Causal mediation analysis with multiple causally-ordered mediators

(1)

Causal mediation analysis with multiple

causally-ordered mediators

Rhian Daniel, Bianca De Stavola and Simon Cousens

with thanks to

Stijn Vansteelandt and Dave Leon

Centre for Statistical Methodology

London School of Hygiene and Tropical Medicine

Symposium onCausal Mediation Analysis

(2)

Background One mediator Two mediators Identification Example Summary References

Single mediator

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

(3)

Multiple mediators

n

— However, as we have seen over the last two days, many realistic

(4)

Many mediators considered

en bloc

— Multiple mediators are trivially accommodated by the existing

formal literature if vieweden bloc(withMa vector).

— But then the direct effect is aroundallof them, and the (single)

indirect effect is through one, more or all of them, and isnot further

(5)

Many mediators considered

en bloc

— Multiple mediators are trivially accommodated by the existing

formal literature if vieweden bloc(withMa vector).

— But then the direct effect is aroundallof them, and the (single)

indirect effect is through one, more or all of them, and isnot further

(6)

Two mediators but only one ‘of interest’

— The causal inference literature does focus on ‘two mediators’,L

andM, in settings withintermediate confounding(cf Vanessa’s talk

yesterday).

— ButM is the mediator of interest, with decomposition again into

(7)

Two mediators but only one ‘of interest’

— The causal inference literature does focus on ‘two mediators’,L

andM, in settings withintermediate confounding(cf Vanessa’s talk

yesterday).

— ButM is the mediator of interest, with decomposition again into

(8)

Two mediators, both of interest (1)

— What if both mediators are ‘of interest’?

— We could then be interested in afiner decomposition, with

(9)

Two mediators, both of interest (1)

— What if both mediators are ‘of interest’?

— We could then be interested in afiner decomposition, with

(10)

Two mediators, both of interest (2)

— This setting has been studied, but either (a)without a focus on

decomposingthe total effect [Avin et al (2005), Albert and Nelson

(11)

Two mediators, not causally ordered

— . . . or (b) in the setting withno effect ofM1onM2[MacKinnon

(2000), Preacher and Hayes (2008), Imai and Yamamoto (in press)] (many examples yesterday).

(12)

Our setting

— This talk is about apath-specific decompositionof the total causal

(13)

More generally

n

— More generally there could bencausally-ordered mediators.

—Traditional path analysisgeneralises easily to this setting, but under strong assumptions, including linearity everywhere.

— Can these be relaxed somewhat by generalising ideas from causal inference?

(14)

More generally

n

(15)

More generally

n

(16)

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

(17)

Outline

1

Background

2

Quick revision: effect decomposition with one mediator

3

4

Identification

5

6

Summary

(18)

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 mediated interaction either to the direct or indirect effect.

(19)

Effect decomposition, single mediator

=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

(20)

Effect decomposition, single mediator

=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

(21)

Effect decomposition, single mediator

=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

(22)

Effect decomposition, single mediator

=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

(23)

Effect decomposition, single mediator

=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

(24)

Effect decomposition, single mediator

=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

(25)

Effect decomposition, single mediator

=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

(26)

Effect decomposition, single mediator

=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

(27)

Effect decomposition, single mediator

=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

(28)

Outline

1

Background

2

3

4

Identification

5

6

Summary

(29)

Counterfactuals

— With one mediator, we need:

M(x),Y(x,m),Y(x,M(x0))

— With two, we need:

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

and

M2(x,M1(x0))

and

Y(x,M1(x0),M2(x00,M1(x000)))

— Natural path-specific effects are defined as contrasts between these for carefully chosen values ofx,x0,x00,x000.

