causal mediation analysis with multiple causally-ordered...

Causal mediation analysis with multiplecausally-ordered mediators

Rhian Daniel, Bianca De Stavola and Simon Cousenswith thanks to Stijn Vansteelandt and Dave Leon

Centre for Statistical MethodologyLondon School of Hygiene and Tropical Medicine

Symposium on Causal Mediation AnalysisGent 28–29 January 2013

Rhian Daniel/Multiple mediators 1/35

Background One mediator Two mediators Identification Example Summary References

Single mediator

— Almost all the causal inference literature on mediation is based ona single mediator.

Multiple mediators

— However, as we have seen over the last two days, many realisticapplications involve multiple mediators.

Many mediators considered en bloc

— Multiple mediators are trivially accommodated by the existingformal literature if viewed en bloc (with M a vector).

— But then the direct effect is around all of them, and the (single)indirect effect is through one, more or all of them, and is not furtherdisentangled.

Many mediators considered en bloc

— Multiple mediators are trivially accommodated by the existingformal literature if viewed en bloc (with M a vector).— But then the direct effect is around all of them, and the (single)indirect effect is through one, more or all of them, and is not furtherdisentangled.

Two mediators but only one ‘of interest’

— The causal inference literature does focus on ‘two mediators’, Land M, in settings with intermediate confounding (cf Vanessa’s talkyesterday).

— But M is the mediator of interest, with decomposition again intotwo effects—through and not through M.

Two mediators but only one ‘of interest’

— The causal inference literature does focus on ‘two mediators’, Land M, in settings with intermediate confounding (cf Vanessa’s talkyesterday).— But M is the mediator of interest, with decomposition again intotwo effects—through and not through M.

Two mediators, both of interest (1)

— What if both mediators are ‘of interest’?

— We could then be interested in a finer decomposition, withpath-specific effects through M1 alone, M2 alone, both and neither.

— What if both mediators are ‘of interest’?— We could then be interested in a finer decomposition, withpath-specific effects through M1 alone, M2 alone, both and neither.

— This setting has been studied, but either (a) without a focus ondecomposing the total effect [Avin et al (2005), Albert and Nelson(2011)]. . .

Two mediators, not causally ordered

— . . . or (b) in the setting with no effect of M1 on M2 [MacKinnon(2000), Preacher and Hayes (2008), Imai and Yamamoto (in press)](many examples yesterday).

Our setting

— This talk is about a path-specific decomposition of the total causaleffect in the setting where M1 affects M2 (causally-ordered mediators).

More generally

— More generally there could be n causally-ordered mediators.

— Traditional path analysis generalises easily to this setting, butunder strong assumptions, including linearity everywhere.— Can these be relaxed somewhat by generalising ideas from causalinference?

More generally

— More generally there could be n causally-ordered mediators.— Traditional path analysis generalises easily to this setting, butunder strong assumptions, including linearity everywhere.

— Can these be relaxed somewhat by generalising ideas from causalinference?

More generally

— More generally there could be n causally-ordered mediators.— Traditional path analysis generalises easily to this setting, butunder strong assumptions, including linearity everywhere.— Can these be relaxed somewhat by generalising ideas from causalinference?

Outline

1 Background

2 Quick revision: effect decomposition with one mediator

3 Path-specific effect estimands with two mediators

4 Identification

5 Example: Izhevsk study

6 Summary

7 References

Outline

1 Background

4 Identification

6 Summary

7 References

Effect decomposition, single mediator

— With one mediator, there are two possible decompositions:

TCE = E{Y (1,M(1))− Y (0,M(0))}= E{Y (1,M(1))− Y (1,M(0)) + Y (1,M(0))− Y (0,M(0))}= TNIE + PNDE= E{Y (1,M(1))− Y (0,M(1)) + Y (0,M(1))− Y (0,M(0))}= TNDE + PNIE

NB: TCE = total causal effect, PNDE = pure natural direct effect, TNDE = total natural direct effect, PNIE = pure

natural indirect effect, TNIE = total natural indirect effect

— VanderWeele [Epidemiology, 2013] shows that:

TCE = PNDE + PNIE + ‘mediated interaction’

— So the two decompositions amount to apportioning the mediatedinteraction either to the direct or indirect effect.

Outline

1 Background

4 Identification

6 Summary

7 References

Counterfactuals

— With one mediator, we need:

M(x),Y (x ,m),Y (x ,M(x ′))

— With two, we need:

M1(x),M2(x ,m1),Y (x ,m1,m2)

andM2(x ,M1(x ′))

andY (x ,M1(x ′),M2(x ′′,M1(x ′′′)))

— Natural path-specific effects are defined as contrasts betweenthese for carefully chosen values of x , x ′, x ′′, x ′′′.

