population-average estimands

WEB APPENDIX 1 POSSIBLE DECOMPOSITIONS AND PROPORTION MEDIATED

For a binary exposure X , there are six possible decompositions of the total effect into the natural direct and indirect effects defined in the main text. Each decomposition corresponds to different values of the hypothetical exposure levels for each natural (in)direct effect, so that their product on the risk ratio scale equals the total effect. To ease notation when deriving the expressions for each decomposition, denote the average nested potential outcome by

μ_x(0), x⁽¹⁾, x⁽²⁾≡ E

{

^Y

(

^{x(0), M1}

(

^x(1)

)

^{, M2}

(

^{x(2), M1}

(

^x(1)

) ) ) }

^.

Then the natural direct effect is RR_NDE

(

x⁽¹⁾, x⁽²⁾

)

=μ_{1, x}(1), x⁽²⁾

μ_{0, x}(1), x⁽²⁾

,

the natural indirect effect via M₁ is RR_NIE1

(

x⁽⁰⁾, x⁽²⁾

)

=μ_x⁽⁰⁾_{,1, x}⁽²⁾

μ_x⁽⁰⁾_{,0, x}⁽²⁾,

and the natural indirect effect via ^M2 is RR_NIE2

(

x⁽⁰⁾, x⁽¹⁾

)

=μ_x(0), x⁽¹⁾,1

μ_x(0), x⁽¹⁾,0

.

Let μ_x=μ_{x , x , x} for x=0,1 , by composition. The decomposition of the total effect (on the risk difference scale) into the proportions mediated via each of the two mediators and the proportion that remains unmediated via either mediator, depends on the fixed hypothetical exposure levels. We derive the results for each of the six potential decompositions described in Web Appendix Table 1.

Decomposition scenario 1: Corresponding to the first row in Web Appendix Table 1, the total effect ^RRTE=μ₁/μ₀ can be written as:

μ₁−μ₀ ¿μ₁−μ_1,0,1+μ_1,0,1−μ_1,0,0+μ_1,0,0−μ₀ μ₁−μ₀

μ₀ ¿μ₁−μ_1,0,1

μ₀ +μ_1,0,1−μ_1,0,0

μ₀ +μ_1,0,0−μ₀ μ₀

RR_TE−1

¿

¿μ₁−μ_1,0,1 μ₀

μ_1,0,1

μ_1,0,1+μ_1,0,1−μ_1,0,0 μ₀

μ_1,0,0

μ_1,0,0+μ_1,0,0−μ₀

μ₀ ¿

(

^μ1^μ,0¹,1

−1

)

^μ1μ,0,10

+

(

^μμ11^,0,0^,1,0

−1

)

^μ1μ,00^,0

+

(

^μ1,0,μ0 ⁰

−1

)

¿ ¿

¿

¿

[

^{RRNIE1(1,1)−1}

]

^μ_μ^1,0,1

1,0,0

μ_1,0,0

μ₀ +

[

^{RRNIE2(1,0)−1}

]

^μ_μ^1,0,0

0

+

[

^RRNDE(0,0)−1

]

¿

[

^RRNIE1(1,1)−1

]

^RRNIE2(1,0)RR_NDE(0,0)+

[

^RRNIE2(1,0)−1

]

^RRNDE(0,0)+

[

^{RRNDE(0,0)−1}

]

1

RR_NIE

1(1,1)−1

RR_TE−1 RR_NIE2(1,0)RR_NDE(0,0)+RR_NIE

2(1,0)−1

RR_TE−1 RR_NDE(0,0)+RR_NDE(0,0)−1 RR_TE−1 ¿

The proportions mediated (PM) via M₁ and via M₂ are PM_M1=RR_NIE1(1,1)−1

RR_TE−1 RR_NIE2(1,0)RR_NDE(0,0)and PM_M2=RR_NIE2(1,0)−1

RR_TE−1 RR_NDE(0,0) ,

respectively, and the proportion unmediated via either mediator is RR_NDE(0,0)−1 RR_TE−1 .