(30)

Counterfactuals

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

and

M2(x,M1(x0))

and

Y(x,M1(x0),M2(x00,M1(x000)))

(31)

Counterfactuals

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

and

M2(x,M1(x0))

and

Y(x,M1(x0),M2(x00,M1(x000)))

(32)

Counterfactuals

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

and

M2(x,M1(x0))

and

Y(x,M1(x0),M2(x00,M1(x000)))

(33)

Counterfactuals

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

and

M2(x,M1(x0))

and

Y(x,M1(x0),M2(x00,M1(x000)))

(34)

Counterfactuals

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

and

M2(x,M1(x0))

and

Y(x,M1(x0),M2(x00,M1(x000)))

— Natural path-specific effects are defined as contrasts between thesefor carefully chosen values ofx,x0,x00,x000.

(35)

Counterfactuals

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

and

M2(x,M1(x0))

and

Y(x,M1(x0),M2(x00,M1(x000)))

— Natural path-specific effects are defined as contrasts between thesefor carefully chosen values ofx,x0,x00,x000.

(36)

Direct effect

— Anatural direct effect(through neitherM1norM2) is of the form:

— The first argument changes and all other arguments stay the same, making it a direct effect.

— There are 8 choices for how to fixx0_,_x00_,_x000_.

— We can choose(x0_,_x00_,_x000_{) = (}₀_,₀_,₀₎_{. We call this NDE-000.}

— Similarly, can choose(x0,x00,x000) = (, ,). We call this .

(37)

Direct effect

E{Y(1,M1(x0),M2(x00,M1(x000)))−Y(0,M1(x0),M2(x00,M1(x000)))}

— There are 8 choices for how to fixx0,x00,x000.

— We can choose(x0,x00,x000) = (0,0,0). We call this NDE-000. — Similarly, can choose(x0_,_x00_,_x000_{) = (}_{, ,}₎_{. We call this}

(38)

Direct effect

— Thefirst argumentchanges and all other arguments stay the

same, making it a direct effect.

(39)

Direct effect

— The first argument changes and allother argumentsstay the

same, making it a direct effect.

(40)

Direct effect

(41)

Direct effect

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

— We can choose(x0,x00,x000) = (0,0,0). We call thisNDE-000.

— Similarly, can choose(x0_,_x00_,_x000_{) = (}_{, ,}₎_{. We call this}

(42)

Direct effect

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

— We can choose(x0,x00,x000) = (0,0,0). We call this NDE-000. — Similarly, can choose(x0_,_x00_,_x000_{) = (}₀_,₀_,₁₎_{. We call this}

(43)

Direct effect

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

— We can choose(x0,x00,x000) = (0,0,0). We call this NDE-000. — Similarly, can choose(x0_,_x00_,_x000_{) = (}₀_,₁_,₀₎_{. We call this}

(44)

Direct effect

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

— We can choose(x0,x00,x000) = (0,0,0). We call this NDE-000. — Similarly, can choose(x0_,_x00_,_x000_{) = (}₀_,₁_,₁₎_{. We call this}

(45)

Direct effect

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

— We can choose(x0,x00,x000) = (0,0,0). We call this NDE-000. — Similarly, can choose(x0_,_x00_,_x000_{) = (}₁_,₀_,₀₎_{. We call this}

(46)

Direct effect

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

— We can choose(x0,x00,x000) = (0,0,0). We call this NDE-000. — Similarly, can choose(x0_,_x00_,_x000_{) = (}₁_,₀_,₁₎_{. We call this}

(47)

Direct effect

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

— We can choose(x0,x00,x000) = (0,0,0). We call this NDE-000. — Similarly, can choose(x0_,_x00_,_x000_{) = (}₁_,₁_,₀₎_{. We call this}

(48)

Direct effect

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

— We can choose(x0,x00,x000) = (0,0,0). We call this NDE-000. — Similarly, can choose(x0_,_x00_,_x000_{) = (}₁_,₁_,₁₎_{. We call this}

(49)

Indirect effect through

M

1

only

— Anatural indirect effect throughM1onlyis of the form:

— The second argument changes and all other arguments stay the

same, making it an indirect effect throughM1only.

— There are 8 choices for how to fixx,x00,x000.

— We can choose(x,x00,x000) = (0,0,0). We call this NIE1-000.