Counterfactuals

M(x),Y (x ,m),Y (x ,M(x ′))

M1(x),M2(x ,m1),Y (x ,m1,m2)

andM2(x ,M1(x ′))

andY (x ,M1(x ′),M2(x ′′,M1(x ′′′)))

Counterfactuals

M(x),Y (x ,m),Y (x ,M(x ′))

M1(x),M2(x ,m1),Y (x ,m1,m2)

andM2(x ,M1(x ′))

andY (x ,M1(x ′),M2(x ′′,M1(x ′′′)))

Counterfactuals

M(x),Y (x ,m),Y (x ,M(x ′))

M1(x),M2(x ,m1),Y (x ,m1,m2)

andM2(x ,M1(x ′))

andY (x ,M1(x ′),M2(x ′′,M1(x ′′′)))

Counterfactuals

M(x),Y (x ,m),Y (x ,M(x ′))

M1(x),M2(x ,m1),Y (x ,m1,m2)

andM2(x ,M1(x ′))

andY (x ,M1(x ′),M2(x ′′,M1(x ′′′)))

Counterfactuals

M(x),Y (x ,m),Y (x ,M(x ′))

M1(x),M2(x ,m1),Y (x ,m1,m2)

andM2(x ,M1(x ′))

andY (x ,M1(x ′),M2(x ′′,M1(x ′′′)))

Counterfactuals

M(x),Y (x ,m),Y (x ,M(x ′))

M1(x),M2(x ,m1),Y (x ,m1,m2)

andM2(x ,M1(x ′))

andY (x ,M1(x ′),M2(x ′′,M1(x ′′′)))

Direct effect

— A natural direct effect (through neither M1 nor M2) is of the form:

— The first argument changes and all other arguments stay thesame, 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 ′′′) = (, , ). We call this.

Direct effect

E{Y (1,M1(x ′),M2(x ′′,M1(x ′′′)))−Y (0,M1(x ′),M2(x ′′,M1(x ′′′)))}

— The first argument changes and all other arguments stay thesame, 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 ′′′) = (, , ). We call this.

Direct effect

— The first argument changes and all other arguments stay thesame, 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 ′′′) = (, , ). We call this.

Direct effect

— The first argument changes and all other arguments stay thesame, 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 ′′′) = (, , ). We call this.

Direct effect

— The first argument changes and all other arguments stay thesame, 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 ′′′) = (, , ). We call this.

Direct effect

E{Y (1,M1(0 ),M2( 0 ,M1( 0 )))−Y (0,M1(0 ),M2( 0 ,M1( 0 )))}

— The first argument changes and all other arguments stay thesame, 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 ′′′) = (, , ). We call this.

Direct effect

E{Y (1,M1(0 ),M2( 0 ,M1( 1 )))−Y (0,M1(0 ),M2( 0 ,M1( 1 )))}

— The first argument changes and all other arguments stay thesame, 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 thisNDE-001.

Direct effect

E{Y (1,M1(0 ),M2( 1 ,M1( 0 )))−Y (0,M1(0 ),M2( 1 ,M1( 0 )))}

Direct effect

E{Y (1,M1(0 ),M2( 1 ,M1( 1 )))−Y (0,M1(0 ),M2( 1 ,M1( 1 )))}

Direct effect

E{Y (1,M1(1 ),M2( 0 ,M1( 0 )))−Y (0,M1(1 ),M2( 0 ,M1( 0 )))}

Direct effect

E{Y (1,M1(1 ),M2( 0 ,M1( 1 )))−Y (0,M1(1 ),M2( 0 ,M1( 1 )))}

Direct effect

E{Y (1,M1(1 ),M2( 1 ,M1( 0 )))−Y (0,M1(1 ),M2( 1 ,M1( 0 )))}

Direct effect

E{Y (1,M1(1 ),M2( 1 ,M1( 1 )))−Y (0,M1(1 ),M2( 1 ,M1( 1 )))}

Indirect effect through M1 only

— A natural indirect effect through M1 only is of the form:

— The second argument changes and all other arguments stay thesame, making it an indirect effect through M1 only.— There are 8 choices for how to fix x , x ′′, x ′′′.— We can choose (x , x ′′, x ′′′) = (0,0,0). We call this NIE1-000.— Similarly, can choose (x , x ′′, x ′′′) = (, , ). We call this .

E{Y (x ,M1(1),M2(x ′′,M1(x ′′′)))−Y (x ,M1(0),M2(x ′′,M1(x ′′′)))}

— The second argument changes and all other arguments stay thesame, making it an indirect effect through M1 only.— There are 8 choices for how to fix x , x ′′, x ′′′.— We can choose (x , x ′′, x ′′′) = (0,0,0). We call this NIE1-000.— Similarly, can choose (x , x ′′, x ′′′) = (, , ). We call this .

— The second argument changes and all other arguments stay thesame, making it an indirect effect through M1 only.

— There are 8 choices for how to fix x , x ′′, x ′′′.— We can choose (x , x ′′, x ′′′) = (0,0,0). We call this NIE1-000.— Similarly, can choose (x , x ′′, x ′′′) = (, , ). We call this .

— The second argument changes and all other arguments stay thesame, making it an indirect effect through M1 only.

— There are 8 choices for how to fix x , x ′′, x ′′′.— We can choose (x , x ′′, x ′′′) = (0,0,0). We call this NIE1-000.— Similarly, can choose (x , x ′′, x ′′′) = (, , ). We call this .

— The second argument changes and all other arguments stay thesame, making it an indirect effect through M1 only.— There are 8 choices for how to fix x , x ′′, x ′′′.

— We can choose (x , x ′′, x ′′′) = (0,0,0). We call this NIE1-000.— Similarly, can choose (x , x ′′, x ′′′) = (, , ). We call this .

E{Y (0,M1(1),M2( 0 ,M1( 0 )))−Y (0,M1(0),M2( 0 ,M1( 0 )))}

— The second argument changes and all other arguments stay thesame, making it an indirect effect through M1 only.— There are 8 choices for how to fix x , x ′′, x ′′′.— We can choose (x , x ′′, x ′′′) = (0,0,0). We call this NIE1-000.

