ( 1 ))) − Y ( 0 , M 1 ( 0 ) , M 2 ( 1 , M 1 ( 1 ))) } — The first argument changes and all other arguments stay the same, making it a direct effect. — There are 8 choices for how to fix x ′ , x ′′ , x ′′′ . — We can choose ( x ′ , x ′′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NDE-000. — Similarly, can choose ( x ′ , x ′′ , x ′′′ ) = ( 0 , 1 , 1 ) . We call this NDE-011. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 12/55

  19. Setting the scene Case study Q&A Wrapping up References Direct effect — A natural direct effect (through neither M 1 nor M 2 ) is of the form: E { Y ( 1 , M 1 ( 1 ) , M 2 ( 0 , M 1 ( 0 ))) − Y ( 0 , M 1 ( 1 ) , M 2 ( 0 , M 1 ( 0 ))) } — The first argument changes and all other arguments stay the same, making it a direct effect. — There are 8 choices for how to fix x ′ , x ′′ , x ′′′ . — We can choose ( x ′ , x ′′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NDE-000. — Similarly, can choose ( x ′ , x ′′ , x ′′′ ) = ( 1 , 0 , 0 ) . We call this NDE-100. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 12/55

  20. Setting the scene Case study Q&A Wrapping up References Direct effect — A natural direct effect (through neither M 1 nor M 2 ) is of the form: E { Y ( 1 , M 1 ( 1 ) , M 2 ( 0 , M 1 ( 1 ))) − Y ( 0 , M 1 ( 1 ) , M 2 ( 0 , M 1 ( 1 ))) } — The first argument changes and all other arguments stay the same, making it a direct effect. — There are 8 choices for how to fix x ′ , x ′′ , x ′′′ . — We can choose ( x ′ , x ′′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NDE-000. — Similarly, can choose ( x ′ , x ′′ , x ′′′ ) = ( 1 , 0 , 1 ) . We call this NDE-101. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 12/55

  21. Setting the scene Case study Q&A Wrapping up References Direct effect — A natural direct effect (through neither M 1 nor M 2 ) is of the form: E { Y ( 1 , M 1 ( 1 ) , M 2 ( 1 , M 1 ( 0 ))) − Y ( 0 , M 1 ( 1 ) , M 2 ( 1 , M 1 ( 0 ))) } — The first argument changes and all other arguments stay the same, making it a direct effect. — There are 8 choices for how to fix x ′ , x ′′ , x ′′′ . — We can choose ( x ′ , x ′′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NDE-000. — Similarly, can choose ( x ′ , x ′′ , x ′′′ ) = ( 1 , 1 , 0 ) . We call this NDE-110. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 12/55

  22. Setting the scene Case study Q&A Wrapping up References Direct effect — A natural direct effect (through neither M 1 nor M 2 ) is of the form: E { Y ( 1 , M 1 ( 1 ) , M 2 ( 1 , M 1 ( 1 ))) − Y ( 0 , M 1 ( 1 ) , M 2 ( 1 , M 1 ( 1 ))) } — The first argument changes and all other arguments stay the same, making it a direct effect. — There are 8 choices for how to fix x ′ , x ′′ , x ′′′ . — We can choose ( x ′ , x ′′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NDE-000. — Similarly, can choose ( x ′ , x ′′ , x ′′′ ) = ( 1 , 1 , 1 ) . We call this NDE-111. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 12/55

  23. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 1 only — A natural indirect effect through M 1 only is of the form: Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 13/55

  24. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 1 only — A natural indirect effect through M 1 only is of the form: E { Y ( x , M 1 ( 1 ) , M 2 ( x ′′ , M 1 ( x ′′′ ))) − Y ( x , M 1 ( 0 ) , M 2 ( x ′′ , M 1 ( x ′′′ ))) } Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 13/55

  25. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 1 only — A natural indirect effect through M 1 only is of the form: E { Y ( x , M 1 ( 1 ) , M 2 ( x ′′ , M 1 ( x ′′′ ))) − Y ( x , M 1 ( 0 ) , M 2 ( x ′′ , M 1 ( x ′′′ ))) } — The second argument changes and all other arguments stay the same, making it an indirect effect through M 1 only. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 13/55

