19.2 Causal Inference Approach to Mediation

Traditional mediation models assume that regression-based estimates provide valid causal inference. However, causal mediation analysis (CMA) extends beyond traditional models by explicitly defining mediation in terms of potential outcomes and counterfactuals.

19.2.1 Example: Traditional Mediation Analysis

We begin with a classic three-step mediation approach.

# Load data
myData <- read.csv("data/mediationData.csv")

# Step 1 (Total Effect: X → Y) [No longer required]
model.0 <- lm(Y ~ X, data = myData)
summary(model.0)
#> 
#> Call:
#> lm(formula = Y ~ X, data = myData)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -5.0262 -1.2340 -0.3282  1.5583  5.1622 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)   2.8572     0.6932   4.122 7.88e-05 ***
#> X             0.3961     0.1112   3.564 0.000567 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 1.929 on 98 degrees of freedom
#> Multiple R-squared:  0.1147, Adjusted R-squared:  0.1057 
#> F-statistic:  12.7 on 1 and 98 DF,  p-value: 0.0005671

# Step 2 (Effect of X on M)
model.M <- lm(M ~ X, data = myData)
summary(model.M)
#> 
#> Call:
#> lm(formula = M ~ X, data = myData)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -4.3046 -0.8656  0.1344  1.1344  4.6954 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)  1.49952    0.58920   2.545   0.0125 *  
#> X            0.56102    0.09448   5.938 4.39e-08 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 1.639 on 98 degrees of freedom
#> Multiple R-squared:  0.2646, Adjusted R-squared:  0.2571 
#> F-statistic: 35.26 on 1 and 98 DF,  p-value: 4.391e-08

# Step 3 (Effect of M on Y, controlling for X)
model.Y <- lm(Y ~ X + M, data = myData)
summary(model.Y)
#> 
#> Call:
#> lm(formula = Y ~ X + M, data = myData)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -3.7631 -1.2393  0.0308  1.0832  4.0055 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)   1.9043     0.6055   3.145   0.0022 ** 
#> X             0.0396     0.1096   0.361   0.7187    
#> M             0.6355     0.1005   6.321 7.92e-09 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 1.631 on 97 degrees of freedom
#> Multiple R-squared:  0.373,  Adjusted R-squared:  0.3601 
#> F-statistic: 28.85 on 2 and 97 DF,  p-value: 1.471e-10

# Step 4: Bootstrapping for ACME
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       0.2119         0.51  <2e-16 ***
#> ADE              0.0396      -0.1750         0.28   0.760    
#> Total Effect     0.3961       0.1743         0.64   0.004 ** 
#> Prop. Mediated   0.9000       0.5042         1.94   0.004 ** 
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Sample Size Used: 100 
#> 
#> 
#> Simulations: 500

Total Effect: $\hat{c} = 0.3961$ → effect of $X$ on $Y$ without controlling for $M$ .
Direct Effect (ADE): $\hat{c'} = 0.0396$ → effect of $X$ on $Y$ after accounting for $M$ .
ACME (Average Causal Mediation Effect):
- ACME = $\hat{c} - \hat{c'} = 0.3961 - 0.0396 = 0.3565$
- Equivalent to product of paths: $\hat{a} \times \hat{b} = 0.56102 \times 0.6355 = 0.3565$ .

These calculations do not rely on strong causal assumptions. For a causal interpretation, we need a more rigorous framework.

19.2.2 Two Approaches in Causal Mediation Analysis

The mediation package Imai, Keele, and Yamamoto (2010) enables causal mediation analysis. It supports two inference types:

Model-Based Inference
- Assumptions:
  - Treatment is randomized (or approximated via matching).
  - Sequential Ignorability: No unobserved confounding in:
    1. Treatment → Mediator
    2. Treatment → Outcome
    3. Mediator → Outcome
  - This assumption is hard to justify in observational studies.
Design-Based Inference
- Relies on experimental design to isolate the causal mechanism.

Notation

We follow the standard potential outcomes framework:

$M_i(t)$ = mediator under treatment condition $t$
$T_i \in {0,1}$ = treatment assignment
$Y_i(t, m)$ = outcome under treatment $t$ and mediator value $m$
$X_i$ = observed pre-treatment covariates

The treatment effect for an individual $i$ : $\tau_i = Y_i(1,M_i(1)) - Y_i (0,M_i(0))$ which decomposes into:

Causal Mediation Effect (ACME):

$\delta_i (t) = Y_i (t,M_i(1)) - Y_i(t,M_i(0))$

Direct Effect (ADE):

$\zeta_i (t) = Y_i (1, M_i(1)) - Y_i(0, M_i(0))$

Summing up:

$\tau_i = \delta_i (t) + \zeta_i (1-t)$

Sequential Ignorability Assumption

For CMA to be valid, we assume:

$\begin{aligned} \{ Y_i (t', m), M_i (t) \} &\perp T_i |X_i = x\\ Y_i(t',m) &\perp M_i(t) | T_i = t, X_i = x \end{aligned}$

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.