— Similarly, can choose (x , x ′′, x ′′′) = (, , ). We call this .

E{Y (0,M1(1),M2( 0 ,M1( 1 )))−Y (0,M1(0),M2( 0 ,M1( 1 )))}

— The second argument changes and all other arguments stay thesame, making it an indirect effect through M1 only.— There are 8 choices for how to fix x , x ′′, x ′′′.— We can choose (x , x ′′, x ′′′) = (0,0,0). We call this NIE1-000.— Similarly, can choose (x , x ′′, x ′′′) = (0,0,1). We call this NIE1-001.

E{Y (0,M1(1),M2( 1 ,M1( 0 )))−Y (0,M1(0),M2( 1 ,M1( 0 )))}

E{Y (0,M1(1),M2( 1 ,M1( 1 )))−Y (0,M1(0),M2( 1 ,M1( 1 )))}

E{Y (1,M1(1),M2( 0 ,M1( 0 )))−Y (1,M1(0),M2( 0 ,M1( 0 )))}

E{Y (1,M1(1),M2( 0 ,M1( 1 )))−Y (1,M1(0),M2( 0 ,M1( 1 )))}

E{Y (1,M1(1),M2( 1 ,M1( 0 )))−Y (1,M1(0),M2( 1 ,M1( 0 )))}

E{Y (1,M1(1),M2( 1 ,M1( 1 )))−Y (1,M1(0),M2( 1 ,M1( 1 )))}

— The third argument changes and all other arguments stay thesame, making it an indirect effect through M2 only.— There are 8 choices for how to fix x , x ′, x ′′′.— We can choose (x , x ′, x ′′′) = (0,0,0). We call this NIE2-000.— Similarly, can choose (x , x ′, x ′′′) = (, , ). We call this .

E{Y (x ,M1(x ′),M2(1,M1(x ′′′)))−Y (x ,M1(x ′),M2(0,M1(x ′′′)))}

— The third argument changes and all other arguments stay thesame, making it an indirect effect through M2 only.— There are 8 choices for how to fix x , x ′, x ′′′.— We can choose (x , x ′, x ′′′) = (0,0,0). We call this NIE2-000.— Similarly, can choose (x , x ′, x ′′′) = (, , ). We call this .

— The third argument changes and all other arguments stay thesame, making it an indirect effect through M2 only.

— There are 8 choices for how to fix x , x ′, x ′′′.— We can choose (x , x ′, x ′′′) = (0,0,0). We call this NIE2-000.— Similarly, can choose (x , x ′, x ′′′) = (, , ). We call this .

— The third argument changes and all other arguments stay thesame, making it an indirect effect through M2 only.

— There are 8 choices for how to fix x , x ′, x ′′′.— We can choose (x , x ′, x ′′′) = (0,0,0). We call this NIE2-000.— Similarly, can choose (x , x ′, x ′′′) = (, , ). We call this .

— The third argument changes and all other arguments stay thesame, making it an indirect effect through M2 only.— There are 8 choices for how to fix x , x ′, x ′′′.

— We can choose (x , x ′, x ′′′) = (0,0,0). We call this NIE2-000.— Similarly, can choose (x , x ′, x ′′′) = (, , ). We call this .

E{Y (0,M1(0 ),M2(1,M1( 0 )))−Y (0,M1(0 ),M2(0,M1( 0 )))}

— The third argument changes and all other arguments stay thesame, making it an indirect effect through M2 only.— There are 8 choices for how to fix x , x ′, x ′′′.— We can choose (x , x ′, x ′′′) = (0,0,0). We call this NIE2-000.

— Similarly, can choose (x , x ′, x ′′′) = (, , ). We call this .

E{Y (0,M1(0 ),M2(1,M1( 1 )))−Y (0,M1(0 ),M2(0,M1( 1 )))}

— The third argument changes and all other arguments stay thesame, making it an indirect effect through M2 only.— There are 8 choices for how to fix x , x ′, x ′′′.— We can choose (x , x ′, x ′′′) = (0,0,0). We call this NIE2-000.— Similarly, can choose (x , x ′, x ′′′) = (0,0,1). We call this NIE2-001.

E{Y (0,M1(1 ),M2(1,M1( 0 )))−Y (0,M1(1 ),M2(0,M1( 0 )))}

E{Y (0,M1(1 ),M2(1,M1( 1 )))−Y (0,M1(1 ),M2(0,M1( 1 )))}

E{Y (1,M1(0 ),M2(1,M1( 0 )))−Y (1,M1(0 ),M2(0,M1( 0 )))}

E{Y (1,M1(0 ),M2(1,M1( 1 )))−Y (1,M1(0 ),M2(0,M1( 1 )))}

E{Y (1,M1(1 ),M2(1,M1( 0 )))−Y (1,M1(1 ),M2(0,M1( 0 )))}

E{Y (1,M1(1 ),M2(1,M1( 1 )))−Y (1,M1(1 ),M2(0,M1( 1 )))}

Indirect effect through both M1 and M2

— A natural indirect effect through both M1 and M2 is of the form:

— The fourth argument changes and all other arguments stay thesame, making it an indirect effect through both M1 and M2.— There are 8 choices for how to fix x , x ′, x ′′.— We can choose (x , x ′, x ′′) = (0,0,0). We call this NIE12-000.— Similarly, can choose (x , x ′, x ′′) = (, , ). We call this .