  26. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 1 only — A natural indirect effect through M 1 only is of the form: E { Y ( x , M 1 ( 1 ) , M 2 ( x ′′ , M 1 ( x ′′′ ))) − Y ( x , M 1 ( 0 ) , M 2 ( x ′′ , M 1 ( x ′′′ ))) } — The second argument changes and all other arguments stay the same, making it an indirect effect through M 1 only. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 13/55

  27. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 1 only — A natural indirect effect through M 1 only is of the form: E { Y ( x , M 1 ( 1 ) , M 2 ( x ′′ , M 1 ( x ′′′ ))) − Y ( x , M 1 ( 0 ) , M 2 ( x ′′ , M 1 ( x ′′′ ))) } — The second argument changes and all other arguments stay the same, making it an indirect effect through M 1 only. — There are 8 choices for how to fix x , x ′′ , x ′′′ . Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 13/55

  28. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 1 only — A natural indirect effect through M 1 only is of the form: E { Y ( 0 , M 1 ( 1 ) , M 2 ( 0 , M 1 ( 0 ))) − Y ( 0 , M 1 ( 0 ) , M 2 ( 0 , M 1 ( 0 ))) } — The second argument changes and all other arguments stay the same, making it an indirect effect through M 1 only. — There are 8 choices for how to fix x , x ′′ , x ′′′ . — We can choose ( x , x ′′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NIE 1 -000. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 13/55

  29. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 1 only — A natural indirect effect through M 1 only is of the form: E { Y ( 0 , M 1 ( 1 ) , M 2 ( 0 , M 1 ( 1 ))) − Y ( 0 , M 1 ( 0 ) , M 2 ( 0 , M 1 ( 1 ))) } — The second argument changes and all other arguments stay the same, making it an indirect effect through M 1 only. — There are 8 choices for how to fix x , x ′′ , x ′′′ . — We can choose ( x , x ′′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NIE 1 -000. — Similarly, can choose ( x , x ′′ , x ′′′ ) = ( 0 , 0 , 1 ) . We call this NIE 1 -001. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 13/55

  30. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 1 only — A natural indirect effect through M 1 only is of the form: E { Y ( 0 , M 1 ( 1 ) , M 2 ( 1 , M 1 ( 0 ))) − Y ( 0 , M 1 ( 0 ) , M 2 ( 1 , M 1 ( 0 ))) } — The second argument changes and all other arguments stay the same, making it an indirect effect through M 1 only. — There are 8 choices for how to fix x , x ′′ , x ′′′ . — We can choose ( x , x ′′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NIE 1 -000. — Similarly, can choose ( x , x ′′ , x ′′′ ) = ( 0 , 1 , 0 ) . We call this NIE 1 -010. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 13/55

  31. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 1 only — A natural indirect effect through M 1 only is of the form: E { Y ( 0 , M 1 ( 1 ) , M 2 ( 1 , M 1 ( 1 ))) − Y ( 0 , M 1 ( 0 ) , M 2 ( 1 , M 1 ( 1 ))) } — The second argument changes and all other arguments stay the same, making it an indirect effect through M 1 only. — There are 8 choices for how to fix x , x ′′ , x ′′′ . — We can choose ( x , x ′′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NIE 1 -000. — Similarly, can choose ( x , x ′′ , x ′′′ ) = ( 0 , 1 , 1 ) . We call this NIE 1 -011. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 13/55

  32. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 1 only — A natural indirect effect through M 1 only is of the form: E { Y ( 1 , M 1 ( 1 ) , M 2 ( 0 , M 1 ( 0 ))) − Y ( 1 , M 1 ( 0 ) , M 2 ( 0 , M 1 ( 0 ))) } — The second argument changes and all other arguments stay the same, making it an indirect effect through M 1 only. — There are 8 choices for how to fix x , x ′′ , x ′′′ . — We can choose ( x , x ′′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NIE 1 -000. — Similarly, can choose ( x , x ′′ , x ′′′ ) = ( 1 , 0 , 0 ) . We call this NIE 1 -100. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 13/55

