Appendix - http;//links.lww.com

Appendix

eAppendix 1:Properties of U given X and M if U and M are normally distributed

We assume the following model for the mediator:

M =α₀+α₁X+α₂C+α₃U +_M,

where the error term_M is normally distributed with constant variance> 0. We assume that U follows a standard normal distribution, and that U and _M are independent. We further assumed thatU is independent ofCandX. Therefore the distribution of(U|X, C) is equal to the distribution ofU. The distribution of(M|X, C)and(U|X, C)is bivariate normal (because U and _M are independent and both normally distributed). Standard theory regarding bivariate normal distributions gives that the conditional distribution ofU givenX,CandM is in this case also normal with mean:

E[U|M, X, C] = μ_U|X,C + σ_U|X,C

σ_M|X,Ccor[M, U|X, C](M−E[M|X, C])

= cor(M, U|X, C)(M −E[M|X, C])

σ_M|X,C ,

becauseμ_U|X,C =μ_U = 0andσ_U|X,C =σ_U = 1. The variance ofU|X, M, C is var[U|X, M, C] = (1−cor[M, U|X, C]²).

Since the mediator depends linearly on X,C andU , the correlation ofU andM given X, C is cor[M, U|X, C] =E[MU|X, C]/(σ_M|X,Cσ_U|X,C) =α₃/σ_M|X,C. This yields:

E[U|M, X, C] = α₃

σ_M|X,C² (M −(α₀+α₁X+α₂C)) var[U|M, X, C] = (1−α²₃/σ_M|X,C² )

Since(U|M, X, C)is normal, it also follows directly using standard theory regarding 1

normal distributions that:

E[exp(β₅U)|X, M, C]) = exp(β₅E[U|X, M, C] +1

2β₅²var[U|X, M, C])

= exp(β₅ α₃

σ_M² _|X,C(M −(α₀+α₁X+α₂C)) + 1

2β₅²(1−α²₃/σ²_M|X,C))

eAppendix 2: Standard errors and confidence intervals for the cor- rected direct and indirected effects estimates

Let S be the covariance matrix of the estimates of the fitted mediation modelα₀· · · , α₂, with elementss_ij, and T be the covariance matrix of the estimates of the fitted outcome modelβ₀· · · , β₄, with elementst_ij.

For the linear outcome model, the corrected CDE is estimated as CDE(m) = ˆβ₁^∗+ ˆβ₃^∗m+ ˆα₁α₃β₅/σ_M² _|X,C

The variance of this estimate is:

var(CDE(m)) =t₁₁+t₃₃m²+ 2t₁₃m+s₁₁[α₃β₅/σ_M² _|X,C]².

The corrected NDE is

NDE = ˆβ₁^∗+ ˆβ₃^∗(ˆα₀+ ˆα₁x₀+ ˆα₂c) + ˆα₁α₃β₅/σ_M² _|X,C

The variance of this estimate can easily be estimated using the delta method, yielding:

var(NDE) =

⎛

⎜⎜

⎜⎜

⎜⎝

βˆ₃^∗

βˆ₃^∗x₀+α₃β₅/σ_M|X,C² βˆ₃^∗c

ˆ 1

α₀+ ˆα₁x₀+ ˆα₂c

⎞

⎟⎟

⎟⎟

⎟⎠

⎛

⎜⎜

⎜⎜

⎝

s₀₀ s₁₀ s₂₀ 0 0 s₁₀ s₁₁ s₂₁ 0 0 s₂₀ s₂₁ s₂₂ 0 0 0 0 0 t₁₁ t₃₁ 0 0 0 t₃₁ t₃₃

⎞

⎟⎟

⎟⎟

⎠

⎛

⎜⎜

⎜⎜

⎜⎝

βˆ₃^∗

βˆ₃^∗x₀+α₃β₅/σ²_M|X,C βˆ₃^∗c

ˆ 1

α₀+ ˆα₁x₀+ ˆα₂c

⎞

⎟⎟

⎟⎟

⎟⎠

The corrected NIE is

NIE = ˆβ₂^∗αˆ₁+ ˆβ₃^∗αˆ₁(x₀+ 1)−αˆ₁α₃β₅/σ_M² _|X,C 2

and estimate of the variance is

var(NIE) =

⎛

⎝ βˆ₂^∗+ ˆβ₃^∗(x₀+ 1)−α₃β₅/σ_M|X,C² ˆ

α₁ ˆ

α₁(x₀+ 1)

