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Mediation Analysis Design

Source: vignettes/mediation_analysis.Rmd
mediation_analysis.Rmd

By randomly assigning units to treatment we can determine whether a treatment affects an outcome but not why or how it might affect it. Identifying causal mechanisms is not a simple task. It involves complex potential outcomes and a mediating variable that is generally not assigned at random and is not a pre-treatment covariate (since it’s affected by the treatment). Researchers often use regression-based approaches to identify causal mechanisms but these rely on assumptions that sometimes can’t be met.

For this analysis we assume that there is a non-zero average treatment effect (ATE) of ZZ on YY. Our main interest lies in decomposing the ATE into direct and indirect effects. The indirect effect is channeled from the treatment ZZ to the outcome YY through a mediator MM and the direct effect runs directly from ZZ to YY.

Design Declaration

  • Model:

    We specify a population of size NN. Individuals from this population have two potential outcomes related to the mediator.

    • Mi(Zi=0):M_i(Z_i=0): The value for the mediator MM when the unit ii is in the control group, and

    • Mi(Zi=1):M_i(Z_i=1): the value for the mediator MM when the unit ii is treated.

    Additionally, individuals have four potential outcomes related to YY.

    Two can be observed under treatment or control conditions.

    • Yi(Zi=0,Mi(Zi=0)):Y_i(Z_i=0 , M_i(Z_i=0)): the outcome of unit ii when the treatment is absent and the mediator takes the value it would when the treatment is absent.

    • Yi(Zi=1,Mi(Zi=1)):Y_i(Z_i=1 , M_i(Z_i=1)): the outcome of unit ii when the treatment is present and the mediator takes the value it would when the treatment is present.

    And two complex potential outcomes.

    • Yi(Zi=1,Mi(Zi=0)):Y_i(Z_i=1 , M_i(Z_i=0)): the outcome of unit ii when the treatment is present but the mediator takes the value it would when the treatment is absent.

    • Yi(Zi=0,Mi(Zi=1)):Y_i(Z_i=0 , M_i(Z_i=1)): the outcome of unit ii when the treatment is absent but the mediator takes the value it would when the treatment is present.

Thus the data generating process we specify defines YY as a function of MM and ZZ and MM as a function of ZZ.

  • Inquiry:

    We are interested in the average effects of the treatment on the mediator, MM, the direct average effects of the treatment on YY and the effects on YY from ZZ that run through MM.

  • Data strategy:

    We use assign units to treatment using complete random assignment.

  • Answer strategy:

    First, we regress MM on ZZ. Then we regress YY on MM and ZZ.

N <- 200
a <- 1
b <- 0.4
c <- 0
d <- 0.5
rho <- 0

population <- declare_population(N = N, e1 = rnorm(N), e2 = rnorm(n = N, 
    mean = rho * e1, sd = sqrt(1 - rho^2)))
POs_M <- declare_potential_outcomes(M ~ 1 * (a * Z + e1 > 
    0))
POs_Y <- declare_potential_outcomes(Y ~ d * Z + b * M + c * 
    M * Z + e2, conditions = list(M = 0:1, Z = 0:1))
POs_Y_nat_0 <- declare_potential_outcomes(Y_nat0_Z_0 = b * 
    M_Z_0 + e2, Y_nat0_Z_1 = d + b * M_Z_0 + c * M_Z_0 + 
    e2)
POs_Y_nat_1 <- declare_potential_outcomes(Y_nat1_Z_0 = b * 
    M_Z_1 + e2, Y_nat1_Z_1 = d + b * M_Z_1 + c * M_Z_1 + 
    e2)
estimands <- declare_inquiries(FirstStage = mean(M_Z_1 - 
    M_Z_0), Indirect_0 = mean(Y_M_1_Z_0 - Y_M_0_Z_0), Indirect_1 = mean(Y_M_1_Z_1 - 
    Y_M_0_Z_1), Controlled_Direct_0 = mean(Y_M_0_Z_1 - Y_M_0_Z_0), 
    Controlled_Direct_1 = mean(Y_M_1_Z_1 - Y_M_1_Z_0), Natural_Direct_0 = mean(Y_nat0_Z_1 - 
        Y_nat0_Z_0), Natural_Direct_1 = mean(Y_nat1_Z_1 - 
        Y_nat1_Z_0))
assignment <- declare_assignment(Z = complete_ra(N, prob = 0.5))
reveal_M <- declare_reveal(M, Z)
reveal_Y <- declare_reveal(Y, assignment_variable = c("M", 
    "Z"))
reveal_nat0 <- declare_reveal(Y_nat0)
reveal_nat1 <- declare_reveal(Y_nat1)
manipulation <- declare_step(Not_M = 1 - M, handler = fabricate)
mediator_regression <- declare_estimator(M ~ Z, model = lm_robust, 
    inquiry = "FirstStage", label = "Stage 1")
stage2_1 <- declare_estimator(Y ~ Z * M, model = lm_robust, 
    term = c("M"), inquiry = c("Indirect_0"), label = "Stage 2")
stage2_2 <- declare_estimator(Y ~ Z * M, model = lm_robust, 
    term = c("Z"), inquiry = c("Controlled_Direct_0", "Natural_Direct_0"), 
    label = "Direct_0")
stage2_3 <- declare_estimator(Y ~ Z * Not_M, model = lm_robust, 
    term = c("Z"), inquiry = c("Controlled_Direct_1", "Natural_Direct_1"), 
    label = "Direct_1")
mediation_analysis_design <- population + POs_M + POs_Y + 
    POs_Y_nat_0 + POs_Y_nat_1 + estimands + assignment + 
    reveal_M + reveal_Y + reveal_nat0 + reveal_nat1 + manipulation + 
    mediator_regression + stage2_1 + stage2_2 + stage2_3