  33. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 1 only — A natural indirect effect through M 1 only is of the form: E { Y ( 1 , M 1 ( 1 ) , M 2 ( 0 , M 1 ( 1 ))) − Y ( 1 , M 1 ( 0 ) , M 2 ( 0 , M 1 ( 1 ))) } — The second argument changes and all other arguments stay the same, making it an indirect effect through M 1 only. — There are 8 choices for how to fix x , x ′′ , x ′′′ . — We can choose ( x , x ′′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NIE 1 -000. — Similarly, can choose ( x , x ′′ , x ′′′ ) = ( 1 , 0 , 1 ) . We call this NIE 1 -101. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 13/55

  34. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 1 only — A natural indirect effect through M 1 only is of the form: E { Y ( 1 , M 1 ( 1 ) , M 2 ( 1 , M 1 ( 0 ))) − Y ( 1 , M 1 ( 0 ) , M 2 ( 1 , M 1 ( 0 ))) } — The second argument changes and all other arguments stay the same, making it an indirect effect through M 1 only. — There are 8 choices for how to fix x , x ′′ , x ′′′ . — We can choose ( x , x ′′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NIE 1 -000. — Similarly, can choose ( x , x ′′ , x ′′′ ) = ( 1 , 1 , 0 ) . We call this NIE 1 -110. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 13/55

  35. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 1 only — A natural indirect effect through M 1 only is of the form: E { Y ( 1 , M 1 ( 1 ) , M 2 ( 1 , M 1 ( 1 ))) − Y ( 1 , M 1 ( 0 ) , M 2 ( 1 , M 1 ( 1 ))) } — The second argument changes and all other arguments stay the same, making it an indirect effect through M 1 only. — There are 8 choices for how to fix x , x ′′ , x ′′′ . — We can choose ( x , x ′′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NIE 1 -000. — Similarly, can choose ( x , x ′′ , x ′′′ ) = ( 1 , 1 , 1 ) . We call this NIE 1 -111. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 13/55

  36. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 2 only — A natural indirect effect through M 2 only is of the form: Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 14/55

  37. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 2 only — A natural indirect effect through M 2 only is of the form: E { Y ( x , M 1 ( x ′ ) , M 2 ( 1 , M 1 ( x ′′′ ))) − Y ( x , M 1 ( x ′ ) , M 2 ( 0 , M 1 ( x ′′′ ))) } Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 14/55

  38. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 2 only — A natural indirect effect through M 2 only is of the form: E { Y ( x , M 1 ( x ′ ) , M 2 ( 1 , M 1 ( x ′′′ ))) − Y ( x , M 1 ( x ′ ) , M 2 ( 0 , M 1 ( x ′′′ ))) } — The third argument changes and all other arguments stay the same, making it an indirect effect through M 2 only. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 14/55

  39. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 2 only — A natural indirect effect through M 2 only is of the form: E { Y ( x , M 1 ( x ′ ) , M 2 ( 1 , M 1 ( x ′′′ ))) − Y ( x , M 1 ( x ′ ) , M 2 ( 0 , M 1 ( x ′′′ ))) } — The third argument changes and all other arguments stay the same, making it an indirect effect through M 2 only. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 14/55

  40. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 2 only — A natural indirect effect through M 2 only is of the form: E { Y ( x , M 1 ( x ′ ) , M 2 ( 1 , M 1 ( x ′′′ ))) − Y ( x , M 1 ( x ′ ) , M 2 ( 0 , M 1 ( x ′′′ ))) } — The third argument changes and all other arguments stay the same, making it an indirect effect through M 2 only. — There are 8 choices for how to fix x , x ′ , x ′′′ . Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 14/55

  41. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 2 only — A natural indirect effect through M 2 only is of the form: E { Y ( 0 , M 1 ( 0 ) , M 2 ( 1 , M 1 ( 0 ))) − Y ( 0 , M 1 ( 0 ) , M 2 ( 0 , M 1 ( 0 ))) } — The third argument changes and all other arguments stay the same, making it an indirect effect through M 2 only. — There are 8 choices for how to fix x , x ′ , x ′′′ . — We can choose ( x , x ′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NIE 2 -000. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 14/55

  42. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 2 only — A natural indirect effect through M 2 only is of the form: E { Y ( 0 , M 1 ( 0 ) , M 2 ( 1 , M 1 ( 1 ))) − Y ( 0 , M 1 ( 0 ) , M 2 ( 0 , M 1 ( 1 ))) } — The third argument changes and all other arguments stay the same, making it an indirect effect through M 2 only. — There are 8 choices for how to fix x , x ′ , x ′′′ . — We can choose ( x , x ′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NIE 2 -000. — Similarly, can choose ( x , x ′ , x ′′′ ) = ( 0 , 0 , 1 ) . We call this NIE 2 -001. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 14/55