Decomposition scenario 2: Corresponding to the second row in Web Appendix Table 1, the total effect RR_TE=μ₁/μ₀ can be written as:

μ₁−μ₀ ¿μ₁−μ_1,1,0+μ_1,1,0−μ_1,0,0+μ_1,0,0−μ₀ μ₁−μ₀

μ₀ ¿μ₁−μ_1,1,0

μ₀ +μ_1,1,0−μ_1,0,0

μ₀ +μ_1,0,0−μ₀ μ₀ RR_TE−1 ¿μ₁−μ_1,1,0

μ₀

μ_{1,1, 0}

μ_{1,1, 0}+μ_{1,1 0}−μ_1,0,0 μ₀

μ_1,0,0

μ_1,0,0+μ_1,0,0−μ₀ μ₀

¿

¿

[

^RRNIE2(1,1)−1

]

_μ^{μ1,1, 0}

1,0,0

μ_1,0,0

μ₀ +

[

^RRNIE1(1,0)−1

]

^μ1_μ^,0,0

0

+

[

^RRNDE(0,0)−1

]

¿1

RR_NIE2(1,1)−1

RR_TE−1 RR_NIE1(1,0)RR_NDE(0,0)+RR_NIE1(1,0)−1

RR_TE−1 RR_NDE(0,0)+RR_NDE(0,0)−1 RR_TE−1 ¿

The proportions mediated via M₁ and via M₂ are PM_M1=RR_NIE1(1,0)−1

RR_TE−1 RR_NDE(0,0) and PM_M2=RR_NIE2(1,1)−1

RR_TE−1 RR_NIE1(1,0)RR_NDE(0,0) ,

respectively, and the proportion unmediated via either mediator is RR_NDE(0,0)−1 RR_TE−1 .

Decomposition scenario 3: Corresponding to the third row in Web Appendix Table 1, the total effect RR_TE=μ₁/μ₀ can be written as:

μ₁−μ₀ ¿μ₁−μ_1,1,0+μ_1,1,0−μ_0,1,0+μ_0,1,0−μ₀ μ₁−μ₀

μ₀ ¿μ₁−μ_1,1,0

μ₀ +μ_1,1,0−μ_0,1,0

μ₀ +μ_0,1,0−μ₀ μ₀ RR_TE−1 ¿μ₁−μ_1,1,0

μ₀

μ_{1,1, 0}

μ_{1,1, 0}+μ_{1,1 0}−μ_0,1,0 μ₀

μ_0,1,0

μ_0,1,0+μ_0,1,0−μ₀ μ₀

¿

¿

[

^{RRNIE2(1,1)−1}

]

_μ^{μ1,1, 0}

0,1,0

μ_0,1,0

μ₀ +

[

^RRNDE(1,0)−1

]

^μ0,1,_μ ⁰

0

+

[

^{RRNIE1(0,0)−1}

]

¿1

RR_NIE2(1,1)−1

RR_TE−1 RR_NDE(1,0)RR_NIE1(0,0)+RR_NDE(1,0)−1

RR_TE−1 RR_NIE1(0,0)+RR_NIE1(0,0)−1 RR_TE−1 ¿

The proportions mediated via M₁ and via M₂ are PM_M1=RR_NIE

1(0,0)−1

RR_TE−1 and PM_M2=RR_NIE

2(1,1)−1

RR_TE−1 RR_NDE(1,0)RR_NIE1(0,0) , respectively, and

the proportion unmediated via either mediator is RR_NDE(1,0)−1

RR_TE−1 RR_NIE1(0,0) .

Decomposition scenario 4: Corresponding to the fourth row in Web Appendix Table 1, the total effect RR_TE=μ₁/μ₀ can be written as:

μ₁−μ₀ ¿μ₁−μ_0,1,1+μ_0,1,1−μ_0,1,0+μ_0,1,0−μ₀ μ₁−μ₀

μ₀ ¿μ₁−μ_0,1,1

μ₀ +μ_0,1,1−μ_0,1,0

μ₀ +μ_0,1,0−μ₀ μ₀ RR_TE−1 ¿μ₁−μ_{0,1, 1}

μ₀

μ_0,1,1

μ_0,1,1+μ_0,1,1−μ_0,1,0 μ₀

μ_0,1,0

μ_0,1,0+μ_0,1,0−μ₀ μ₀

¿

¿

[

^RRNDE(1,1)−1

]

^μ_μ^0,1,1

0,1,0

μ_0,1,0

μ₀ +

[

^RRNIE2(0,1)−1

]

^μ0,1,_μ ⁰

0

+

[

^RRNIE1(0,0)−1

]

¿1

RR_NDE(1,1)−1

RR_TE−1 RR_NIE2(0,1)RR_NIE1(0,0)+RR_NIE2(0,1)−1

RR_TE−1 RR_NIE1(0,0)+RR_NIE1(0,0)−1 RR_TE−1 ¿

The proportions mediated via M₁ and via M₂ are PM_M1=RR_NIE1(0,0)−1

RR_TE−1 and PM_M2=RR_NIE2(0,1)−1

RR_TE−1 RR_NIE1(0,0) , respectively, and the

proportion unmediated via either mediator is RR_NDE(1,1)−1

RR_TE−1 RR_NIE2(0,1)RR_NIE1(0,0) .