⎞

⎠

⎛

⎝ s₁₁ 0 0 0 t₂₂ t₃₂ 0 t₃₂ t₃₃

⎞

⎠

⎛

⎝ βˆ₂^∗+ ˆβ₃^∗(x₀+ 1)−α₃β₅/σ_M² _|X,C ˆ

α₁ ˆ

α₁(x₀ + 1)

⎞

⎠

For the relative risk model and the odds ratio model, the variance of the logarithm of the direct and indirect effects should be calculated. This yields the same variance estimates for the logarithm of controlled direct effect and the natural indirect effect as described above.

The logarithm of the corrected direct effect is

log(NDE) = ˆβ₁^∗+ ˆβ₃^∗(ˆα₀+ˆα₁x₀+ˆα₂c+ ˆβ₂^∗σ_M² _|X,C)+0.5 ˆβ₃²σ_M² _|X,C((x₀+1)²−x²₀)+ˆα₁α₃β₅/σ_M|X,C²

The variance of this estimate can be estimated by:

var(log(NDE)) =

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎝

βˆ₃^∗

βˆ₃^∗x₀+α₃β₅/σ²_M|X,C βˆ₃^∗c

ˆ 1

β₃^∗σ_M² _|X,C ˆ

α₀+ ˆα₁x₀+ ˆα₂cβˆ₂^∗σ²_M|X,C + ˆβ₃σ_M² _|X,C((x₀+ 1)²−x²₀)

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎠

⎛

⎜⎜

⎜⎜

⎜⎜

⎝

s₀₀ s₁₀ s₂₀ 0 0 0 s₁₀ s₁₁ s₂₁ 0 0 0 s₂₀ s₂₁ s₂₂ 0 0 0 0 0 0 t₁₁ t₂₁ t₃₁ 0 0 0 t₂₁ t₂₂ t₃₂ 0 0 0 t₃₁ t₃₂ t₃₃

⎞

⎟⎟

⎟⎟

⎟⎟

⎠

×

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎝

βˆ₃^∗

βˆ₃^∗x₀ +α₃β₅/σ_M² _|X,C βˆ₃^∗c

ˆ 1

β₃^∗σ²_M|X,C ˆ

α₀+ ˆα₁x₀+ ˆα₂cβˆ₂^∗σ_M² _|X,C + ˆβ₃σ_M|X,C² ((x₀ + 1)²−x²₀)

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎠

3

eAppendix 3: Derivation of the reduced model if U is binary and the response model for Y is linear

Let U be binary with prevalencep_u = Pr(U = 1). Then the reduced linear regression model is:

E[Y|X, M, C] = β₀+β₁X+β₂M +β₃MX +β₄C+β₅E[U|X, M]

= β₀+β₁X+β₂M +β₃MX +β₄C+β₅Pr[U = 1|X, M]

Using Bayes’ rule yields:

Pr[U = 1|X, M, C] = p_uf(M|X, C,1)

p_uf(M|X, C,1) + (1−p_u)f(M|X, C,0)

= p_uf(M|X, C,1)/f(M|X, C,0) p_uf(M|X, C,1)/f(M|X, C,0) + (1−p_u)

withf(.)the density function ofM|X, C, U. Since(M|X, C, U)is normally distributed, with meanα₀+α₁X+α₂C+α₃U, and variance equal toσ_M|X,C,U² , the following holds:

f(M|U = 1, X, C)/f(M|U = 0, X, C) = exp((α₃(M−α₀−α₁X−α₂C)−0.5α₃²)/σ_M² _|X,C,U)

If we define

g(m, α) = exp((α₃(m−α)−0.5α²₃))/σ_M² _|X,C,U), the reduced model can be written as

E[Y|X, M, C] =β₀+β₁X+β₂M+β₃MX+β₄C+β₅ p_ug(m, α₀+α₁X+α₂C) g(m, α₀+α₁X+α₂C) + (1−p_u) which shows that the resulting model is no longer linear inX, C, M andMX.

4