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A Guide on Data Analysis

36 Mediation

36.1 Traditional Approach

Baron and Kenny (1986) is outdated because of step 1, but we could still see the original idea.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 (causal) variable

  • \(Y\) = dependent (outcome) variable

  • \(M\) = mediating variable

Note: Originally, the first path from \(X \to Y\) suggested by (Baron and Kenny 1986) needs to be significant. But there are cases in which 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).

Unmediated model
Unmediated model

where \(c\) is the total effect

where

  • \(c'\) = direct effect (effect of \(X\) on \(Y\) after accounting for the indirect path)

  • \(ab\) = indirect effect

Hence,

\[ \begin{aligned} \text{total effect} &= \text{direct effect} + \text{indirect effect} \\ c &= c' + ab \end{aligned} \]

However, this simple equation does not only hold in cases of

  1. Models with latent variables
  2. Logistic models (only approximately). Hence, you can only calculate \(c\) as the total effect of \(c' + ab\)
  3. Multi-level models (Bauer, Preacher, and Gil 2006)

To measure mediation (i.e., indirect effect),

  1. \(1 - \frac{c'}{c}\) highly unstable (D. P. MacKinnon, Warsi, and Dwyer 1995), especially in cases that \(c\) is small (not re* recommended)
  2. Product method: \(a\times b\)
  3. Difference method: \(c- c'\)

For linear models, we have the following assumptions:

  1. No unmeasured confound between \(X-Y\), \(X-M\) and \(M-Y\) relationships.

  2. \(X \not\rightarrow C\) where \(C\) is a confounder between \(M-Y\) relationship

  3. Reliability: No errors in measurement of \(M\) (also known as reliability assumption) (can consider errors-in-variables models)

Mathematically,

\[ Y = b_0 + b_1 X + \epsilon \]

\(b_1\) does not need to be significant.

  1. 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.

  1. 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 (\(b_4 < b_1\) in step 1) Partially (i.e., partial mediation)
  1. Examine the mediation effect (i.e., whether it is significant)

Notes:

  • Proximal mediation (\(a > b\)) can lead to multicollinearity and reduce statistical power, whereas distal mediation (\(b > a\)) is preferred for maximizing test power.

  • The ideal balance for maximizing power in mediation analysis involves slightly distal mediators (i.e., path \(b\) is somewhat larger than path \(a\)) (Hoyle 1999).

  • Tests for direct effects (c and c’) have lower power compared to the indirect effect (ab), making it possible for ab to be significant while c is not, even in cases where there seems to be complete mediation but no statistical evidence of a direct cause-effect relationship between X and Y without considering M (Kenny and Judd 2014).

  • The testing of \(ab\) offers a power advantage over \(c’\) because it effectively combines two tests. However, claims of complete mediation based solely on the non-significance of \(c’\) should be approached with caution, emphasizing the need for sufficient sample size and power, especially in assessing partial mediation. Or one should never make complete mediation claim (Hayes and Scharkow 2013)

36.1.1 Assumptions

36.1.1.1 Direction

  • Quick fix but not convincing: Measure \(X\) before \(M\) and \(Y\) to prevent \(M\) or \(Y\) causing \(X\); measure \(M\) before \(Y\) to avoid \(Y\) causing \(M\).

  • \(Y\) may cause \(M\) in a feedback model.

    • Assuming \(c' =0\) (full mediation) allows for estimating models with reciprocal causal effects between \(M\) and \(Y\) via IV estimation.

    • E. R. Smith (1982) proposes treating both \(M\) and \(Y\) as outcomes with potential to mediate each other, requiring distinct instrumental variables for each that do not affect the other.

36.1.1.2 Interaction

  • When M interact with X to affect Y, M is both a mediator and a mediator (Baron and Kenny 1986).

  • Interaction between \(XM\) should always be estimated.

  • For the interpretation of this interaction, see (T. VanderWeele 2015)

36.1.1.3 Reliability

  • When mediator contains measurement errors, \(b, c'\) are biased. Possible fix: mediator = latent variable (but loss of power) (Ledgerwood and Shrout 2011)

    • \(b\) is attenuated (closer to 0)

    • \(c'\) is

      • overestimated when \(ab >0\)

      • underestiamted when \(ab<0\)

  • When treatment contains measurement errors, \(a,b\) are biased

    • \(a\) is attenuated

    • \(b\) is

      • overestimated when \(ac'>0\)

      • underestimated when \(ac' <0\)

  • When outcome contains measurement errors,

    • If unstandardized, no bias

    • If standardized, attenuation bias

36.1.1.4 Confounding

36.1.1.4.1 Design Strategies

  • Randomization of treatment variable. If possible, also mediator

  • Control for the confounder (but still only for measureable observables)

36.1.1.4.2 Statistical Strategies

  • Instrumental variable on treatment

    • Specifically for confounder affecting the \(M-Y\) pair, front-door adjustment is possible when there is a variable that completely mediates the effect of the mediator on the outcome and is unaffected by the confounder.