Decomposition scenario 5: Corresponding to the fifth row in Web Appendix Table 1, the total effect RR_TE=μ₁/μ₀ can be written as:

μ₁−μ₀ ¿μ₁−μ_0,1,1+μ_0,1,1−μ_0,0,1+μ_0,0,1−μ₀ μ₁−μ₀

μ₀ ¿μ₁−μ_0,1,1

μ₀ +μ_0,1,1−μ_0,0,0

μ₀ +μ_0,0,0−μ₀ μ₀ RR_TE−1 ¿μ₁−μ_{0,1, 1}

μ₀

μ_0,1,1

μ_0,1,1+μ_0,1,1−μ_0,0,1 μ₀

μ_0,0,1

μ_0,0,1+μ_0,0,1−μ₀ μ₀

¿

¿

[

^RRNDE(1,1)−1

]

^μ_μ^0,1,1

0,0,1

μ_0,0,1

μ₀ +

[

^RRNIE1(0,1)−1

]

^μ0_μ^,0,1

0

+

[

^RRNIE2(0,0)−1

]

¿1

RR_NDE(1,1)−1

RR_TE−1 RR_NIE1(0,1)RR_NIE2(0,0)+RR_NIE1(0,1)−1

RR_TE−1 RR_NIE2(0,0)+RR_NIE2(0,0)−1 RR_TE−1 ¿

The proportions mediated via M₁ and via M₂ are PM_M1=RR_NIE1(0,1)−1

RR_TE−1 RR_NIE2(0,0) and PM_M2=RR_NIE2(0,0)−1

RR_TE−1 , respectively, and the

proportion unmediated via either mediator is RR_NDE(1,1)−1

RRTE−1 RR_NIE1(0,1)RR_NIE2(0,0) .

Decomposition scenario 6: Corresponding to the sixth row in Web Appendix Table 1, the total effect RR_TE=μ₁/μ₀ can be written as:

μ₁−μ₀ ¿μ₁−μ_1,0,1+μ_1,0,1−μ_0,0,1+μ_0,0,1−μ₀ μ₁−μ₀

μ₀ ¿μ₁−μ_{1,0, 1}

μ₀ +μ_1,0,1−μ_0,0,1

μ₀ +μ_0,0,1−μ₀ μ₀ RR_TE−1 ¿μ1−μ1,0,1

μ₀

μ1,0,1

μ_1,0,1+μ1,0,1−μ0,0,1

μ₀

μ0,0,1

μ_0,0,1+μ0,0, 1−μ0

μ₀

¿

¿

[

^RRNIE1(1,1)−1

]

_μ^μ1,0,1

0,0,1

μ_{0,0, 1}

μ₀ +

[

^RRNDE(0,1)−1

]

^μ0_μ^,0,1

0

+

[

^RRNIE2(0,0)−1

]

¿1

RR_{N IE1}(1,1)−1

RR_TE−1 RR_NDE(0,1)RR_NIE2(0,0)+RR_NDE(0,1)−1

RR_TE−1 RR_NIE2(0,0)+RR_NIE2(0,0)−1 RR_TE−1 ¿

The proportions mediated via M₁ and via M₂ are

PM_M1=RR_NIE1(1,1)−1

RR_TE−1 RR_NDE(0,1)RR_NIE2(0,0) and PM_M2=RR_NIE2(0,0)−1

RR_TE−1 , respectively, and

the proportion unmediated via either mediator is RR_NDE(0,1)−1

RR_TE−1 RR_NIE2(0,0) .

WEB APPENDIX 2 ESTIMATION PROCEDURE FOR NATURAL EFFECTS MODELS

In this section, we restate the estimation procedure presented in steps 1 to 6 of the main text of Steen et al. (2017).

