34.1 Traditional
(Baron and Kenny 1986) is outdated because of step 1, but we could still see the original idea.
3 regressions
Step 1: \(X \to Y\)
Step 2: \(X \to M\)
Step 3: \(X + M \to Y\)
where
\(X\) = independent variable
\(Y\) = dependent variable
\(M\) = mediating variable
- Originally, the first path from \(X \to Y\) suggested by (Baron and Kenny 1986) needs to be significant. But there are cases that you could have indirect of \(X\) on \(Y\) without significant direct effect of \(X\) on \(Y\) (e.g., when the effect is absorbed into M, or there are two counteracting effects \(M_1, M_2\) that cancel out each other effect).
Mathematically,
\[ Y = b_0 + b_1 X + \epsilon \]
\(b_1\) does not need to be significant.
- We examine the effect of \(X\) on \(M\). This step requires that there is a significant effect of \(X\) on \(M\) to continue with the analysis
Mathematically,
\[ M = b_0 + b_2 X + \epsilon \]
where \(b_2\) needs to be significant.
- In this step, we want to the effect of \(M\) on \(Y\) “absorbs” most of the direct effect of \(X\) on \(Y\) (or at least makes the effect smaller).
Mathematically,
\[ Y = b_0 + b_4 X + b_3 M + \epsilon \]
\(b_4\) needs to be either smaller or insignificant.
| The effect of \(X\) on \(Y\) | then, \(M\) … mediates between \(X\) and \(Y\) |
|---|---|
| completely disappear (\(b_4\) insignificant) | Fully (i.e., full mediation) |
| partially disappear (\(b_4\) smaller than in step 1) | Partially (i.e., partial mediation) |
- Examine the mediation effect (i.e., whether it is significant)
Fist approach: Sobel’s test (Sobel 1982)
Second approach: bootstrapping (Preacher and Hayes 2004) (preferable)
More details can be found here
34.1.1 Example 1
from Virginia’s library
myData <-
read.csv('http://static.lib.virginia.edu/statlab/materials/data/mediationData.csv')
# Step 1 (no longer necessary)
model.0 <- lm(Y ~ X, myData)
# Step 2
model.M <- lm(M ~ X, myData)
# Step 3
model.Y <- lm(Y ~ X + M, myData)
# Step 4 (boostrapping)
library(mediation)
results <- mediate(
model.M,
model.Y,
treat = 'X',
mediator = 'M',
boot = TRUE,
sims = 500
)
summary(results)
#>
#> Causal Mediation Analysis
#>
#> Nonparametric Bootstrap Confidence Intervals with the Percentile Method
#>
#> Estimate 95% CI Lower 95% CI Upper p-value
#> ACME 0.3565 0.2153 0.53 <2e-16 ***
#> ADE 0.0396 -0.1993 0.32 0.71
#> Total Effect 0.3961 0.1838 0.68 <2e-16 ***
#> Prop. Mediated 0.9000 0.4781 1.86 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Sample Size Used: 100
#>
#>
#> Simulations: 500Total Effect = 0.3961 = \(b_1\) (step 1) = total effect of \(X\) on \(Y\) without \(M\)
Direct Effect = ADE = 0.0396 = \(b_4\) (step 3) = direct effect of \(X\) on \(Y\) accounting for the indirect effect of \(M\)
ACME = Average Causal Mediation Effects = \(b_1 - b_4\) = 0.3961 - 0.0396 = 0.3565 = \(b_2 \times b_3\) = 0.56102 * 0.6355 = 0.3565
Using mediation package suggested by (Imai, Keele, and Tingley 2010) (Imai, Keele, and Yamamoto 2010). More on details of the package can be found here
2 types of Inference in this package:
Model-based inference:
Assumptions:
Treatment is randomized (could use matching methods to achieve this).
Sequential Ignorability: conditional on covariates, there is other confounders that affect the relationship between (1) treatment-mediator, (2) treatment-outcome, (3) mediator-outcome. Typically hard to argue in observational data. This assumption is for the identification of ACME (i.e., average causal mediation effects).
Design-based inference
Notations: we stay consistent with package instruction
\(M_i(t)\) = mediator
\(T_i\) = treatment status \((0,1)\)
\(Y_i(t,m)\) = outcome where \(t\) = treatment, and \(m\) = mediating variables.
\(X_i\) = vector of observed pre-treatment confounders
Treatment effect (per unit \(i\)) = \(\tau_i = Y_i(1,M_i(1)) - Y_i (0,M_i(0))\) which has 2 effects
Causal mediation effects: \(\delta_i (t) \equiv Y_i (t,M_i(1)) - Y_i(t,M_i(0))\)
Direct effects: \(\zeta (t) \equiv Y_i (1, M_i(1)) - Y_i(0, M_i(0))\)
summing up to the treatment effect: \(\tau_i = \delta_i (t) + \zeta_i (1-t)\)
More on sequential ignorability
\[ \{ Y_i (t', m) , M_i (t) \} \perp T_i |X_i = x \]
\[ Y_i(t',m) \perp M_i(t) | T_i = t, X_i = x \]
where
\(0 < P(T_i = t | X_i = x)\)
\(0 < P(M_i = m | T_i = t , X_i =x)\)
First condition is the standard strong ignorability condition where treatment assignment is random conditional on pre-treatment confounders.
Second condition is stronger where the mediators is also random given the observed treatment and pre-treatment confounders. This condition is satisfied only when there is no unobserved pre-treatment confounders, and post-treatment confounders, and multiple mediators that are correlated.
My understanding is that until the moment I write this note, there is no way to test the sequential ignorability assumption. Hence, researchers can only do sensitivity analysis to argue for their result.