E{Y (x ,M1(x ′),M2(x ′′,M1(1)))− Y (x ,M1(x ′),M2(x ′′,M1(0)))}

— The fourth argument changes and all other arguments stay thesame, making it an indirect effect through both M1 and M2.— There are 8 choices for how to fix x , x ′, x ′′.— We can choose (x , x ′, x ′′) = (0,0,0). We call this NIE12-000.— Similarly, can choose (x , x ′, x ′′) = (, , ). We call this .

— The fourth argument changes and all other arguments stay thesame, making it an indirect effect through both M1 and M2.

— There are 8 choices for how to fix x , x ′, x ′′.— We can choose (x , x ′, x ′′) = (0,0,0). We call this NIE12-000.— Similarly, can choose (x , x ′, x ′′) = (, , ). We call this .

— The fourth argument changes and all other arguments stay thesame, making it an indirect effect through both M1 and M2.

— There are 8 choices for how to fix x , x ′, x ′′.— We can choose (x , x ′, x ′′) = (0,0,0). We call this NIE12-000.— Similarly, can choose (x , x ′, x ′′) = (, , ). We call this .

— The fourth argument changes and all other arguments stay thesame, making it an indirect effect through both M1 and M2.— There are 8 choices for how to fix x , x ′, x ′′.

— We can choose (x , x ′, x ′′) = (0,0,0). We call this NIE12-000.— Similarly, can choose (x , x ′, x ′′) = (, , ). We call this .

E{Y (0,M1(0 ),M2( 0 ,M1(1)))− Y (0,M1(0 ),M2( 0 ,M1(0)))}

— The fourth argument changes and all other arguments stay thesame, making it an indirect effect through both M1 and M2.— There are 8 choices for how to fix x , x ′, x ′′.— We can choose (x , x ′, x ′′) = (0,0,0). We call this NIE12-000.

— Similarly, can choose (x , x ′, x ′′) = (, , ). We call this .

E{Y (0,M1(0 ),M2( 1 ,M1(1)))− Y (0,M1(0 ),M2( 1 ,M1(0)))}

— The fourth argument changes and all other arguments stay thesame, making it an indirect effect through both M1 and M2.— There are 8 choices for how to fix x , x ′, x ′′.— We can choose (x , x ′, x ′′) = (0,0,0). We call this NIE12-000.— Similarly, can choose (x , x ′, x ′′) = (0,0,1). We call this NIE12-001.

E{Y (0,M1(1 ),M2( 0 ,M1(1)))− Y (0,M1(1 ),M2( 0 ,M1(0)))}

E{Y (0,M1(1 ),M2( 1 ,M1(1)))− Y (0,M1(1 ),M2( 1 ,M1(0)))}

E{Y (1,M1(0 ),M2( 0 ,M1(1)))− Y (1,M1(0 ),M2( 0 ,M1(0)))}

E{Y (1,M1(0 ),M2( 1 ,M1(1)))− Y (1,M1(0 ),M2( 1 ,M1(0)))}

E{Y (1,M1(1 ),M2( 0 ,M1(1)))− Y (1,M1(1 ),M2( 0 ,M1(0)))}

E{Y (1,M1(1 ),M2( 1 ,M1(1)))− Y (1,M1(1 ),M2( 1 ,M1(0)))}

Decomposition

— We have defined 8 types (cf pure/total) of each of 4 path-specificeffects (cf direct/indirect).

— We could therefore form 84 = 4096 sums of the form

NDE + NIE1 + NIE2 + NIE12

— 24 of these sums are equal to the TCE. For example

NDE-000 + NIE1-100 + NIE2-110 + NIE12-111 = TCE

— More generally, with n mediators, there are 2(2n−1)·2npossible

sums, (2n)! of which are equal to the TCE.

n (2n)!1 22 243 40,3204 2.092× 1013

5 2.631× 1035

Decomposition

— We have defined 8 types (cf pure/total) of each of 4 path-specificeffects (cf direct/indirect).— We could therefore form 84 = 4096 sums of the form

n (2n)!1 22 243 40,3204 2.092× 1013

5 2.631× 1035

Decomposition

n (2n)!1 22 243 40,3204 2.092× 1013

5 2.631× 1035

Decomposition

n (2n)!1 22 243 40,3204 2.092× 1013

5 2.631× 1035

Decomposition

n (2n)!1 2

2 243 40,3204 2.092× 1013

5 2.631× 1035

Decomposition

n (2n)!1 22 24

3 40,3204 2.092× 1013

5 2.631× 1035

Decomposition

n (2n)!1 22 243 40,320

4 2.092× 1013

5 2.631× 1035

Decomposition

n (2n)!1 22 243 40,3204 2.092× 1013

5 2.631× 1035

Decomposition

n (2n)!1 22 243 40,3204 2.092× 1013

5 2.631× 1035

Outline

1 Background

4 Identification

6 Summary

7 References

Nonparametric identification: single mediator

— Throughout, we omit background confounders C from ourdiagrams, but they are always implicitly there.

— Recall that for nonparametric identification of PNDE, TNDE, PNIEand TNIE in the one-mediator setting, we require no unmeasuredconfounding of X and M, M and Y , and X and Y .— And no intermediate confounders, L.