1. Fit a suitable model for the probability (density) of the first mediator M1 conditional on exposure X and covariates C. For example, a saturated logistic regression model for binary M1 may be:

logit

{

^Pr

(

^{M1=1∨X , C}

) }

^{=α10+α1xX+α1CC+α1xcXC}

where logit(z) = log{z/(1 − z)}. Denote the resulting estimated density by f^

(

M1∨X=x ,C

)

, x=0,1.

2. Fit a suitable mean model for the outcome Y conditional on exposure X, both mediators M1 and M2, and covariates C. For example, a logistic regression model for binary Y with all possible interactions between the exposure and mediators may be:

logit

{

^E

(

^{Y∨X , M1, M2, C}

) }

^{=β0+βxX+β1M1+β2M2+βcC+βx1X M1+βx2XM2+β12M1M2+βx12X M1M2.}

Denote the fitted model by ^E

(

^{Y∨X , M}1, M₂,C

)

^.

3. Construct an expanded dataset for every individual as shown in Web Appendix Table 2.

The observed data for each individual is duplicated four times, with each row

corresponding to a different combination of the hypothetical exposures {x⁽⁰⁾, x⁽¹⁾, x⁽²⁾} for the counterfactual mediators. In the first row, all three exposures equal the observed value X. In the second row, the exposure x⁽⁰⁾ is set to its counterfactual value 1−X. In the third row, the exposure x⁽¹⁾ is set to its counterfactual value 1 − X. In the fourth row, both

exposures x⁽⁰⁾ and x⁽¹⁾ are set to their counterfactual values 1 − X. In all rows, x⁽²⁾ is set to the observed value X.

4. For each row of the expanded dataset in Web Appendix Table 2, compute weights using the estimated density in step 1:

w^

(

x⁽¹⁾, x⁽²⁾

)

=

f^

(

^M1∨X=x⁽¹⁾,C

)

f^

(

^M1∨X=x⁽²⁾,C

)

5. For each row of the expanded dataset in Web Appendix Table 2, impute the nested

counterfactuals Y

(

^{x(0), M}1

(

x⁽¹⁾

)

, M₂

(

^{x(2), M}1

(

x⁽¹⁾

) ) )

as predictions

^E

(

^{Y∨X=x(0), M}1, M₂,C

)

from the fitted outcome model in step 2.

6. Fit the conditional natural effects model to the expanded dataset in Web Appendix Table 2 using weighted regression. Regress the imputed potential outcomes from the previous step on {x⁽⁰⁾, x⁽¹⁾, x⁽²⁾} and C, weighting by the computed values of w^

(

x⁽¹⁾, x⁽²⁾

)

. For example, a conditional logistic natural effects model with main effects only is:

logit

[

^E

{

^Y

(

^{x(0), M1}

(

^x(1)

)

^{, M2}

(

^{x(2), M1}

(

^x(1)

) ) )

^∨C

} ]

^{=θ0+θ1x(0)+θ2x(1)+θ3x(2)+θcC .}

Nonparametric bootstrap confidence intervals may be constructed by randomly resampling observations with replacement and repeating all the above steps for each bootstrap sample.

WORKING MODEL FOR THE DENSITY OF M2

The above procedure requires a working model for M1 given X and C. Alternatively, a suitable working model for M2 given X, M1, and C can be used. Below we describe changes to be made in each step of the above procedure. For example, suppose that M2 is an ordinal variable taking

three possible values {0, 1, 2}. In step 1, fit a saturated multinomial logistic regression model for M2:

log

{

^PrPr

( (

^{MM22=m∨=0∨MM11X , CX , C}

) ) }=α20m+α2mxX+α21mM1+α2mcC+α2xm1X M1+α2mxcXC+α21cm M1C+α2mx1cX M1C ;m=1,2

Denote the resulting estimated density by f^

(

M2∨M1, X=x ,C

)

; x=0,1.

In step 3, construct an expanded dataset as shown in Web Appendix Table 3, by swapping the (values in the) columns x⁽¹⁾ and x⁽²⁾ in Web Appendix Table 2. In step 4, calculate the weights in each row of the expanded dataset as:

w^

(

x⁽²⁾, x⁽¹⁾

)

=

f^

(

^M2∨M₁, X=x⁽²⁾, C

)

f^

(

^M2∨M₁, X=x⁽¹⁾, C

)

Then carry out all other steps as described above.