Takeaways

We diagnose two versions of this design: one in which the correlation between the error term of the mediator regression and one of the outcome regression (ρ\rho) is greater than zero, and another in which ρ\rho equals zero.

designs <- expand_design(mediation_analysis_designer, rho = c(0,.5))
diagnosis <- diagnose_design(designs, sims = 25)
## Warning: We recommend you choose a number of simulations higher than 30.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning: We recommend you choose a number of simulations higher than 30.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(M ~ Z), model = ~lm_robust): The argument 'model = ' is
## deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("M")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
## Warning in fn(data, ~(Y ~ Z * Not_M), model = ~lm_robust, term = ~c("Z")): The
## argument 'model = ' is deprecated. Please use '.method = ' instead.
rho Inquiry Estimator Outcome Term N Sims Mean Estimand Mean Estimate Bias SD Estimate RMSE Power Coverage
0 Controlled_Direct_0 Direct_0 Y Z 25 0.50 0.49 -0.01 0.25 0.25 0.40 0.96
(0.00) (0.04) (0.04) (0.04) (0.04) (0.07) (0.04)
0 Controlled_Direct_1 Direct_1 Y Z 25 0.50 0.45 -0.05 0.15 0.16 0.80 0.96
(0.00) (0.03) (0.03) (0.03) (0.03) (0.08) (0.04)
0 FirstStage Stage 1 M Z 25 0.35 0.35 0.00 0.04 0.05 1.00 1.00
(0.00) (0.01) (0.01) (0.01) (0.01) (0.00) (0.00)
0 Indirect_0 Stage 2 Y M 25 0.40 0.43 0.03 0.18 0.18 0.60 1.00
(0.00) (0.04) (0.04) (0.02) (0.02) (0.10) (0.00)
0 Indirect_1 NA NA NA 25 0.40 NA NA NA NA NA NA
(0.00) NA NA NA NA NA NA
0 Natural_Direct_0 Direct_0 Y Z 25 0.50 0.49 -0.01 0.25 0.25 0.40 0.96
(0.00) (0.04) (0.04) (0.04) (0.04) (0.07) (0.04)
0 Natural_Direct_1 Direct_1 Y Z 25 0.50 0.45 -0.05 0.15 0.16 0.80 0.96
(0.00) (0.03) (0.03) (0.03) (0.03) (0.08) (0.04)
0.5 Controlled_Direct_0 Direct_0 Y Z 25 0.50 0.07 -0.43 0.31 0.53 0.08 0.68
(0.00) (0.06) (0.06) (0.05) (0.07) (0.05) (0.09)
0.5 Controlled_Direct_1 Direct_1 Y Z 25 0.50 0.28 -0.22 0.15 0.26 0.36 0.72
(0.00) (0.03) (0.03) (0.02) (0.02) (0.09) (0.10)
0.5 FirstStage Stage 1 M Z 25 0.33 0.33 0.00 0.04 0.04 1.00 0.96
(0.01) (0.01) (0.01) (0.01) (0.01) (0.00) (0.04)
0.5 Indirect_0 Stage 2 Y M 25 0.40 1.15 0.75 0.16 0.77 1.00 0.00
(0.00) (0.03) (0.03) (0.02) (0.03) (0.00) (0.00)
0.5 Indirect_1 NA NA NA 25 0.40 NA NA NA NA NA NA
(0.00) NA NA NA NA NA NA
0.5 Natural_Direct_0 Direct_0 Y Z 25 0.50 0.07 -0.43 0.31 0.53 0.08 0.68
(0.00) (0.06) (0.06) (0.05) (0.07) (0.05) (0.09)
0.5 Natural_Direct_1 Direct_1 Y Z 25 0.50 0.28 -0.22 0.15 0.26 0.36 0.72
(0.00) (0.03) (0.03) (0.02) (0.02) (0.09) (0.10)

Our diagnosis indicates that when the error terms are not correlated, the direct and indirect effects can be estimated without bias. By contrast, when ρ\rho does not equal zero, the regression underestimates the effect of the mediator on YY and overstates the direct effects of ZZ on YY.

Unfortunately, the assumption of no correlation is not always guaranteed, since MM is not assigned at random and might be correlated with YY.

Further Reading

Gerber, Alan S., and Donald P. Green. 2012. Field Experiments: Design, Analysis, and Interpretation. New York: W.W. Norton.
Imai, Kosuke, Luke Keele, Dustin Tingley, and Teppei Yamamoto. 2011. “Unpacking the Black Box of Causality: Learning about Causal Mechanisms from Experimental and Observational Studies.” American Political Science Review 105 (4): 765–89.