— Throughout, we omit background confounders C from ourdiagrams, but they are always implicitly there.— Recall that for nonparametric identification of PNDE, TNDE, PNIEand TNIE in the one-mediator setting, we require

no unmeasuredconfounding of X and M, M and Y , and X and Y .— And no intermediate confounders, L.

— Throughout, we omit background confounders C from ourdiagrams, but they are always implicitly there.— Recall that for nonparametric identification of PNDE, TNDE, PNIEand TNIE in the one-mediator setting, we require no unmeasuredconfounding of X and M,

M and Y , and X and Y .— And no intermediate confounders, L.

— Throughout, we omit background confounders C from ourdiagrams, but they are always implicitly there.— Recall that for nonparametric identification of PNDE, TNDE, PNIEand TNIE in the one-mediator setting, we require no unmeasuredconfounding of X and M, M and Y ,

and X and Y .— And no intermediate confounders, L.

— Throughout, we omit background confounders C from ourdiagrams, but they are always implicitly there.— Recall that for nonparametric identification of PNDE, TNDE, PNIEand TNIE in the one-mediator setting, we require no unmeasuredconfounding of X and M, M and Y , and X and Y .

— And no intermediate confounders, L.

— Throughout, we omit background confounders C from ourdiagrams, but they are always implicitly there.— Recall that for nonparametric identification of PNDE, TNDE, PNIEand TNIE in the one-mediator setting, we require no unmeasuredconfounding of X and M, M and Y , and X and Y .— And no intermediate confounders, L.

Extension to two mediators

— Consider the natural extensions of these assumptions.

— No unmeasured confounding, and no intermediate confounding.— Are these sufficient for identification?

— Consider the natural extensions of these assumptions.— No unmeasured confounding,

and no intermediate confounding.— Are these sufficient for identification?

— Consider the natural extensions of these assumptions.— No unmeasured confounding, and no intermediate confounding.

— Are these sufficient for identification?

— Consider the natural extensions of these assumptions.— No unmeasured confounding, and no intermediate confounding.— Are these sufficient for identification?

Identification?

— Consider the 32 path-specific effects we wish to identify:

E{Y (1,M1(0),M2(0,M1(0)))− Y (0,M1(0),M2(0,M1(0)))}

— Each half of each path-specific effect is of the form

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

— If (1) is identified under our extended assumptions, allpath-specific effects are identified.— Using these assumptions, we can re-write (1) as:∫

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

· fM2|C,X ,M1 (m2 |c, x ′′,m′1 ) fM1(x′′′)|C,M1(x′) (m′1 |c,m1 )

· fM1|C,X (m1 |c, x ′ ) fC (c)· dµM2 (m2)dµM1 (m

′1)dµM1 (m1)dµC (c)

Identification?

E{Y (1,M1(0),M2(0,M1(1)))− Y (0,M1(0),M2(0,M1(1)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (1,M1(0),M2(1,M1(0)))− Y (0,M1(0),M2(1,M1(0)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (1,M1(0),M2(1,M1(1)))− Y (0,M1(0),M2(1,M1(1)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (1,M1(1),M2(0,M1(0)))− Y (0,M1(1),M2(0,M1(0)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (1,M1(1),M2(0,M1(1)))− Y (0,M1(1),M2(0,M1(1)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (1,M1(1),M2(1,M1(0)))− Y (0,M1(1),M2(1,M1(0)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (1,M1(1),M2(1,M1(1)))− Y (0,M1(1),M2(1,M1(1)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (0,M1(1),M2(0,M1(0)))− Y (0,M1(0),M2(0,M1(0)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (0,M1(1),M2(0,M1(1)))− Y (0,M1(0),M2(0,M1(1)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (0,M1(1),M2(1,M1(0)))− Y (0,M1(0),M2(1,M1(0)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (0,M1(1),M2(1,M1(1)))− Y (0,M1(0),M2(1,M1(1)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (1,M1(1),M2(0,M1(0)))− Y (1,M1(0),M2(0,M1(0)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (1,M1(1),M2(0,M1(1)))− Y (1,M1(0),M2(0,M1(1)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (1,M1(1),M2(1,M1(0)))− Y (1,M1(0),M2(1,M1(0)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (1,M1(1),M2(1,M1(1)))− Y (1,M1(0),M2(1,M1(1)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (0,M1(0),M2(1,M1(0)))− Y (0,M1(0),M2(0,M1(0)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (0,M1(0),M2(1,M1(1)))− Y (0,M1(0),M2(0,M1(1)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (0,M1(1),M2(1,M1(0)))− Y (0,M1(1),M2(0,M1(0)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (0,M1(1),M2(1,M1(1)))− Y (0,M1(1),M2(0,M1(1)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (1,M1(0),M2(1,M1(0)))− Y (1,M1(0),M2(0,M1(0)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (1,M1(0),M2(1,M1(1)))− Y (1,M1(0),M2(0,M1(1)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (1,M1(1),M2(1,M1(0)))− Y (1,M1(1),M2(0,M1(0)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (1,M1(1),M2(1,M1(1)))− Y (1,M1(1),M2(0,M1(1)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (0,M1(0),M2(0,M1(1)))− Y (0,M1(0),M2(0,M1(0)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (0,M1(0),M2(1,M1(1)))− Y (0,M1(0),M2(1,M1(0)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (0,M1(1),M2(0,M1(1)))− Y (0,M1(1),M2(0,M1(0)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (0,M1(1),M2(1,M1(1)))− Y (0,M1(1),M2(1,M1(0)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (1,M1(0),M2(0,M1(1)))− Y (1,M1(0),M2(0,M1(0)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (1,M1(0),M2(1,M1(1)))− Y (1,M1(0),M2(1,M1(0)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (1,M1(1),M2(0,M1(1)))− Y (1,M1(1),M2(0,M1(0)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (1,M1(1),M2(1,M1(1)))− Y (1,M1(1),M2(1,M1(0)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (1,M1(1),M2(1,M1(1)))− Y (1,M1(1),M2(1,M1(0)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (1,M1(1),M2(1,M1(1)))− Y (1,M1(1),M2(1,M1(0)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

— If (1) is identified under our extended assumptions, allpath-specific effects are identified.