POPULATION-AVERAGE ESTIMANDS

The conditional natural effects encoded in the above model are stratum-specific effects within strata of C. Because our interest lies in population-level average effects, we will further use inverse propensity score weights to transport the results to the entire target population by accounting for the possibly selective nature of individuals with observed exposure X. Therefore, the estimation procedure is augmented with the following step to be carried out before step 1.

1. Fit a propensity score model for exposure X conditional on C to the observed data. For example, a logistic regression model for X with main effects only may be:

logit

{

^Pr(X=1∨C)

}

^=γ0+γ_cC .

Let ^{^p=expit}

(

^{γ^}0+ ^γ_cC

)

denote the (individual) predicted probability of receiving the observed exposure based on the maximum likelihood estimates of the parameters ^{γ^}0

and γ^_c , where expit(z) = exp(z)/{1 + exp(z)}. The estimated individual inverse propensity score weight is then w^^x=X/ ^p+(1−X)/ (1− ^p).

Then in step 6, fit the marginal natural effects model (instead of the conditional model) to the expanded dataset using the (product of) weights w^^xw^

(

x⁽¹⁾, x⁽²⁾

)

. For example, a saturated marginal logistic natural effect model is:

logit

[

^E

{

^Y

(

^{x(0), M1}

(

^x(1)

)

^{, M2}

(

^{x(2), M1}

(

^x(1)

) ) ) } ]

^{=θ0+θ1x(0)+θ2x(1)+θ3x(2)+θ4x(0)x(1)+θ5x(0)x(2)+θ6x(1)x(2)+θ7x(0)x(1)x(2).}

Closed-form expressions for the natural direct and natural indirect effects, on the risk ratio scale, under each of the six possible decompositions are displayed in Web Appendix Table 4 when using the above saturated marginal logistic natural effect model.

WEB APPENDIX 3

SENSITIVITY ANALYSIS FOR CROSS-WORLD INDEPENDENCE ASSUMPTION

Focusing on the natural (in)direct effects of Steen et al. (2017), rather than all possible fine- grained path-specific effects, avoids assumptions about the independence of mediator counterfactuals under different hypothetical exposure levels that are additionally required to identify fine-grained path-specific effects (Daniel et al., 2015). Nonetheless, for completeness, we describe in this section the procedure and results from implementing a sensitivity analysis for violations of the cross-world independence assumption.

We implemented the sensitivity analysis of Albert and Nelson (2011) for non-continuous mediators, using the estimation procedure of Daniel et al. (2015). Following Albert and Nelson (2011), we estimated the conditional distributions of the counterfactual mediators SGA (M1) given each fixed value of the cross-world correlation along a pre-specified equally spaced grid {0, 0.05, …, 1}. We used this method because it was designed for discrete mediators. We then estimated the four separate path-specific effects given each value of the cross-world

correlation, using the Monte Carlo simulation-based parametric G-computation procedure of Daniel et al. (2015). The results for 12 possible decompositions of the total effect are displayed in Web Appendix Figure 2.

Because the path-specific effects can be combined to yield the natural indirect effects encoded in the natural effect model, we compared the combined effect estimates using Daniel et al.

(2015) with the estimates using Steen et al. (2017). The corresponding estimates of the natural

indirect effects under the six decompositions differed from those in Web Appendix Table 5 at either the first or second decimal point. The differences arose due to working models for both mediators in Daniel et al. (2015), compared to a working model for only one mediator in Steen et al. (2017). In summary, the estimated RR using two different procedures yielded very similar results.

Web Appendix Table 1

Six possible decompositions of the total effect into natural indirect effects via each mediator, and the natural direct effect, for two causally ordered mediators and a binary exposure Decomposition

scenario RR_NDE

(

x⁽¹⁾, x⁽²⁾

)

RR_NIE1

(

x⁽⁰⁾, x⁽²⁾

)

RR_NIE2

(

x⁽⁰⁾, x⁽¹⁾

)