  • Weighting methods (e.g., inverse propensity) See Heiss for R code

    • Need strong ignorability assumption (i.e.., all confounders are included and measured without error (Westfall and Yarkoni 2016)). Not fixable, but can be examined with robustness checks.

36.1.2 Indirect Effect Tests

36.1.2.1 Sobel Test

Standard Error

\[ \sqrt{\hat{b}^2 s_{\hat{a}} + \hat{a}^2 s_{b}^2} \]

The test of the indirect effect is

\[ z = \frac{\hat{ab}}{\sqrt{\hat{b}^2 s_{\hat{a}} + \hat{a}^2 s_{b}^2}} \]

Disadvantages

  • Assume \(a\) and \(b\) are independent.

  • Assume \(ab\) is normally distributed.

  • Does not work well for small sample sizes.

  • Power of the test is low and the test is conservative as compared to Bootstrapping.

36.1.2.2 Joint Significance Test

  • Effective for determining if the indirect effect is nonzero (by testing whether \(a\) and \(b\) are both statistically significant), assumes \(a \perp b\).

  • It’s recommended to use it with other tests and has similar performance to a Bootstrapping test (Hayes and Scharkow 2013).

  • The test’s accuracy can be affected by heteroscedasticity (Fossum and Montoya 2023) but not by non-normality.

  • Although helpful in computing power for the test of the indirect effect, it doesn’t provide a confidence interval for the effect.

36.1.2.3 Bootstrapping

  • First used by Bollen and Stine (1990)

  • It allows for the calculation of confidence intervals, p-values, etc.

  • It does not require \(a \perp b\) and corrects for bias in the bootstrapped distribution.

  • It can handle non-normality (in the sampling distribution of the indirect effect), complex models, and small samples.

  • Concerns exist about the bias-corrected bootstrapping being too liberal (Fritz, Taylor, and MacKinnon 2012). Hence, current recommendations favor percentile bootstrap without bias correction for better Type I error rates (Tibbe and Montoya 2022).

  • A special case of bootstrapping is a proposed by where you don’t need access to raw data to generate resampling, you only need \(a, b, var(a), var(b), cov(a,b)\) (which can be taken from lots of primary studies)

result <-
    causalverse::med_ind(
        a = 0.5,
        b = 0.7,
        var_a = 0.04,
        var_b = 0.05,
        cov_ab = 0.01
    )
result$plot
36.1.2.3.1 With Instrument 
library(DiagrammeR)
grViz("
digraph {
  graph []
  node [shape = plaintext]
    X [label = 'Treatment']
    Y [label = 'Outcome']
  edge [minlen = 2]
    X->Y
  { rank = same; X; Y }
}
")

grViz("
digraph {
  graph []
  node [shape = plaintext]
    X [label ='Treatment', shape = box]
    Y [label ='Outcome', shape = box]
    M [label ='Mediator', shape = box]
    IV [label ='Instrument', shape = box]
  edge [minlen = 2]
    IV->X
    X->M  
    M->Y 
    X->Y 
  { rank = same; X; Y; M }
}
")
library(mediation)
data("boundsdata")
library(fixest)

# Total Effect
out1 <- feols(out ~ ttt, data = boundsdata)

# Indirect Effect
out2 <- feols(med ~ ttt, data = boundsdata)

# Direct and Indirect Effect
out3 <- feols(out ~ med + ttt, data = boundsdata)

# Proportion Test
# To what extent is the effect of the treatment mediated by the mediator?
coef(out2)['ttt'] * coef(out3)['med'] / coef(out1)['ttt'] * 100
#>      ttt 
#> 68.63609


# Sobel Test
bda::mediation.test(boundsdata$med, boundsdata$ttt, boundsdata$out) |> 
    tibble::rownames_to_column() |> 
    causalverse::nice_tab(2)
#>   rowname Sobel Aroian Goodman
#> 1 z.value  4.05   4.03    4.07
#> 2 p.value  0.00   0.00    0.00
# Mediation Analysis using boot
library(boot)
set.seed(1)
mediation_fn <- function(data, i){
    # sample the dataset
    df <- data[i,]
    
    
    a_path <- feols(med ~ ttt, data = df)
    a <- coef(a_path)['ttt']
    
    b_path <-  feols(out ~ med + ttt, data = df)
    b <- coef(b_path)['med']
    
    cp <- coef(b_path)['ttt']
    