— Using these assumptions, we can re-write (1) as:∫C

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Identification?

E{Y (1,M1(1),M2(1,M1(1)))− Y (1,M1(1),M2(1,M1(0)))}

E {Y (x ,M1(x ′),M2(x ′′,M1(x ′′′)))} (1)

E {Y |C = c,X = x ,M1 = m1,M2 = m2 }

Unidentified density

— Everything on the previous slide is a function of the distribution ofthe observed data, except for:

fM1(x′′′)|C,M1(x′) (m′1 |c,m1 ) (2)

which (for x ′ 6= x ′′′) involves the joint distribution (conditional on C) ofM1(x) across different worlds.

— Two exceptions:

1 if x ′ = x ′′′ then (2) = I (m1 = m′1).This means that NDE-000, NDE-010, NDE-101, NDE-111,NIE2-000, NIE2-100, NIE2-011 and NIE2-111 are all identified

2 if there is no arrow from M1 to M2, everything simplifies and (2)does not enter

— For all other effects, and when there is an arrow from M1 to M2, wevary (2) in a sensitivity analysis.

fM1(x′′′)|C,M1(x′) (m′1 |c,m1 ) (2)

which (for x ′ 6= x ′′′) involves the joint distribution (conditional on C) ofM1(x) across different worlds.— Two exceptions:

1 if x ′ = x ′′′ then (2) = I (m1 = m′1).

This means that NDE-000, NDE-010, NDE-101, NDE-111,NIE2-000, NIE2-100, NIE2-011 and NIE2-111 are all identified

fM1(x′′′)|C,M1(x′) (m′1 |c,m1 ) (2)

1 if x ′ = x ′′′ then (2) = I (m1 = m′1).

This means that NDE-000, NDE-010, NDE-101, NDE-111,NIE2-000, NIE2-100, NIE2-011 and NIE2-111 are all identified

fM1(x′′′)|C,M1(x′) (m′1 |c,m1 ) (2)

Outline

1 Background

4 Identification

6 Summary

7 References

The Izhevsk study data

— Question: how much of the causal effect of alcohol consumptionon SBP is via GGT (a liver enzyme), via BMI, via both, via neither?— 1275 population-based controls from a case-control study carriedout to assess the effects of hazardous alcohol-drinking on mortality inmen living in Izhevsk, Russia.— Dichotomised exposure, X = heavy drinking, yes/no.— Baseline confounders: age, socio-economic status (SES),smoking status, cigarettes per day.

— Question: how much of the causal effect of alcohol consumptionon SBP is via GGT (a liver enzyme), via BMI, via both, via neither?

— 1275 population-based controls from a case-control study carriedout to assess the effects of hazardous alcohol-drinking on mortality inmen living in Izhevsk, Russia.— Dichotomised exposure, X = heavy drinking, yes/no.— Baseline confounders: age, socio-economic status (SES),smoking status, cigarettes per day.

— Question: how much of the causal effect of alcohol consumptionon SBP is via GGT (a liver enzyme), via BMI, via both, via neither?— 1275 population-based controls from a case-control study carriedout to assess the effects of hazardous alcohol-drinking on mortality inmen living in Izhevsk, Russia.

— Dichotomised exposure, X = heavy drinking, yes/no.— Baseline confounders: age, socio-economic status (SES),smoking status, cigarettes per day.

— Question: how much of the causal effect of alcohol consumptionon SBP is via GGT (a liver enzyme), via BMI, via both, via neither?— 1275 population-based controls from a case-control study carriedout to assess the effects of hazardous alcohol-drinking on mortality inmen living in Izhevsk, Russia.— Dichotomised exposure, X = heavy drinking, yes/no.

— Baseline confounders: age, socio-economic status (SES),smoking status, cigarettes per day.

— Question: how much of the causal effect of alcohol consumptionon SBP is via GGT (a liver enzyme), via BMI, via both, via neither?— 1275 population-based controls from a case-control study carriedout to assess the effects of hazardous alcohol-drinking on mortality inmen living in Izhevsk, Russia.— Dichotomised exposure, X = heavy drinking, yes/no.— Baseline confounders: age, socio-economic status (SES),smoking status, cigarettes per day.

Analysis

— Fully-parametric estimation (linear models with all interactions),approximated by Monte Carlo simulation.— With a sensitivity parameter (κ) representing the proportion of theresidual variance in M1(x) shared across worlds.

Analysis

— Fully-parametric estimation (linear models with all interactions),approximated by Monte Carlo simulation.

— With a sensitivity parameter (κ) representing the proportion of theresidual variance in M1(x) shared across worlds.