  43. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 2 only — A natural indirect effect through M 2 only is of the form: E { Y ( 0 , M 1 ( 1 ) , M 2 ( 1 , M 1 ( 0 ))) − Y ( 0 , M 1 ( 1 ) , M 2 ( 0 , M 1 ( 0 ))) } — The third argument changes and all other arguments stay the same, making it an indirect effect through M 2 only. — There are 8 choices for how to fix x , x ′ , x ′′′ . — We can choose ( x , x ′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NIE 2 -000. — Similarly, can choose ( x , x ′ , x ′′′ ) = ( 0 , 1 , 0 ) . We call this NIE 2 -010. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 14/55

  44. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 2 only — A natural indirect effect through M 2 only is of the form: E { Y ( 0 , M 1 ( 1 ) , M 2 ( 1 , M 1 ( 1 ))) − Y ( 0 , M 1 ( 1 ) , M 2 ( 0 , M 1 ( 1 ))) } — The third argument changes and all other arguments stay the same, making it an indirect effect through M 2 only. — There are 8 choices for how to fix x , x ′ , x ′′′ . — We can choose ( x , x ′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NIE 2 -000. — Similarly, can choose ( x , x ′ , x ′′′ ) = ( 0 , 1 , 1 ) . We call this NIE 2 -011. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 14/55

  45. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 2 only — A natural indirect effect through M 2 only is of the form: E { Y ( 1 , M 1 ( 0 ) , M 2 ( 1 , M 1 ( 0 ))) − Y ( 1 , M 1 ( 0 ) , M 2 ( 0 , M 1 ( 0 ))) } — The third argument changes and all other arguments stay the same, making it an indirect effect through M 2 only. — There are 8 choices for how to fix x , x ′ , x ′′′ . — We can choose ( x , x ′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NIE 2 -000. — Similarly, can choose ( x , x ′ , x ′′′ ) = ( 1 , 0 , 0 ) . We call this NIE 2 -100. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 14/55

  46. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 2 only — A natural indirect effect through M 2 only is of the form: E { Y ( 1 , M 1 ( 0 ) , M 2 ( 1 , M 1 ( 1 ))) − Y ( 1 , M 1 ( 0 ) , M 2 ( 0 , M 1 ( 1 ))) } — The third argument changes and all other arguments stay the same, making it an indirect effect through M 2 only. — There are 8 choices for how to fix x , x ′ , x ′′′ . — We can choose ( x , x ′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NIE 2 -000. — Similarly, can choose ( x , x ′ , x ′′′ ) = ( 1 , 0 , 1 ) . We call this NIE 2 -101. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 14/55

  47. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 2 only — A natural indirect effect through M 2 only is of the form: E { Y ( 1 , M 1 ( 1 ) , M 2 ( 1 , M 1 ( 0 ))) − Y ( 1 , M 1 ( 1 ) , M 2 ( 0 , M 1 ( 0 ))) } — The third argument changes and all other arguments stay the same, making it an indirect effect through M 2 only. — There are 8 choices for how to fix x , x ′ , x ′′′ . — We can choose ( x , x ′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NIE 2 -000. — Similarly, can choose ( x , x ′ , x ′′′ ) = ( 1 , 1 , 0 ) . We call this NIE 2 -110. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 14/55

  48. Setting the scene Case study Q&A Wrapping up References Indirect effect through M 2 only — A natural indirect effect through M 2 only is of the form: E { Y ( 1 , M 1 ( 1 ) , M 2 ( 1 , M 1 ( 1 ))) − Y ( 1 , M 1 ( 1 ) , M 2 ( 0 , M 1 ( 1 ))) } — The third argument changes and all other arguments stay the same, making it an indirect effect through M 2 only. — There are 8 choices for how to fix x , x ′ , x ′′′ . — We can choose ( x , x ′ , x ′′′ ) = ( 0 , 0 , 0 ) . We call this NIE 2 -000. — Similarly, can choose ( x , x ′ , x ′′′ ) = ( 1 , 1 , 1 ) . We call this NIE 2 -111. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 14/55