(50)

Indirect effect through

M

1

only

E{Y(x,M1(1),M2(x00,M1(x000)))−Y(x,M1(0),M2(x00,M1(x000)))}

(51)

Indirect effect through

M

1

only

— Thesecond argumentchanges and all other arguments stay the

(52)

Indirect effect through

M

1

only

— The second argument changes and allother argumentsstay the

(53)

Indirect effect through

M

1

only

(54)

Indirect effect through

M

1

only

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

— We can choose(x,x00,x000) = (0,0,0). We call thisNIE1-000.

(55)

Indirect effect through

M

1

only

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

(56)

Indirect effect through

M

1

only

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

(57)

Indirect effect through

M

1

only

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

(58)

Indirect effect through

M

1

only

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

(59)

Indirect effect through

M

1

only

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

(60)

Indirect effect through

M

1

only

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

(61)

Indirect effect through

M

1

only

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

(62)

Indirect effect through

M

2

only

— The third argument changes and all other arguments stay the

(63)

Indirect effect through

M

2

only

E{Y(x,M1(x0),M2(1,M1(x000)))−Y(x,M1(x0),M2(0,M1(x000)))}

(64)

Indirect effect through

M

2

only

— Thethird argumentchanges and all other arguments stay the

(65)

Indirect effect through

M

2

only

— The third argument changes and allother argumentsstay the

(66)

Indirect effect through

M

2

only

(67)

Indirect effect through

M

2

only

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

(68)

Indirect effect through

M

2

only

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

(69)

Indirect effect through

M

2

only

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

(70)

Indirect effect through

M

2

only

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

(71)

Indirect effect through

M

2

only

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

(72)

Indirect effect through

M

2

only

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

(73)

Indirect effect through

M

2

only

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

(74)

Indirect effect through

M

2

only

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

(75)

Indirect effect through both

M

1

and

M

2

— Anatural indirect effect through bothM1andM2is of the form:

— The fourth argument changes and all other arguments stay the

same, making it an indirect effect through bothM1andM2.

(76)

Indirect effect through both

M

1

and

M

2

E{Y(x,M1(x0),M2(x00,M1(1)))−Y(x,M1(x0),M2(x00,M1(0)))}

(77)

Indirect effect through both

M

1

and

M

2

— Thefourth argumentchanges and all other arguments stay the

(78)

Indirect effect through both

M

1

and

M

2

— The fourth argument changes and allother argumentsstay the

(79)

Indirect effect through both

M

1

and

M

2

(80)

Indirect effect through both

M

1

and

M

2

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

(81)

Indirect effect through both

M

1

and

M

2

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

(82)

Indirect effect through both

M

1

and

M

2

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

(83)

Indirect effect through both

M

1

and

M

2

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

(84)

Indirect effect through both

M

1

and

M

2

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

(85)

Indirect effect through both

M

1

and

M

2

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

(86)

Indirect effect through both

M

1

and

M

2

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

(87)

Indirect effect through both

M

1

and

M

2

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

(88)

Decomposition

— We have defined8 types(cf pure/total) of each of4 path-specific

effects(cf direct/indirect).

— We could therefore form84₌₄₀₉₆_{sums of the form}

NDE+NIE1+NIE2+NIE12

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

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

— More generally, withnmediators, there are2(2n−1)·2npossible

sums,(2n_)!_{of which are equal to the TCE.}

n (2n_)! 1 2 2 24 3 40,320 4 2.092×1013 5 2.631×1035

(89)

Decomposition

NDE+NIE1+NIE2+NIE12

n (2n_)! 1 2 2 24 3 40,320 4 2.092×1013 5 2.631×1035

(90)

Decomposition

NDE+NIE1+NIE2+NIE12

n (2n_)! 1 2 2 24 3 40,320 4 2.092×1013 5 2.631×1035

(91)