1 x⁽¹⁾=0 x⁽²⁾=0 x⁽⁰⁾=1 x⁽²⁾=1 x⁽⁰⁾=1 x⁽¹⁾=0 2 x⁽¹⁾=0 x⁽²⁾=0 x⁽⁰⁾=1 x⁽²⁾=0 x⁽⁰⁾=1 x⁽¹⁾=1 3 x⁽¹⁾=1 x⁽²⁾=0 x⁽⁰⁾=0 x⁽²⁾=0 x⁽⁰⁾=1 x⁽¹⁾=1 4 x⁽¹⁾=1 x⁽²⁾=1 x⁽⁰⁾=0 x⁽²⁾=0 x⁽⁰⁾=0 x⁽¹⁾=1 5 x⁽¹⁾=1 x⁽²⁾=1 x⁽⁰⁾=0 x⁽²⁾=1 x⁽⁰⁾=0 x⁽¹⁾=0 6 x⁽¹⁾=0 x⁽²⁾=1 x⁽⁰⁾=1 x⁽²⁾=1 x⁽⁰⁾=0 x⁽¹⁾=0

Web Appendix Table 2

Expanded dataset for each individual when a working model for M1 is used to estimate the parameters in a natural effect model. The observed mediators M1, M2 and covariates C are omitted for simplicity.

x⁽⁰⁾ x⁽¹⁾ x⁽²⁾ Estimated weight Imputed potential outcome

X X X 1 ^E

(

^{Y∨X , M}1, M₂,C

)

1-X X X 1 ^E

(

^{Y∨1−X , M}1, M₂, C

)

X 1-X X f^

(

^M1∨1−X ,C

)

^{/ ^f}

(

^M1∨X ,C

)

^{^E}

(

^{Y∨X , M}1, M₂,C

)

1-X 1-X X f^

(

M1∨1−X ,C

)

^{/ ^}f

(

M1∨X ,C

)

^{^E}

(

^{Y∨1−X , M}1, M₂, C

)

Web Appendix Table 3

Expanded dataset for each individual when a working model for M2 is used to estimate the parameters in a natural effect model. The observed mediators M1, M2 and covariates C are omitted for simplicity.

x⁽⁰⁾ x⁽¹⁾ x⁽²⁾ Estimated weight Imputed potential outcome

X X X 1 ^E

(

^{Y∨X , M}1, M₂,C

)

1-X X X 1 ^E

(

^{Y∨1−X , M}1, M₂, C

)

X X 1-X f^

(

^M2∨M₁,1−X , C

)

^{/ ^f}

(

^M2∨M₁X , C

)

^{^E}

(

^{Y∨X , M}1, M₂,C

)

1-X X 1-X f^

(

M₂∨M₁,1−X , C

)

/ ^f

(

M₂∨M₁X , C

)

^{^E}

(

^{Y∨1−X , M}1, M₂, C

)

Web Appendix Table 4

The natural direct effect and the natural indirect effects on the RR scale under six different decomposition scenarios when fitting a saturated marginal logistic natural effect model. The association of placental abruption and perinatal mortality is assumed to

operate through the causally ordered mediators small for gestational age (SGA) births and preterm delivery.

Decomposition scenario

RR_NDE RR_NIE1 RR_NIE2 RR_NDE RR_NIE1 (SGA births) RR_NIE2 (Preterm delivery)

x⁽¹⁾ x⁽²⁾ x⁽⁰⁾ x⁽²⁾ x⁽⁰⁾ x⁽¹⁾

1 0 0 1 1 1 0 expit

(

^θ0+θ₁

)

expit

(

^θ0

)

expit

(

^{θ0+θ+θ1+θ}5+θ²⁺₆+θ^θ3₇^+θ4

)

expit

(

^θ0+θ₁+θ₃+θ₅

)

expit

(

^θ0+θ₁+θ₃+θ₅

)

expit

(

^θ0+θ₁

)

2 0 0 1 0 1 1 expit

(

^θ0+θ₁

)

expit

(

^θ0

)

expit

(

^θ0+θ₁+θ₂+θ₄

)

expit

(

^θ0+θ₁

)

expit

(

^{θ0+θ+θ1+θ}5+θ²⁺₆+θ^θ3₇^+θ4

)

expit

(

^θ0+θ₁+θ₂+θ₄

)

3 1 0 0 0 1 1 expit

(

^θ0+θ₁+θ₂+θ₄

)

expit

(

^θ0+θ₂

)

expit

(

^θ0+θ₂

)

expit

(

^θ0

)

expit

(

^{θ0+θ+θ1+θ}5+θ²⁺₆+θ^θ3₇^+θ4

)

expit

(

^θ0+θ₁+θ₂+θ₄

)