Analysis

— Fully-parametric estimation (linear models with all interactions),approximated by Monte Carlo simulation.— With a sensitivity parameter (κ) representing the proportion of theresidual variance in M1(x) shared across worlds.

ResultsCaveat: CIs are wide!

κ = 10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

direct through BMI only

through GGT only through BMI and GGT

1 1 1 1 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 8 8 8 8

5 8 2 8 2 6 5 8 1 1 8 6 3 1 8 8 1 4 3 7 1 7 1 4

8 5 8 2 6 2 8 5 8 6 1 1 8 8 3 1 4 1 7 3 7 1 4 1

2 2 5 6 8 8 1 1 5 8 6 8 1 3 1 4 8 8 1 1 3 4 7 7

κ = 00

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

1 1 1 1 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 8 8 8 8

5 8 2 8 2 6 5 8 1 1 8 6 3 1 8 8 1 4 3 7 1 7 1 4

8 5 8 2 6 2 8 5 8 6 1 1 8 8 3 1 4 1 7 3 7 1 4 1

2 2 56

8 8 1 1 5 8 6 8 1 3 1 4 8 8 1 1 3 4 7 7

κ = 0.50

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

1 1 1 1 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 8 8 8 8

5 8 2 8 2 6 5 8 1 1 8 6 3 1 8 8 1 4 3 7 1 7 1 4

8 5 8 2 6 2 8 5 8 6 1 1 8 8 3 1 4 1 7 3 7 1 4 1

2 2 56

8 8 1 1 5 86 8 1 3 1 4 8 8 1 1 3 4 7 7

Outline

1 Background

4 Identification

6 Summary

7 References

Concluding remarks

— Mediation, particularly effect decomposition, is a subtle business!

— With multiple mediators things get worse!— But as always, better to be aware of this.— In most settings, the choice between possible decompositions islikely to be somewhat arbitrary.— With two mediators, we can look at all 24 decompositions andhope that they are broadly similar (implying no strong mediated interactions?).— But what would we do if they weren’t?— And if there are > 2 mediators?! Sample from the possibledecompositions?— We should also consider the impact of cross-world dependencesensitivity parameters.— We can allow for a restricted pattern of intermediate confoundingusing a generalisation of Petersen’s assumption [Petersen et al,2006].— Would semiparametric estimation methods of these estimands beviable?— How far beyond traditional path analysis can we really go?

Concluding remarks

— Mediation, particularly effect decomposition, is a subtle business!— With multiple mediators things get worse!

— But as always, better to be aware of this.— In most settings, the choice between possible decompositions islikely to be somewhat arbitrary.— With two mediators, we can look at all 24 decompositions andhope that they are broadly similar (implying no strong mediated interactions?).— But what would we do if they weren’t?— And if there are > 2 mediators?! Sample from the possibledecompositions?— We should also consider the impact of cross-world dependencesensitivity parameters.— We can allow for a restricted pattern of intermediate confoundingusing a generalisation of Petersen’s assumption [Petersen et al,2006].— Would semiparametric estimation methods of these estimands beviable?— How far beyond traditional path analysis can we really go?

Concluding remarks

— Mediation, particularly effect decomposition, is a subtle business!— With multiple mediators things get worse!— But as always, better to be aware of this.

— In most settings, the choice between possible decompositions islikely to be somewhat arbitrary.— With two mediators, we can look at all 24 decompositions andhope that they are broadly similar (implying no strong mediated interactions?).— But what would we do if they weren’t?— And if there are > 2 mediators?! Sample from the possibledecompositions?— We should also consider the impact of cross-world dependencesensitivity parameters.— We can allow for a restricted pattern of intermediate confoundingusing a generalisation of Petersen’s assumption [Petersen et al,2006].— Would semiparametric estimation methods of these estimands beviable?— How far beyond traditional path analysis can we really go?

Concluding remarks

— Mediation, particularly effect decomposition, is a subtle business!— With multiple mediators things get worse!— But as always, better to be aware of this.— In most settings, the choice between possible decompositions islikely to be somewhat arbitrary.

— With two mediators, we can look at all 24 decompositions andhope that they are broadly similar (implying no strong mediated interactions?).— But what would we do if they weren’t?— And if there are > 2 mediators?! Sample from the possibledecompositions?— We should also consider the impact of cross-world dependencesensitivity parameters.— We can allow for a restricted pattern of intermediate confoundingusing a generalisation of Petersen’s assumption [Petersen et al,2006].— Would semiparametric estimation methods of these estimands beviable?— How far beyond traditional path analysis can we really go?

Concluding remarks

— Mediation, particularly effect decomposition, is a subtle business!— With multiple mediators things get worse!— But as always, better to be aware of this.— In most settings, the choice between possible decompositions islikely to be somewhat arbitrary.— With two mediators, we can look at all 24 decompositions andhope that they are broadly similar (implying no strong mediated interactions?).

— But what would we do if they weren’t?— And if there are > 2 mediators?! Sample from the possibledecompositions?— We should also consider the impact of cross-world dependencesensitivity parameters.— We can allow for a restricted pattern of intermediate confoundingusing a generalisation of Petersen’s assumption [Petersen et al,2006].— Would semiparametric estimation methods of these estimands beviable?— How far beyond traditional path analysis can we really go?

