Parametric Regression-Based Causal Mediation Analysis of Binary Outcomes and Binary Mediators: Moving Beyond the Rareness or Commonness of the Outcome

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Parametric Regression-Based Causal Mediation Analysis of Binary Outcomes and Binary Mediators: Moving Beyond the Rareness or Commonness of the Outcome

Mariia Samoilenko and Geneviève Lefebvre

Table of contents

Web Appendix 1 page 1

Web Table 1 page 39

Web Table 2 page 40

Web Table 3 page 42

Web Table 4 page 44

Web Table 5 page 46

Web Table 6 page 47

Web References page 48

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WEB APPENDIX 1

General formulas used in the delta method for the exact regression-based mediation effects The formulas for calculating the confidence intervals of the natural effects on the odd ratio, risk ratio, and risk difference scales depend upon the nested counterfactual outcome probability

P

(

^Y(^{a , M}⁽^a^¿⁾)⁼¹

|

^C=c

)

and its gradient with respect to the coefficients of the mediator and outcome models (β , θ) . The following equations will be used to derive the expression for the aforementioned gradient as well as those for the standard errors of estimates.

For any scalar function φ(α) differentiable in _Rⁿ_, where α=(^α1, α₂, … , α_n)^' , the partial derivatives of expit(^φ^(α⁾) and 1−expit(^φ⁽^α⁾) are:

∂

∂ α_i

(

1^exp+exp^⁡⁽^φ(α⁡(φ(⁾⁾α))

)

⁼₍₁_+exp^exp^⁡⁽^φ(_⁡_(φ^α(α⁾⁾)))²^∙

∂ φ(α)

∂ α_i ,

∂

∂ α_i

(

¹^+exp¹(φ(α))

)

⁼₍1+exp^−exp⁽(^φφ⁽(α^α⁾)⁾))²^∙

∂ φ(α)

∂ α_i ,

(A1)

where i=1, 2,… , n .

For any scalar functions f(α) and g(α) , differentiable in _Rⁿ and taking values in (0, 1) :

∂

∂ α_i

(

^ln

⁽

¹⁻^f⁽^f^α^(α⁾ ⁾

⁾ )

⁼^{∂ α}^∂ⁱ⁽^ln⁽^f^(α⁾⁾^−ln⁽^1−f⁽^α⁾⁾⁾

¿ 1 f(α)

∂ f(α)

∂ α_i + 1 1−f(α)

∂ f(α)

∂ α_i = 1

f(α)(1−f(α))

∂ f(α)

∂ α_i ,

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∂

∂ α_i

(

^ln

⁽ ⁽

¹⁻^f⁽^f^α^(α⁾ ⁾

⁾

^/

⁽

^1−g^g⁽^α^(α⁾ ⁾

⁾ ⁾ )

⁼^{∂ α}^∂ⁱ

⁽

^ln

⁽

^1−f^f⁽^α⁽⁾^α⁾

⁾

^−ln

⁽

¹⁻^g^(α^g⁽⁾^α⁾

⁾ ⁾

⁼^f^(α⁾⁽¹⁻¹ ^f⁽^α⁾⁾ ^{∂ f}^∂α⁽^αⁱ⁾⁻^(A2)^g⁽^α⁾⁽¹⁻¹ ^g^(α⁾⁾ ^{∂ g}^{∂ α}^(αⁱ ⁾^,

∂

∂ α_i

(

^ln

(

^f^g⁽(^αα⁾)

) )

⁼^{∂ α}^∂i

(lnf (α)−lng(α))= 1

f(α)

∂ f (α)

∂ α_i − 1 g(α)

∂ g(α)

∂ α_i , (A3)

where α=(α₁, α₂, … , α_n)'∈Rⁿ , i=1, 2,… , n .

Nested probability formulas based on the mediator and outcome logistic models Let C denote the set of adjustment variables. Under identifying and modeling

assumptions (1-2) from the main text, the nested probability P

(

^Y(^{a , M}⁽^a^¿⁾)⁼¹

|

^C=c

)

^for

exposure levels a and a^¿ is expressed as follows:

P

(

^Y(^{a , M}(a^¿))⁼¹

|

^C=c

)

¿P(Y=1|A=a , M=1,C=c)P(M=1|A=a^¿, C=c)

+P(Y=1|A=a , M=0,C=c)P(M=0|A=a^¿, C=c)

¿ exp(^θ0+θ₁a+θ₂+θ₃a+θ₄^' c)

1+exp(^θ0+θ₁a+θ₂+θ₃a+θ₄^' c₂)^∙

exp(^β0+β₁a^¿+β₂^' c)

1+exp(^β0+β₁a^¿+β₂^'c)

+exp(^θ0+θ₁a+θ₄^' c)

1+exp(^θ0+θ₁a+θ₄^' c)^∙

1

1+exp(^β0+β₁a^¿+β₂^' c)^.