Key Challenge

⚠️ Sequential Ignorability is not testable. Researchers should conduct sensitivity analysis.

We now fit a causal mediation model using mediation.

library(mediation)
set.seed(2014)
data("framing", package = "mediation")

# Step 1: Fit mediator model (M ~ T, X)
med.fit <-
    lm(emo ~ treat + age + educ + gender + income, data = framing)

# Step 2: Fit outcome model (Y ~ M, T, X)
out.fit <-
    glm(
        cong_mesg ~ emo + treat + age + educ + gender + income,
        data = framing,
        family = binomial("probit")
    )

# Step 3: Causal Mediation Analysis (Quasi-Bayesian)
med.out <-
    mediate(
        med.fit,
        out.fit,
        treat = "treat",
        mediator = "emo",
        robustSE = TRUE,
        sims = 100
    )  # Use sims = 10000 in practice
summary(med.out)
#> 
#> Causal Mediation Analysis 
#> 
#> Quasi-Bayesian Confidence Intervals
#> 
#>                          Estimate 95% CI Lower 95% CI Upper p-value    
#> ACME (control)             0.0791       0.0351         0.15  <2e-16 ***
#> ACME (treated)             0.0804       0.0367         0.16  <2e-16 ***
#> ADE (control)              0.0206      -0.0976         0.12    0.70    
#> ADE (treated)              0.0218      -0.1053         0.12    0.70    
#> Total Effect               0.1009      -0.0497         0.23    0.14    
#> Prop. Mediated (control)   0.6946      -6.3109         3.68    0.14    
#> Prop. Mediated (treated)   0.7118      -5.7936         3.50    0.14    
#> ACME (average)             0.0798       0.0359         0.15  <2e-16 ***
#> ADE (average)              0.0212      -0.1014         0.12    0.70    
#> Prop. Mediated (average)   0.7032      -6.0523         3.59    0.14    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Sample Size Used: 265 
#> 
#> 
#> Simulations: 100

Alternative: Nonparametric Bootstrap

med.out <-
    mediate(
        med.fit,
        out.fit,
        boot = TRUE,
        treat = "treat",
        mediator = "emo",
        sims = 100,
        boot.ci.type = "bca"
    )
summary(med.out)
#> 
#> Causal Mediation Analysis 
#> 
#> Nonparametric Bootstrap Confidence Intervals with the BCa Method
#> 
#>                          Estimate 95% CI Lower 95% CI Upper p-value    
#> ACME (control)             0.0848       0.0424         0.14  <2e-16 ***
#> ACME (treated)             0.0858       0.0410         0.14  <2e-16 ***
#> ADE (control)              0.0117      -0.0726         0.13    0.58    
#> ADE (treated)              0.0127      -0.0784         0.14    0.58    
#> Total Effect               0.0975       0.0122         0.25    0.06 .  
#> Prop. Mediated (control)   0.8698       1.7460       151.20    0.06 .  
#> Prop. Mediated (treated)   0.8804       1.6879       138.91    0.06 .  
#> ACME (average)             0.0853       0.0434         0.14  <2e-16 ***
#> ADE (average)              0.0122      -0.0756         0.14    0.58    
#> Prop. Mediated (average)   0.8751       1.7170       145.05    0.06 .  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Sample Size Used: 265 
#> 
#> 
#> Simulations: 100

If we suspect moderation, we include an interaction term.

med.fit <-
    lm(emo ~ treat + age + educ + gender + income, data = framing)

out.fit <-
    glm(
        cong_mesg ~ emo * treat + age + educ + gender + income,
        data = framing,
        family = binomial("probit")
    )

med.out <-
    mediate(
        med.fit,
        out.fit,
        treat = "treat",
        mediator = "emo",
        robustSE = TRUE,
        sims = 100
    )

summary(med.out)
#> 
#> Causal Mediation Analysis 
#> 
#> Quasi-Bayesian Confidence Intervals
#> 
#>                           Estimate 95% CI Lower 95% CI Upper p-value    
#> ACME (control)             0.07417      0.02401         0.14  <2e-16 ***
#> ACME (treated)             0.09496      0.02702         0.16  <2e-16 ***
#> ADE (control)             -0.01353     -0.11855         0.11    0.76    
#> ADE (treated)              0.00726     -0.11007         0.11    0.90    
#> Total Effect               0.08143     -0.05646         0.19    0.26    
#> Prop. Mediated (control)   0.64510    -14.31243         3.13    0.26    
#> Prop. Mediated (treated)   0.98006    -17.83202         4.01    0.26    
#> ACME (average)             0.08457      0.02738         0.15  <2e-16 ***
#> ADE (average)             -0.00314     -0.11457         0.12    1.00    
#> Prop. Mediated (average)   0.81258    -16.07223         3.55    0.26    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Sample Size Used: 265 
#> 
#> 
#> Simulations: 100
test.TMint(med.out, conf.level = .95)  # Tests for interaction effect
#> 
#>  Test of ACME(1) - ACME(0) = 0
#> 
#> data:  estimates from med.out
#> ACME(1) - ACME(0) = 0.020796, p-value = 0.3
#> alternative hypothesis: true ACME(1) - ACME(0) is not equal to 0
#> 95 percent confidence interval:
#>  -0.01757310  0.07110837