  49. Setting the scene Case study Q&A Wrapping up References Indirect effect through both M 1 and M 2 — A natural indirect effect through both M 1 and M 2 is of the form: Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 15/55

  50. Setting the scene Case study Q&A Wrapping up References Indirect effect through both M 1 and M 2 — A natural indirect effect through both M 1 and M 2 is of the form: E { Y ( x , M 1 ( x ′ ) , M 2 ( x ′′ , M 1 ( 1 ))) − Y ( x , M 1 ( x ′ ) , M 2 ( x ′′ , M 1 ( 0 ))) } Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 15/55

  51. Setting the scene Case study Q&A Wrapping up References Indirect effect through both M 1 and M 2 — A natural indirect effect through both M 1 and M 2 is of the form: E { Y ( x , M 1 ( x ′ ) , M 2 ( x ′′ , M 1 ( 1 ))) − Y ( x , M 1 ( x ′ ) , M 2 ( x ′′ , M 1 ( 0 ))) } — The fourth argument changes and all other arguments stay the same, making it an indirect effect through both M 1 and M 2 . Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 15/55

  52. Setting the scene Case study Q&A Wrapping up References Indirect effect through both M 1 and M 2 — A natural indirect effect through both M 1 and M 2 is of the form: E { Y ( x , M 1 ( x ′ ) , M 2 ( x ′′ , M 1 ( 1 ))) − Y ( x , M 1 ( x ′ ) , M 2 ( x ′′ , M 1 ( 0 ))) } — The fourth argument changes and all other arguments stay the same, making it an indirect effect through both M 1 and M 2 . Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 15/55

  53. Setting the scene Case study Q&A Wrapping up References Indirect effect through both M 1 and M 2 — A natural indirect effect through both M 1 and M 2 is of the form: E { Y ( x , M 1 ( x ′ ) , M 2 ( x ′′ , M 1 ( 1 ))) − Y ( x , M 1 ( x ′ ) , M 2 ( x ′′ , M 1 ( 0 ))) } — The fourth argument changes and all other arguments stay the same, making it an indirect effect through both M 1 and M 2 . — There are 8 choices for how to fix x , x ′ , x ′′ . Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 15/55

  54. Setting the scene Case study Q&A Wrapping up References Indirect effect through both M 1 and M 2 — A natural indirect effect through both M 1 and M 2 is of the form: E { Y ( 0 , M 1 ( 0 ) , M 2 ( 0 , M 1 ( 1 ))) − Y ( 0 , M 1 ( 0 ) , M 2 ( 0 , M 1 ( 0 ))) } — The fourth argument changes and all other arguments stay the same, making it an indirect effect through both M 1 and M 2 . — There are 8 choices for how to fix x , x ′ , x ′′ . — We can choose ( x , x ′ , x ′′ ) = ( 0 , 0 , 0 ) . We call this NIE 12 -000. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 15/55

  55. Setting the scene Case study Q&A Wrapping up References Indirect effect through both M 1 and M 2 — A natural indirect effect through both M 1 and M 2 is of the form: E { Y ( 0 , M 1 ( 0 ) , M 2 ( 1 , M 1 ( 1 ))) − Y ( 0 , M 1 ( 0 ) , M 2 ( 1 , M 1 ( 0 ))) } — The fourth argument changes and all other arguments stay the same, making it an indirect effect through both M 1 and M 2 . — There are 8 choices for how to fix x , x ′ , x ′′ . — We can choose ( x , x ′ , x ′′ ) = ( 0 , 0 , 0 ) . We call this NIE 12 -000. — Similarly, can choose ( x , x ′ , x ′′ ) = ( 0 , 0 , 1 ) . We call this NIE 12 -001. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 15/55

  56. Setting the scene Case study Q&A Wrapping up References Indirect effect through both M 1 and M 2 — A natural indirect effect through both M 1 and M 2 is of the form: E { Y ( 0 , M 1 ( 1 ) , M 2 ( 0 , M 1 ( 1 ))) − Y ( 0 , M 1 ( 1 ) , M 2 ( 0 , M 1 ( 0 ))) } — The fourth argument changes and all other arguments stay the same, making it an indirect effect through both M 1 and M 2 . — There are 8 choices for how to fix x , x ′ , x ′′ . — We can choose ( x , x ′ , x ′′ ) = ( 0 , 0 , 0 ) . We call this NIE 12 -000. — Similarly, can choose ( x , x ′ , x ′′ ) = ( 0 , 1 , 0 ) . We call this NIE 12 -010. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 15/55