Let g(a , a^¿, c) denote this last expression:

g(a , a^¿, c)= exp⁡(θ0+θ1a+θ2+θ3a+θ4'

c)

1+exp⁡(θ₀+θ₁a+θ₂+θ₃a+θ₄^' c)∙ exp⁡(β0+β1a^¿+β2'

c) 1+exp⁡(β₀+β₁a^¿+β₂^'c)

(A4)

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+exp⁡(θ₀+θ₁a+θ₄^'c)

1+exp⁡(θ₀+θ₁a+θ₄^' c)∙ 1

1+exp(^β0+β₁a^¿+β₂^'c)^.

Then, using general formulas (A1), the partial derivatives of g(a , a^¿, c) with respect to each of the coefficients are:

∂ g(a , a^¿, c)

∂ β₀

¿ exp⁡(θ₀+θ₁a+θ₂+θ₃a+θ₄^' c)

1+exp⁡(θ₀+θ₁a+θ₂+θ₃a+θ^'₄c)∙ exp⁡(β₀+β₁a^¿+β₂^' c)

(^1+exp^⁡⁽^β0+β₁a^¿+β₂^' c))²

−exp⁡(θ₀+θ₁a+θ₄^' c)

1+exp⁡(θ₀+θ₁a+θ₄^' c)∙ exp⁡(β₀+β₁a^¿+β₂^' c)

(¹⁺^exp^⁡⁽^β0+β₁a^¿+β₂^'c))²

¿ exp(^β0+β₁a^¿+β₂^' c)

(

^1+exp(^β0+β₁a^¿+β₂^'c)

)

²

×

(

^1+expêxp^⁡^(θ^⁡^(θ⁰^+θ0+¹θâ+θ₁a+θ²^+θ₂+θ³â+₃a+^θ⁴^'θ^c₄^'⁾c)− exp⁡(θ₀+θ₁a+θ₄^' c) 1+exp⁡(θ₀+θ₁a+θ^'₄c)

)

^,

∂ g(a , a^¿, c)

∂ β₁ =a^¿∙∂ g(a , a^¿, c)

∂ β₀ ,

∂ g(a , a^¿, c)

∂ β_2i =c_i∙∂ g(a , a^¿, c)

∂ β₀ ,i=1, … , k ,

∂ g(a , a^¿, c)

∂θ₀

¿ exp⁡(θ₀+θ₁a+θ₂+θ₃a+θ₄^' c)

(¹⁺^exp^⁡^(θ0+θ₁a+θ₂+θ₃a+θ₄^' c))²^∙

exp⁡(β₀+β₁a^¿+β₂^' c) 1+exp⁡(β₀+β₁a^¿+β₂^'c)

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+exp⁡(θ₀+θ₁a+θ^'₄c)

(^1+exp^⁡^(θ0+θ₁a+θ^'₄c))²^∙

1

1+exp(^β0+β₁a^¿+β₂^'c)^,

∂ g(a , a^¿, c)

∂θ₁ =a ∙∂ g(a , a^¿, c)

∂ θ₀ ,

∂ g(a , a^¿, c)

∂θ₂

¿ exp⁡(θ₀+θ₁a+θ₂+θ₃a+θ₄^' c)

(¹⁺^exp^⁡^(θ0+θ₁a+θ₂+θ₃a+θ₄^' c))²^∙

exp(^β0+β₁a^¿+β₂^'c)

1+exp(^β0+β₁a^¿+β₂^' c)^,

∂ g(a , a^¿, c)

∂θ₃ =a ∙∂ g(a , a^¿, c)

∂ θ₂ ,

∂ g(a , a^¿, c)

∂ θ_4i =c_i∙∂ g(a , a^¿, c)

∂ θ₀ ,i=1, … , k ,

where k is the cardinal number of C .