Since sequential ignorability is untestable, we examine how unmeasured confounding affects ACME estimates.

# Load required package
library(mediation)

# Simulate some example data
set.seed(123)
n <- 100
data <- data.frame(
  treat = rbinom(n, 1, 0.5),       # Binary treatment
  med = rnorm(n),                   # Continuous mediator
  outcome = rnorm(n)                 # Continuous outcome
)

# Fit the mediator model (med ~ treat)
med_model <- lm(med ~ treat, data = data)

# Fit the outcome model (outcome ~ treat + med)
outcome_model <- lm(outcome ~ treat + med, data = data)

# Perform mediation analysis
med_out <- mediate(med_model, 
                   outcome_model, 
                   treat = "treat", 
                   mediator = "med", 
                   sims = 100)

# Conduct sensitivity analysis
sens_out <- medsens(med_out, sims = 100)

# Print and plot results
summary(sens_out)
#> 
#> Mediation Sensitivity Analysis for Average Causal Mediation Effect
#> 
#> Sensitivity Region
#> 
#>        Rho    ACME 95% CI Lower 95% CI Upper R^2_M*R^2_Y* R^2_M~R^2_Y~
#>  [1,] -0.9 -0.6194      -1.3431       0.1043         0.81       0.7807
#>  [2,] -0.8 -0.3898      -0.8479       0.0682         0.64       0.6168
#>  [3,] -0.7 -0.2790      -0.6096       0.0516         0.49       0.4723
#>  [4,] -0.6 -0.2067      -0.4552       0.0418         0.36       0.3470
#>  [5,] -0.5 -0.1525      -0.3406       0.0355         0.25       0.2409
#>  [6,] -0.4 -0.1083      -0.2487       0.0321         0.16       0.1542
#>  [7,] -0.3 -0.0700      -0.1723       0.0323         0.09       0.0867
#>  [8,] -0.2 -0.0354      -0.1097       0.0389         0.04       0.0386
#>  [9,] -0.1 -0.0028      -0.0648       0.0591         0.01       0.0096
#> [10,]  0.0  0.0287      -0.0416       0.0990         0.00       0.0000
#> [11,]  0.1  0.0603      -0.0333       0.1538         0.01       0.0096
#> [12,]  0.2  0.0928      -0.0317       0.2173         0.04       0.0386
#> [13,]  0.3  0.1275      -0.0333       0.2882         0.09       0.0867
#> [14,]  0.4  0.1657      -0.0369       0.3684         0.16       0.1542
#> [15,]  0.5  0.2100      -0.0422       0.4621         0.25       0.2409
#> [16,]  0.6  0.2642      -0.0495       0.5779         0.36       0.3470
#> [17,]  0.7  0.3364      -0.0601       0.7329         0.49       0.4723
#> [18,]  0.8  0.4473      -0.0771       0.9717         0.64       0.6168
#> [19,]  0.9  0.6768      -0.1135       1.4672         0.81       0.7807
#> 
#> Rho at which ACME = 0: -0.1
#> R^2_M*R^2_Y* at which ACME = 0: 0.01
#> R^2_M~R^2_Y~ at which ACME = 0: 0.0096
plot(sens_out)

If ACME confidence intervals contain 0, the effect is not robust to confounding.

Alternatively, using $R^2$ interpretation, we need to specify the direction of confounder that affects the mediator and outcome variables in plot using sign.prod = "positive" (i.e., same direction) or sign.prod = "negative" (i.e., opposite direction).

plot(sens.out, sens.par = "R2", r.type = "total", sign.prod = "positive")

Summary: Causal Mediation vs. Traditional Mediation

Aspect	Traditional Mediation	Causal Mediation
Model Assumption	Linear regressions	Potential outcomes framework
Assumptions Needed	No omitted confounders	Sequential ignorability
Inference Method	Product of coefficients	Counterfactual reasoning
Bootstrapping?	Common	Essential
Sensitivity Analysis?	Rarely used	Strongly recommended

References

Imai, Kosuke, Luke Keele, and Teppei Yamamoto. 2010. “Identification, Inference and Sensitivity Analysis for Causal Mediation Effects.”