  57. Setting the scene Case study Q&A Wrapping up References Indirect effect through both M 1 and M 2 — A natural indirect effect through both M 1 and M 2 is of the form: E { Y ( 0 , M 1 ( 1 ) , M 2 ( 1 , M 1 ( 1 ))) − Y ( 0 , M 1 ( 1 ) , M 2 ( 1 , M 1 ( 0 ))) } — The fourth argument changes and all other arguments stay the same, making it an indirect effect through both M 1 and M 2 . — There are 8 choices for how to fix x , x ′ , x ′′ . — We can choose ( x , x ′ , x ′′ ) = ( 0 , 0 , 0 ) . We call this NIE 12 -000. — Similarly, can choose ( x , x ′ , x ′′ ) = ( 0 , 1 , 1 ) . We call this NIE 12 -011. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 15/55

  58. Setting the scene Case study Q&A Wrapping up References Indirect effect through both M 1 and M 2 — A natural indirect effect through both M 1 and M 2 is of the form: E { Y ( 1 , M 1 ( 0 ) , M 2 ( 0 , M 1 ( 1 ))) − Y ( 1 , M 1 ( 0 ) , M 2 ( 0 , M 1 ( 0 ))) } — The fourth argument changes and all other arguments stay the same, making it an indirect effect through both M 1 and M 2 . — There are 8 choices for how to fix x , x ′ , x ′′ . — We can choose ( x , x ′ , x ′′ ) = ( 0 , 0 , 0 ) . We call this NIE 12 -000. — Similarly, can choose ( x , x ′ , x ′′ ) = ( 1 , 0 , 0 ) . We call this NIE 12 -100. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 15/55

  59. Setting the scene Case study Q&A Wrapping up References Indirect effect through both M 1 and M 2 — A natural indirect effect through both M 1 and M 2 is of the form: E { Y ( 1 , M 1 ( 0 ) , M 2 ( 1 , M 1 ( 1 ))) − Y ( 1 , M 1 ( 0 ) , M 2 ( 1 , M 1 ( 0 ))) } — The fourth argument changes and all other arguments stay the same, making it an indirect effect through both M 1 and M 2 . — There are 8 choices for how to fix x , x ′ , x ′′ . — We can choose ( x , x ′ , x ′′ ) = ( 0 , 0 , 0 ) . We call this NIE 12 -000. — Similarly, can choose ( x , x ′ , x ′′ ) = ( 1 , 0 , 1 ) . We call this NIE 12 -101. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 15/55

  60. Setting the scene Case study Q&A Wrapping up References Indirect effect through both M 1 and M 2 — A natural indirect effect through both M 1 and M 2 is of the form: E { Y ( 1 , M 1 ( 1 ) , M 2 ( 0 , M 1 ( 1 ))) − Y ( 1 , M 1 ( 1 ) , M 2 ( 0 , M 1 ( 0 ))) } — The fourth argument changes and all other arguments stay the same, making it an indirect effect through both M 1 and M 2 . — There are 8 choices for how to fix x , x ′ , x ′′ . — We can choose ( x , x ′ , x ′′ ) = ( 0 , 0 , 0 ) . We call this NIE 12 -000. — Similarly, can choose ( x , x ′ , x ′′ ) = ( 1 , 1 , 0 ) . We call this NIE 12 -110. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 15/55

  61. Setting the scene Case study Q&A Wrapping up References Indirect effect through both M 1 and M 2 — A natural indirect effect through both M 1 and M 2 is of the form: E { Y ( 1 , M 1 ( 1 ) , M 2 ( 1 , M 1 ( 1 ))) − Y ( 1 , M 1 ( 1 ) , M 2 ( 1 , M 1 ( 0 ))) } — The fourth argument changes and all other arguments stay the same, making it an indirect effect through both M 1 and M 2 . — There are 8 choices for how to fix x , x ′ , x ′′ . — We can choose ( x , x ′ , x ′′ ) = ( 0 , 0 , 0 ) . We call this NIE 12 -000. — Similarly, can choose ( x , x ′ , x ′′ ) = ( 1 , 1 , 1 ) . We call this NIE 12 -111. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 15/55