Thus, the gradient of the scalar function g(a , a^¿, c) with respect to the vector (β , θ)=(^β0, β₁, β₂₁,… , β₂_k,θ₀, θ₁, θ₂,θ₃, θ₄₁, …,θ₄_k)^'

is

∇(^g⁽^{a , a}^¿^{, c}⁾)⁼

(

^{∂ g}⁽^{a , a}∂ β^¿^{, c}⁾,∂ g(a , a^¿, c)

∂ θ

)

^' ^(A5)

where

∂ g(a , a^¿, c)

∂ β =

(

^{∂ g}⁽^{∂ β}^{a , a}0^¿^{, c}⁾

,∂ g(a ,a^¿, c)

β₁ ,∂ g(a , a^¿, c)

∂ β₂₁ , …,∂ g(a , a^¿, c)

∂ β₂_k

)

^,

∂ g(a , a^¿, c)

∂ θ =

(

^{∂ g}⁽^{a , a}^{∂ θ}0^¿^{, c}⁾

,∂ g(a ,a^¿, c)

∂θ₁ ,∂ g(a , a^¿, c)

∂θ₂ ,∂ g(a , a^¿, c)

∂ θ₃ ,∂ g(a , a^¿, c)

∂θ₄₁ , … ,∂ g(a , a^¿, c)

∂ θ₄_k

)

^.

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Delta method for mediation exact regression-based odds ratios We define the following notation:

^g(a , a^¿, c)=g(a ,a^¿, c)

|

(β , θ)=(^{^}β ,θ^),

∇(g^{^}(_{a , a}^¿_{, c}))=∇(g(_{a , a}^¿_{, c}))

|

(β , θ)=(^{^}β ,θ^),

where ∇(g(a , a^¿, c)) is defined in (A5).

We can express the regression-based exact ¿_{a , a}^NDE^¿_|_c and ¿_{a , a}^NIE^¿_|_c in terms of g(a , a^¿, c) defined in (A4) as follows:

¿_{a , a}^NDE^¿_|_c=

g(a , a^¿, c) 1−g(a , a^¿, c)

g(a^¿, a^¿, c) 1−g(a^¿, a^¿, c)

,¿_{a , a}^NIE^¿_c=

g(a , a , c) 1−g(a , a , c)

g(a , a^¿, c) 1−g(a , a^¿, c)

.

To construct the 95% confidence intervals (CIs) for the exact natural effect OR by first-order multivariate delta method (1), we have applied the same approach as in (2), that is we have expressed standard errors (se) for ln

(

^¿^{^}a ,a^¿|c

NDE

)

^and ^ln

(

^¿^{^}a ,a^¿|c

NIE

)

according to the following approximate formulas:

se

(

^ln

(

^¿^{^}a ,a^¿|c

NDE

) )

^≈

√

^∇

⁽

^ln

⁽

^¿^{^}^{a ,a}^NDE^¿^|^c

⁾ ⁾

^'^{∙ ∑∙}^∇

⁽

^ln

⁽

^¿^{^}^{a ,a}^NDE^¿^|^c

⁾ ⁾

^❑^,

se

(

^ln

(

^¿^{^}a ,a^¿|c

NIE

) )

^≈

√

^∇

⁽

^ln

⁽

^¿^{^}^{a ,a}^NIE^¿^|^c

⁾ ⁾

^'^{∙ ∑∙}^∇

⁽

^ln

⁽

^¿^{^}^{a ,a}^NIE^¿^|^c

⁾ ⁾

^❑^,

where ∑=diag

{

^∑^{^}β,∑θ^

}

is a block matrix, ∑_^_β and ^∑θ^ are the covariance matrices for the vectors ^β and θ^ , respectively. The gradients of ln

(

^¿^{^}a ,a^¿|c

NDE

)

^and ^ln

(

^¿^{^}a ,a^¿c NIE

)

^are

expressed using ∇(g^{^}(a , a^¿, c)) as follows:

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∇

(

^ln

⁽

^¿^{^}^{a , a}^NDE^¿^|^c

⁾ )

¿ ∇(g^(a ,a^¿, c))

g^(a , a^¿, c)(^{1−^}^g(a , a^¿, c))⁻

∇(g^(a^¿, a^¿, c))

^g(a^¿, a^¿,)(^{1−^}^g(a^¿, a^¿, c))^,

∇

(

^ln

(

^¿^{^}a , a^¿|c NIE

) )

¿ ∇(^{^}^g⁽^{a , a , c}⁾)

g^(a , a , c)(1− ^g(a , a , c))⁻

∇(^g(a , a^¿, c))

g^(a , a^¿, c)(^{1−^}^g(a , a^¿, c))^.

