TABLE 1.
Generalized Form of the Front-Door Formula |
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Notation: X = Exposure (binary; x and x*), M = Mediator (binary), Y = Outcome (binary), C = Measured confounders, X’ = using X for confounding adjustment at the Y-stage of the front-door formula, U = uncontrolled confounders between X and Y. |
E(YMx) = E(YX’Mx) = Σm, cP(M = m X = x, C = c)Σx’E (Y M = m, X’ = x’, C = c)P(X’ = x’, C = c) |
Path-specific frontdoor effect (PSFDE): the contrast between E(YMx) and E(YMx*), |
with E(YMx) = E(YX’Mx) = Σx’E(Yx’Mx)P(x’) = E(YxMx)P(x) + E(Yx*Mx)P(x*) |
where the equality hold if no U-M interaction. |
Step 1: Obtain Empirical Parameters |
Step 1a. Model for the mediator given the exposure and the confounders: |
P(M = m X = x, C = c) |
Step 1b. Model for the outcome given the exposure, the mediator, and the confounders: |
E(Y M = m, X’ = x’, C = c) |
Step 2: Simulate the Potential Mediator and the Potential Outcome |
Step 2a. Create two copies of the original sample |
Step 2b. Simulate the exposure variable that is marginally independent of the confounders |
Step 2c. Simulate the mediator variable as a function of its parents (the simulated exposure in step 2b and the confounders) using empirical parameters obtained in step 1a. |
Step 2d. Simulate the outcome variable as a function of its parents (the original exposure [not the simulated exposure in step 2b]), the simulated mediator in step 2c, and the confounders) using empirical parameters obtained in step 1b. |
Step 3: Fit the Final Marginal Structural Model |
Regress the simulated outcome in step 2d on the simulated exposure in step 2b to obtain point estimates of marginal effect using the pooled data with two copies of the original sample. Bootstrap can be used to obtain 95% confidence intervals. |
Details in the distinction between Pearl’s original formula, Fulcher et al.’s generalization, and our approach are described in eText 1; http://links.lww.com/EDE/B916.