  62. Setting the scene Case study Q&A Wrapping up References Outline Setting the scene 1 Quick summary of yesterday Today’s case study Mediation analysis with multiple mediators Sequential mediation analysis Interventional effects for multiple mediators Case study 2 Q&A 3 References 4 Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 16/55

  63. Setting the scene Case study Q&A Wrapping up References Sequential mediation analysis • For more about the different possible decompositions of the TCE into the many path-specific effects defined above, and assumptions under which this can be achieved, see Daniel et al, Biometrics (2015). Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 17/55

  64. Setting the scene Case study Q&A Wrapping up References Sequential mediation analysis • For more about the different possible decompositions of the TCE into the many path-specific effects defined above, and assumptions under which this can be achieved, see Daniel et al, Biometrics (2015). • But for today, we’ll focus on a simpler, more practical and intuitive idea presented by VanderWeele et al (2014), known as sequential mediation analysis. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 17/55

  65. Setting the scene Case study Q&A Wrapping up References The idea M 1 C M 2 X Y • First we consider M 1 and M 2 to be joint mediators. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 18/55

  66. Setting the scene Case study Q&A Wrapping up References The idea M 1 C M 2 X Y • First we consider M 1 and M 2 to be joint mediators. • This allows us to use single mediator analysis, with ( M 1 , M 2 ) as the mediator. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 18/55

  67. Setting the scene Case study Q&A Wrapping up References The idea M 1 C M 2 X Y • We thus estimate NDE joint = E { Y ( 1 , M 1 ( 0 ) , M 2 ( 0 , M 1 ( 0 ))) − Y ( 0 , M 1 ( 0 ) , M 2 ( 0 , M 1 ( 0 ))) } and NIE joint = E { Y ( 1 , M 1 ( 1 ) , M 2 ( 1 , M 1 ( 1 ))) − Y ( 1 , M 1 ( 0 ) , M 2 ( 0 , M 1 ( 0 ))) } with TCE = NDE joint + NIE joint Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 19/55

  68. Setting the scene Case study Q&A Wrapping up References The idea M 1 C M 2 X Y • Next we consider M 1 to be the only mediator of interest, and we ignore M 2 . Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 20/55

  69. Setting the scene Case study Q&A Wrapping up References The idea M 1 C M 2 X Y • Next we consider M 1 to be the only mediator of interest, and we ignore M 2 . • This allows us to use single mediator analysis, with M 1 as the mediator. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 20/55

  70. Setting the scene Case study Q&A Wrapping up References The idea M 1 C M 2 X Y • Next we consider M 1 to be the only mediator of interest, and we ignore M 2 . • This allows us to use single mediator analysis, with M 1 as the mediator. • The direct effect then includes the effect via neither M 1 nor M 2 and the effect through M 2 alone, whereas the indirect effect includes the effect via M 1 alone and the effect via both M 1 and M 2 . Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 20/55

  71. Setting the scene Case study Q&A Wrapping up References The idea M 1 C M 2 X Y • In other words, we estimate NDE not M 1 = E { Y ( 1 , M 1 ( 0 ) , M 2 ( 1 , M 1 ( 0 ))) − Y ( 0 , M 1 ( 0 ) , M 2 ( 0 , M 1 ( 0 ))) } and NIE M 1 = E { Y ( 1 , M 1 ( 1 ) , M 2 ( 1 , M 1 ( 1 ))) − Y ( 1 , M 1 ( 0 ) , M 2 ( 1 , M 1 ( 0 ))) } with TCE = NDE M 1 + NIE M 1 Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 21/55

  72. Setting the scene Case study Q&A Wrapping up References The idea M 1 C M 2 X Y • We then note that we can obtain (one of) the indirect effect(s) through M 2 alone by taking the difference between NIE joint and NIE M 1 : NIE joint − NIE M 1 = E { Y ( 1 , M 1 ( 0 ) , M 2 ( 1 , M 1 ( 0 ))) − Y ( 1 , M 1 ( 0 ) , M 2 ( 0 , M 1 ( 0 ))) } = NIE M 2 − 100 Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 22/55