Thus, ln

(

^¿^{^}a ,a^¿|c

NDE

)

^{± Φ}⁻¹⁽^0.975)^{∙ se}

(

^ln

⁽

^¿^{^}^{a , a}^NDE^¿^|^c

⁾ )

^,

ln

(

^¿^{^}a ,a^¿|c

NIE

)

^{± Φ}⁻¹⁽^{0.975)∙ se}

(

^ln

⁽

^¿^{^}^{a , a}^NIE^¿^|^c

⁾ )

are the 95% CIs for ln

(

^¿a ,a^¿|c

NDE

)

^and ^ln

(

^¿a ,a^¿|c

NIE

)

, respectively. Therefore, 95% CIs for ¿_{a , a}^NDE^¿_|_c and ¿_{a , a}^NIE^¿_|_c are given by

¿^_{a , a}^NDE^¿_|_c∙exp

(

^±Φ⁻¹^(0.975^{)∙ se}

(

^ln

⁽

^¿^{^}^{a ,a}^NDE^¿^|^c

⁾ ) )

^,

¿^_{a , a}^NIE^¿_|_c∙exp

(

^±Φ⁻¹⁽^0.975⁾^{∙ se}

⁽

^ln

⁽

^¿^{^}^{a ,a}^NIE^¿^|^c

⁾ ⁾ )

^.

Finally, we have for the total effect odds ratio ¿_{a , a}^TE^¿_|_c that ln

(

^¿^{^}a ,a^¿|c

TE

)

^=ln

(

^¿^{^}a , a^¿|c

NDE

)

^+ln

(

^¿^{^}a ,a^¿|c NIE

)

^,

∇

(

^ln

(

^¿^{^}a , a^¿|c

TE

) )

^=∇

(

^ln

(

^¿^{^}a , a^¿|c

NDE

) )

⁺^∇

(

^ln

(

^¿^{^}a ,a^¿|c NIE

) )

and

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se

(

^ln

(

^¿^{^}a ,a^¿|c

TE

) )

^≈

√

^∇

⁽

^ln

⁽

^¿^{^}^TE^{a ,a}^¿^|^c

⁾ ⁾

^{' ∙∑∙}^∇

⁽

^ln

⁽

^¿^{^}^{a , a}^TE^¿^|^c

⁾ ⁾

^.

Thus, a 95% CI for ln

(

^¿a ,a^¿|c

TE

)

^is

ln

(

^¿^{^}a ,a^¿|c

TE

)

^{± Φ}⁻¹⁽^0.975⁾^{∙ se}

(

^ln

⁽

^¿^{^}^{a , a}^TE^¿^|^c

⁾ )

^,

and, consequently, approximate 95% CI for ¿_{a , a}^TE^¿_|_c is given by

¿^_{a , a}^TE^¿_|_c∙exp

(

^±Φ⁻¹^(0.975^{)∙ se}

⁽

^ln

⁽

^¿^{^}^{a ,a}^TE^¿^|^c

⁾ ⁾ )

^.

Delta method for exact mediation regression-based risk ratios We have for the RR scale that

RR_{a ,a}^NDE^¿_|_c= g(a , a^¿, c)

g(a^¿, a^¿, c)^{, RR}^{a ,a}^¿^|^c

NIE = g(a , a , c) g(a , a^¿,c)^,

and, correspondingly, ln

(

^RR^{^}a , a^¿|c

NDE

)

⁼^ln⁽^{^}^g⁽^{a , a}^¿^{, c}⁾⁾^−ln⁽^{^}^g⁽^a^¿^{, a}^¿^{, c}⁾⁾^,

ln

(

^RR^{^}a , a^¿|c

NIE

)

⁼^ln⁽^{^}^g⁽^{a , a , c}⁾⁾⁻^ln⁽^g^{^}⁽^{a ,a}^¿^{, c}⁾⁾^.