Decomposition

NDE+NIE1+NIE2+NIE12

— More generally, withnmediators, there are2(2n−1)·2n possible

n (2n_)! 1 2 2 24 3 40,320 4 2.092×1013 5 2.631×1035

(92)

Decomposition

NDE+NIE1+NIE2+NIE12

n (2n_)! 1 2 2 24 3 40,320 4 2.092×1013 5 2.631×1035

(93)

Decomposition

NDE+NIE1+NIE2+NIE12

n (2n_)! 1 2 2 24 3 40,320 4 2.092×1013 5 2.631×1035

(94)

Decomposition

NDE+NIE1+NIE2+NIE12

n (2n_)! 1 2 2 24 3 40,320 4 2.092×1013 5 2.631×1035

(95)

Decomposition

NDE+NIE1+NIE2+NIE12

n (2n_)!

1 2

2 24

3 40,320

(96)

Decomposition

NDE+NIE1+NIE2+NIE12

n (2n_)!

1 2

2 24

(97)

Outline

1

Background

2

3

4

Identification

5

6

Summary

(98)

Nonparametric identification: single mediator

— Throughout, we omitbackground confoundersCfrom our

diagrams, but they are always implicitly there.

— Recall that for nonparametric identification of PNDE, TNDE, PNIE

and TNIE in the one-mediator setting, we requireno unmeasured

confoundingofX andM,M andY, andX andY.

(99)

Nonparametric identification: single mediator

— Recall that for nonparametric identification of PNDE, TNDE, PNIE and TNIE in the one-mediator setting, we require

no unmeasured

confoundingofX andM,M andY, andX andY.

(100)

Nonparametric identification: single mediator

U₁

M andY, andX andY.

(101)

Nonparametric identification: single mediator

U₁ U₂

andX andY.

(102)

Nonparametric identification: single mediator

U₁ U₂

U₃

(103)

Nonparametric identification: single mediator

(104)

Extension to two mediators

— Consider thenatural extensionsof these assumptions.

—No unmeasured confounding, andno intermediate confounding.

(105)

Extension to two mediators

U₁ U₂

U₃

—No unmeasured confounding,

andno intermediate confounding.

(106)

U₁ U₂

U₄

U₃

(107)

Extension to two mediators

U₁ U₂

U₅ U₄

U₃

(108)

U₁ U₂

U₅ U₄

U₆

U₃

(109)

Extension to two mediators

(110)

(111)

Extension to two mediators

(112)

(113)

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(x0),M2(x00,M1(x000)))} (1)

— If (1) is identified under ourextended assumptions, all

path-specific effects are identified.

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

Z C Z M1 Z M1 Z M2 E{Y|C=c,X =x,M1=m1,M2=m2} ·fM2|C,X,M1 (m2|c,x 00_,_m0 1) fM1(x000)|C,M1(x0)(m 0 1|c,m1) ·fM1|C,X (m1|c,x 0₎_f C(c) ·dµM2(m2)dµM1(m 0 1)dµM1(m1)dµC(c)

(114)

Identification?

E{Y(x,M1(x0),M2(x00,M1(x000)))} (1)

(115)

Identification?

E{Y(x,M1(x0),M2(x00,M1(x000)))} (1)

(116)

Identification?

E{Y(x,M1(x0),M2(x00,M1(x000)))} (1)

(117)

Identification?

E{Y(x,M1(x0),M2(x00,M1(x000)))} (1)

(118)

Identification?

E{Y(x,M1(x0),M2(x00,M1(x000)))} (1)

(119)

Identification?

E{Y(x,M1(x0),M2(x00,M1(x000)))} (1)

(120)

Identification?

E{Y(x,M1(x0),M2(x00,M1(x000)))} (1)

(121)

Identification?

E{Y(x,M1(x0),M2(x00,M1(x000)))} (1)

(122)

Identification?

E{Y(x,M1(x0),M2(x00,M1(x000)))} (1)

(123)

Identification?

E{Y(x,M1(x0),M2(x00,M1(x000)))} (1)

(124)

Identification?

— Each half of each path-specific effect i