Concluding remarks

— Mediation, particularly effect decomposition, is a subtle business!— With multiple mediators things get worse!— But as always, better to be aware of this.— In most settings, the choice between possible decompositions islikely to be somewhat arbitrary.— With two mediators, we can look at all 24 decompositions andhope that they are broadly similar (implying no strong mediated interactions?).— But what would we do if they weren’t?

— And if there are > 2 mediators?! Sample from the possibledecompositions?— We should also consider the impact of cross-world dependencesensitivity parameters.— We can allow for a restricted pattern of intermediate confoundingusing a generalisation of Petersen’s assumption [Petersen et al,2006].— Would semiparametric estimation methods of these estimands beviable?— How far beyond traditional path analysis can we really go?

Concluding remarks

— Mediation, particularly effect decomposition, is a subtle business!— With multiple mediators things get worse!— But as always, better to be aware of this.— In most settings, the choice between possible decompositions islikely to be somewhat arbitrary.— With two mediators, we can look at all 24 decompositions andhope that they are broadly similar (implying no strong mediated interactions?).— But what would we do if they weren’t?— And if there are > 2 mediators?! Sample from the possibledecompositions?

— We should also consider the impact of cross-world dependencesensitivity parameters.— We can allow for a restricted pattern of intermediate confoundingusing a generalisation of Petersen’s assumption [Petersen et al,2006].— Would semiparametric estimation methods of these estimands beviable?— How far beyond traditional path analysis can we really go?

Concluding remarks

— Mediation, particularly effect decomposition, is a subtle business!— With multiple mediators things get worse!— But as always, better to be aware of this.— In most settings, the choice between possible decompositions islikely to be somewhat arbitrary.— With two mediators, we can look at all 24 decompositions andhope that they are broadly similar (implying no strong mediated interactions?).— But what would we do if they weren’t?— And if there are > 2 mediators?! Sample from the possibledecompositions?— We should also consider the impact of cross-world dependencesensitivity parameters.

— We can allow for a restricted pattern of intermediate confoundingusing a generalisation of Petersen’s assumption [Petersen et al,2006].— Would semiparametric estimation methods of these estimands beviable?— How far beyond traditional path analysis can we really go?

Concluding remarks

— Mediation, particularly effect decomposition, is a subtle business!— With multiple mediators things get worse!— But as always, better to be aware of this.— In most settings, the choice between possible decompositions islikely to be somewhat arbitrary.— With two mediators, we can look at all 24 decompositions andhope that they are broadly similar (implying no strong mediated interactions?).— But what would we do if they weren’t?— And if there are > 2 mediators?! Sample from the possibledecompositions?— We should also consider the impact of cross-world dependencesensitivity parameters.— We can allow for a restricted pattern of intermediate confoundingusing a generalisation of Petersen’s assumption [Petersen et al,2006].

— Would semiparametric estimation methods of these estimands beviable?— How far beyond traditional path analysis can we really go?

Concluding remarks

— Mediation, particularly effect decomposition, is a subtle business!— With multiple mediators things get worse!— But as always, better to be aware of this.— In most settings, the choice between possible decompositions islikely to be somewhat arbitrary.— With two mediators, we can look at all 24 decompositions andhope that they are broadly similar (implying no strong mediated interactions?).— But what would we do if they weren’t?— And if there are > 2 mediators?! Sample from the possibledecompositions?— We should also consider the impact of cross-world dependencesensitivity parameters.— We can allow for a restricted pattern of intermediate confoundingusing a generalisation of Petersen’s assumption [Petersen et al,2006].— Would semiparametric estimation methods of these estimands beviable?

— How far beyond traditional path analysis can we really go?

Concluding remarks

— Mediation, particularly effect decomposition, is a subtle business!— With multiple mediators things get worse!— But as always, better to be aware of this.— In most settings, the choice between possible decompositions islikely to be somewhat arbitrary.— With two mediators, we can look at all 24 decompositions andhope that they are broadly similar (implying no strong mediated interactions?).— But what would we do if they weren’t?— And if there are > 2 mediators?! Sample from the possibledecompositions?— We should also consider the impact of cross-world dependencesensitivity parameters.— We can allow for a restricted pattern of intermediate confoundingusing a generalisation of Petersen’s assumption [Petersen et al,2006].— Would semiparametric estimation methods of these estimands beviable?— How far beyond traditional path analysis can we really go?

Outline

1 Background

4 Identification

6 Summary

7 References

References (1)

Avin C, Shpitser I, Pearl JIdentifiability of path-specific effects.Proceedings of the Nineteenth Joint Conference on ArtificialIntelligence, pp 357–363, 2005.

Albert JM, Nelson SGeneralized causal mediation analysis.Biometrics, 67:1028–1038, 2011.

MacKinnon DPContrasts in multiple mediator models.In: Multivariate applications in substance use research, pp141–160, 2000.

References (2)

Preacher KJ, Hayes AFAsymptotic and resampling strategies for assessing andcomparing indirect effects in multiple mediator models.Behavior Research Methods, 40:879–891, 2008.

Imai K, Yamamoto TIdentification and sensitivity analysis for multiple causalmechanisms: revisiting evidence from framing experiments.Political Analysis, in press.

References (3)

VanderWeele TJA three-way decomposition of a total effect into direct, indirect,and interactive effects.Epidemiology, Epub ahead of print.

Petersen ML, Sinisi SE, van der Laan MJEstimation of direct causal effects.Epidemiology, 17:276–284, 2006.

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