The equation (A3) implies that

∇

(

^ln

(

^RR^{^}a ,a^¿|c

NDE

) )

⁼^∇⁽^{^}^g⁽^{a , a}

¿, c))

^g(a , a^¿,c) ⁻

∇(g^(a^¿, a^¿, c))

^g(a^¿, a^¿, c) ^,

∇

(

^ln

(

^RR^{^}a ,a^¿|c

NIE

) )

⁼^∇_^_g⁽^g^{^}₍_{a , a , c}⁽^{a , a , c}₎⁾⁾⁻^∇⁽^{^}^g⁽^{a , a}

¿, c))

^g(a , a^¿, c) ^. Thus,

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se

(

^ln

(

^{^}^RRa , a^¿|c

NDE

) )

^≈

√

^∇

⁽

^ln

⁽

^RR^{^}^{a ,a}^NDE^¿^|^c

⁾ ⁾

^{' ∙∑∙}^∇

⁽

^ln

⁽

^{^}^RR^{a ,a}^NDE^¿^|^c

⁾ ⁾

^,

se

(

^ln

(

^{^}^RRa , a^¿|c

NIE

) )

^≈

√

^∇

⁽

^ln

⁽

^RR^{^}^{a ,a}^NIE^¿^|^c

⁾ ⁾

^{' ∙∑∙}^∇

⁽

^ln

⁽

^{^}^RR^{a ,a}^NIE^¿^|^c

⁾ ⁾

^,

and 95% CIs for RR_{a ,a}^NDE^¿_|_c and RR_{a ,a}^NIE^¿_|_c can be approximated by

^RR_{a ,a}^NDE^¿_|_c∙exp

(

^{± Φ}⁻¹⁽^0.975⁾^{∙ se}

⁽

^ln

⁽

^{^}^RR^{a ,a}^NDE^¿^|^c

⁾ ⁾ )

^,

^RR_{a ,a}^NIE^¿_|_c∙exp

(

^{± Φ}⁻¹⁽^0.975⁾^{∙ se}

⁽

^ln

⁽

^{^}^RR^{a ,a}^NIE^¿^|^c

⁾ ⁾ )

^.

Finally, we have for the total effect RR_{a ,a}^TE^¿_|_c=RR_{a ,a}^NDE^¿_|_c∙ RR_{a , a}^NIE^¿_|_c : ln

(

^RR^{^}^{a , a}^¿|c

TE

)

⁼^ln

(

^{^}^RR^{a , a}^¿|c

NDE

)

^+ln

(

^RR^{^}^{a ,a}^¿|c NIE

)

^.

∇

(

^ln

(

^RR^{^}a ,a^¿|c

TE

) )

⁼^∇

(

^ln

(

^{^}^RRa , a^¿|c

NDE

) )

⁺^∇

(

^ln

(

^RR^{^}a ,a^¿|c NIE

) )

^,

se

(

^ln

(

^{^}^RRa , a^¿|c

TE

) )

^≈

√

^∇

⁽

^ln

⁽

^RR^{^}^{a ,a}^TE^¿^|^c

⁾ ⁾

^{' ∙∑∙}^∇

⁽

^ln

⁽

^{^}^RR^{a ,a}^TE^¿^|^c

⁾ ⁾

^.

The 95% CI for RR_{a ,a}^TE^¿_|_c can be approximated by

^RR_{a ,a}^TE^¿_∨c∙exp

(

^{± Φ}⁻¹⁽^0.975)^{∙ se}

⁽

^ln

⁽

^RR^{^}^TE^{a ,a}^¿^|^c

⁾ ⁾ )

^.

Delta method for exact mediation regression-based risk differences We have for the RD scale:

RD_{a ,a}^NDE^¿_|_c=g(a , a^¿, c)−g(a^¿, a^¿, c), RD_{a ,a}^NIE^¿_|_c=g(a , a , c)−g(a , a^¿, c), RD^TE_{a ,a}^¿_|_c=RD_{a , a}^NDE^¿_|_c+RD_{a , a}^NIE^¿_|_c,

∇

(

^{^}^RDa ,a^¿|c

NDE

)

^=∇⁽^g^{^}⁽^{a , a}^¿^{, c}⁾⁾⁻^∇⁽^{^}^g⁽^a^¿^{, a}^¿^{, c}⁾⁾^,

Parametric Regression-Based Causal Mediation Analysis of Binary Outcomes and Binary Mediators: Moving Beyond the Rareness or Commonness of the Outcome

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