4 1 1 0 0 0 1

expit

(

^{θ0+θ+θ1+θ}5+θ²⁺₆+θ^θ3₇^+θ4

)

expit θ θ +θ +θ

expit

(

^θ0+θ₂

)

expit

(

^θ0

)

expit

(

^θ0+θ₂+θ₃+θ₆

)

expit

(

^θ0+θ₂

)

5 1 1 0 1 0 0

expit

(

^{θ0+θ+θ1+θ}5+θ²⁺₆+θ^θ3₇^+θ4

)

expit

(

^θ0+θ₂+θ₃+θ₆

)

expit

(

^θ0+θ₂+θ₃+θ₆

)

expit

(

^θ0+θ₃

)

expit

(

^θ0+θ₃

)

expit

(

^θ0

)

6 0 1 1 1 0 0 expit

(

^θ0+θ₁+θ₃+θ₅

)

expit

(

^θ0+θ₃

)

expit

(

^{θ0+θ+θ1+θ}5+θ²⁺₆+θ^θ3₇^+θ4

)

expit

(

^θ0+θ₁+θ₃+θ₅

)

expit

(

^θ0+θ₃

)

expit

(

^θ0

)

Web Appendix Table 5

Results of causal mediation analysis of the association between placental abruption and perinatal mortality with small for gestational age birth ( M₁¿ and preterm delivery ( M₂¿ as multiple mediators under six different decomposition scenarios:

United States singleton births, 1998-2002 Decomposition

scenario Effect decomposition Abruption x⁽⁰⁾

SGA birth x⁽¹⁾

Preterm delivery x⁽²⁾

Risk ratio

95% confidence interval

1 Total effect 11.88 11.63, 12.14

Natural direct effect x⁽¹⁾=0 x⁽²⁾=0 5.44 5.24, 5.65

NIE through SGA birth x⁽⁰⁾=¿ 1 x⁽²⁾=¿ 1 1.01 1.01, 1.02

NIE through preterm delivery x⁽⁰⁾=¿ 1 x⁽¹⁾=0 2.15 2.09, 2.22

Mediated through M₁ (%) 2 1, 2

Mediated through ^M2 (%) 58 56, 59

Proportion unmediated (%) 41 39, 42

2 Total effect 11.88 11.63, 12.14

Natural direct effect x⁽¹⁾=0 x⁽²⁾=0 5.44 5.24, 5.65

NIE through SGA birth x⁽⁰⁾=¿ 1 x⁽²⁾=0 1.03 1.02, 1.03

NIE through preterm delivery x⁽⁰⁾=¿ 1 x⁽¹⁾=1 2.13 2.06, 2.19

Mediated through ^M1 (%) 1 1, 2

Mediated through M₂ (%) 58 56, 59

Proportion unmediated (%) 41 39, 42

3 Total effect 11.88 11.63, 12.14

Natural direct effect x⁽¹⁾=1 x⁽²⁾=0 5.17 4.98, 5.35

NIE through SGA birth x⁽⁰⁾=0 x⁽²⁾=0 1.08 1.08, 1.09

NIE through preterm delivery x⁽⁰⁾=¿ 1 x⁽¹⁾=1 2.13 2.06, 2.19

Mediated through M₁ (%) 1 1, 1

Mediated through ^M (%) 58 56, 59

4 Total effect 11.88 11.63, 12.14

Natural direct effect x⁽¹⁾=1 x⁽²⁾=¿ 1 3.02 2.96, 3.09

NIE through SGA birth x⁽⁰⁾=0 x⁽²⁾=0 1.08 1.08, 1.09

NIE through preterm delivery x⁽⁰⁾=0 x⁽¹⁾=1 3.64 3.61, 3.66

Mediated through M₁ (%) 1 1, 1

Mediated through ^M2 (%) 26 26, 27

Proportion unmediated (%) 73 72, 74

5 Total effect 11.88 11.63, 12.14

Natural direct effect x⁽¹⁾=1 x⁽²⁾=¿ 1 3.02 2.96, 3.09

NIE through SGA birth x⁽⁰⁾=0 x⁽²⁾=¿ 1 1.05 1.05, 1.06

NIE through preterm delivery x⁽⁰⁾=0 x⁽¹⁾=0 3.73 3.70, 3.76

Mediated through ^M1 (%) 2 2, 2

Mediated through M₂ (%) 25 25, 26

Proportion unmediated (%) 73 72, 74

6 Total effect 11.88 11.63, 12.14

Natural direct effect x⁽¹⁾=0 x⁽²⁾=¿ 1 3.14 3.07, 3.21

NIE through SGA birth x⁽⁰⁾=¿ 1 x⁽²⁾=¿ 1 1.01 1.01, 1.02

NIE through preterm delivery x⁽⁰⁾=¿ 0 x⁽¹⁾=0 3.73 3.70, 3.76

Mediated through M₁ (%) 2 1, 2

Mediated through ^M2 (%) 25 25, 26

Proportion unmediated (%) 73 73, 74

Abbreviations: CI, confidence interval; NIE, natural indirect effect; SGA, small for gestational age 95% confidence intervals were estimated based on the bootstrap resampling method (k=5000)

Web Appendix Table 6

Results of causal mediation analysis of the association between placental abruption and perinatal mortality with small for gestational age birth ( M₁¿ and preterm delivery ( M₂¿ as multiple mediators (using a working model for the distribution of

M₂ instead of a working model for the distribution of ^M1 ) under six different decomposition scenarios:

United States singleton births, 1998-2002 Decomposition

scenario Effect decomposition Abruption x⁽⁰⁾

SGA birth x⁽¹⁾

Preterm delivery x⁽²⁾

Risk ratio

95% confidence interval

1 Total effect 11.88 11.63, 12.14

Natural direct effect x⁽¹⁾=0 x⁽²⁾=0 5.44 5.24, 5.65

NIE through SGA birth x⁽⁰⁾=¿ 1 x⁽²⁾=¿ 1 1.02 1.02, 1.03

NIE through preterm delivery x⁽⁰⁾=¿ 1 x⁽¹⁾=0 2.13 2.07, 2.20

Mediated through ^M1 (%) 3 2, 3

Mediated through M₂ (%) 57 55, 58

Proportion unmediated (%) 41 39, 42

2 Total effect 11.88 11.63, 12.14

Natural direct effect x⁽¹⁾=0 x⁽²⁾=0 5.44 5.24, 5.65

NIE through SGA birth x⁽⁰⁾=¿ 1 x⁽²⁾=0 1.05 1.05, 1.06

NIE through preterm delivery x⁽⁰⁾=¿ 1 x⁽¹⁾=1 2.07 2.01, 2.13

Mediated through M₁ (%) 3 2, 3

Mediated through ^M2 (%) 56 55, 58

Proportion unmediated (%) 41 39, 42

3 Total effect 11.88 11.63, 12.14

Natural direct effect x⁽¹⁾=1 x⁽²⁾=0 4.65 4.48, 4.82

NIE through SGA birth x⁽⁰⁾=0 x⁽²⁾=0 1.23 1.22, 1.25

NIE through preterm delivery x⁽⁰⁾=¿ 1 x⁽¹⁾=1 2.07 2.01, 2.13

Mediated through ^M (%) 2 2, 2

Proportion unmediated (%) 41 40, 43

4 Total effect 11.88 11.63, 12.14

Natural direct effect x⁽¹⁾=1 x⁽²⁾=¿ 1 3.02 2.96, 3.09

NIE through SGA birth x⁽⁰⁾=0 x⁽²⁾=0 1.23 1.22, 1.25

NIE through preterm delivery x⁽⁰⁾=0 x⁽¹⁾=1 3.19 3.16, 3.22

Mediated through ^M1 (%) 2 2, 2

Mediated through M₂ (%) 25 24, 25

Proportion unmediated (%) 73 72, 74

5 Total effect 11.88 11.63, 12.14

Natural direct effect x⁽¹⁾=1 x⁽²⁾=¿ 1 3.02 2.96, 3.09

NIE through SGA birth x⁽⁰⁾=0 x⁽²⁾=¿ 1 1.09 1.09, 1.1

NIE through preterm delivery x⁽⁰⁾=0 x⁽¹⁾=0 3.60 3.58, 3.63

Mediated through M₁ (%) 3 3, 3

Mediated through ^M2 (%) 24 23, 25

Proportion unmediated (%) 73 72, 74

6 Total effect 11.88 11.63, 12.14

Natural direct effect x⁽¹⁾=0 x⁽²⁾=¿ 1 3.22 3.15, 3.29

NIE through SGA birth x⁽⁰⁾=¿ 1 x⁽²⁾=¿ 1 1.02 1.02, 1.03

NIE through preterm delivery x⁽⁰⁾=¿