  73. Setting the scene Case study Q&A Wrapping up References Sequential mediation analysis • Sequential mediation analysis doesn’t require any further results on identification nor any new methods for estimation, since it is simply an application of single mediator analysis twice: once with M 1 and M 2 as joint mediators, and then with M 1 as the only mediator. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 23/55

  74. Setting the scene Case study Q&A Wrapping up References Sequential mediation analysis • Sequential mediation analysis doesn’t require any further results on identification nor any new methods for estimation, since it is simply an application of single mediator analysis twice: once with M 1 and M 2 as joint mediators, and then with M 1 as the only mediator. • Writing M for ( M 1 , M 2 ) , the assumptions for identification therefore include that there should be no unmeasured confounders of X and M , X and Y , M and Y , X and M 1 , M 1 and Y , and no confounders (measured or unmeasured) of M and Y or of M 1 and Y that are affected by X . Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 23/55

  75. Setting the scene Case study Q&A Wrapping up References Sequential mediation analysis • Sequential mediation analysis doesn’t require any further results on identification nor any new methods for estimation, since it is simply an application of single mediator analysis twice: once with M 1 and M 2 as joint mediators, and then with M 1 as the only mediator. • Writing M for ( M 1 , M 2 ) , the assumptions for identification therefore include that there should be no unmeasured confounders of X and M , X and Y , M and Y , X and M 1 , M 1 and Y , and no confounders (measured or unmeasured) of M and Y or of M 1 and Y that are affected by X . • This means that in order to apply sequential mediation analysis, we need to know the order of the mediators (i.e. M 1 affects M 2 but not vice versa) and the mediators cannot share any unmeasured common causes (since this would violate the no unmeasured confounding assumption for M 1 and Y ). Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 23/55

  76. Setting the scene Case study Q&A Wrapping up References Sequential mediation analysis • Sequential mediation analysis doesn’t require any further results on identification nor any new methods for estimation, since it is simply an application of single mediator analysis twice: once with M 1 and M 2 as joint mediators, and then with M 1 as the only mediator. • Writing M for ( M 1 , M 2 ) , the assumptions for identification therefore include that there should be no unmeasured confounders of X and M , X and Y , M and Y , X and M 1 , M 1 and Y , and no confounders (measured or unmeasured) of M and Y or of M 1 and Y that are affected by X . • This means that in order to apply sequential mediation analysis, we need to know the order of the mediators (i.e. M 1 affects M 2 but not vice versa) and the mediators cannot share any unmeasured common causes (since this would violate the no unmeasured confounding assumption for M 1 and Y ). • In many practical applications, these assumptions are implausible. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 23/55

  77. Setting the scene Case study Q&A Wrapping up References Sequential mediation analysis • Sequential mediation analysis doesn’t require any further results on identification nor any new methods for estimation, since it is simply an application of single mediator analysis twice: once with M 1 and M 2 as joint mediators, and then with M 1 as the only mediator. • Writing M for ( M 1 , M 2 ) , the assumptions for identification therefore include that there should be no unmeasured confounders of X and M , X and Y , M and Y , X and M 1 , M 1 and Y , and no confounders (measured or unmeasured) of M and Y or of M 1 and Y that are affected by X . • This means that in order to apply sequential mediation analysis, we need to know the order of the mediators (i.e. M 1 affects M 2 but not vice versa) and the mediators cannot share any unmeasured common causes (since this would violate the no unmeasured confounding assumption for M 1 and Y ). • In many practical applications, these assumptions are implausible. • So we now turn to an alternative, based on interventional effects. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 23/55

  78. Setting the scene Case study Q&A Wrapping up References Outline Setting the scene 1 Quick summary of yesterday Today’s case study Mediation analysis with multiple mediators Sequential mediation analysis Interventional effects for multiple mediators Case study 2 Q&A 3 References 4 Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 24/55

  79. Setting the scene Case study Q&A Wrapping up References Our proposal • In Vansteelandt and Daniel (2017), we proposed an extension of the single mediator interventional effects to multiple mediator settings. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 25/55

  80. Setting the scene Case study Q&A Wrapping up References Our proposal • In Vansteelandt and Daniel (2017), we proposed an extension of the single mediator interventional effects to multiple mediator settings. • The effects we define will sum to the total causal effect. Rhian Daniel/Counterfactual-based mediation analysisWorkshop 2 25/55

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