    # indirect effect
    ind_ef <- a*b
    total_ef <- a*b + cp
    return(c(ind_ef, total_ef))
    
}

boot_med <- boot(boundsdata, mediation_fn, R = 100, parallel = "multicore", ncpus = 2)
boot_med 
#> 
#> ORDINARY NONPARAMETRIC BOOTSTRAP
#> 
#> 
#> Call:
#> boot(data = boundsdata, statistic = mediation_fn, R = 100, parallel = "multicore", 
#>     ncpus = 2)
#> 
#> 
#> Bootstrap Statistics :
#>       original        bias    std. error
#> t1* 0.04112035  0.0006346725 0.009539903
#> t2* 0.05991068 -0.0004462572 0.029556611

summary(boot_med) |> 
    causalverse::nice_tab()
#>     R original bootBias bootSE bootMed
#> 1 100     0.04        0   0.01    0.04
#> 2 100     0.06        0   0.03    0.06

# confidence intervals (percentile is always recommended)
boot.ci(boot_med, type = c("norm", "perc"))
#> BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
#> Based on 100 bootstrap replicates
#> 
#> CALL : 
#> boot.ci(boot.out = boot_med, type = c("norm", "perc"))
#> 
#> Intervals : 
#> Level      Normal             Percentile     
#> 95%   ( 0.0218,  0.0592 )   ( 0.0249,  0.0623 )  
#> Calculations and Intervals on Original Scale
#> Some percentile intervals may be unstable

# point estimates (Indirect, and Total Effects)
colMeans(boot_med$t)
#> [1] 0.04175502 0.05946442

Alternatively, one can use the robmed package

Power test or use app

library(pwr2ppl)

# indirect path ab power
medjs(
    # X on M (path a)
    rx1m1 = .3,
    # correlation between X and Y (path c')
    rx1y  = .1,
    # correlation between M and Y (path b)
    rym1  = .3,
    # sample size
    n     = 100,
    alpha = 0.05,
    # number of mediators
    mvars = 1,
    # should use 10000
    rep   = 1000
)

36.1.3 Multiple Mediation

The most general package to handle multiple cases is manymome

See vignette for an example

36.1.3.1 Multiple Mediators

36.1.3.1.0.1 Lavaan 
# Load required packages
library(MASS)  # for mvrnorm
library(lavaan)

# Function to generate synthetic data with correctly correlated errors for mediators
generate_data <- function(n = 10000, a1 = 0.5, a2 = -0.35, b1 = 0.7, b2 = 0.48, corr = TRUE, correlation_value = 0.7) {
    set.seed(12345)
    X <- rnorm(n)
    
    # Generate correlated errors using a multivariate normal distribution
    if (corr) {
        Sigma <- matrix(c(1, correlation_value, correlation_value, 1), nrow = 2)  # Higher covariance matrix for errors
        errors <- mvrnorm(n, mu = c(0, 0), Sigma = Sigma)  # Generate correlated errors
    } else {
        errors <- mvrnorm(n, mu = c(0, 0), Sigma = diag(2))  # Independent errors
    }
    
    M1 <- a1 * X + errors[, 1]
    M2 <- a2 * X + errors[, 2]
    Y <- b1 * M1 + b2 * M2 + rnorm(n)  # Y depends on M1 and M2
    
    data.frame(X = X, M1 = M1, M2 = M2, Y = Y)
}

# Ground truth for comparison
ground_truth <- data.frame(Parameter = c("b1", "b2"), GroundTruth = c(0.7, 0.48))

# Function to extract relevant estimates, standard errors, and model fit
extract_estimates_b1_b2 <- function(fit) {
    estimates <- parameterEstimates(fit)
    estimates <- estimates[estimates$lhs == "Y" & estimates$rhs %in% c("M1", "M2"), c("rhs", "est", "se")]
    estimates$Parameter <- ifelse(estimates$rhs == "M1", "b1", "b2")
    estimates <- estimates[, c("Parameter", "est", "se")]
    fit_stats <- fitMeasures(fit, c("aic", "bic", "rmsea", "chisq"))
    return(list(estimates = estimates, fit_stats = fit_stats))
}

# Case 1: Correlated errors for mediators (modeled correctly)
Data_corr <- generate_data(n = 10000, corr = TRUE, correlation_value = 0.7)
model_corr <- '
  Y ~ b1 * M1 + b2 * M2 + c * X
  M1 ~ a1 * X
  M2 ~ a2 * X
  M1 ~~ M2  # Correlated mediators (errors)
'
fit_corr <- sem(model = model_corr, data = Data_corr)
results_corr <- extract_estimates_b1_b2(fit_corr)

# Case 2: Uncorrelated errors for mediators (modeled correctly)
Data_uncorr <- generate_data(n = 10000, corr = FALSE)
model_uncorr <- '
  Y ~ b1 * M1 + b2 * M2 + c * X
  M1 ~ a1 * X
  M2 ~ a2 * X
'
fit_uncorr <- sem(model = model_uncorr, data = Data_uncorr)
results_uncorr <- extract_estimates_b1_b2(fit_uncorr)

# Case 3: Correlated errors, but not modeled as correlated
fit_corr_incorrect <- sem(model = model_uncorr, data = Data_corr)
results_corr_incorrect <- extract_estimates_b1_b2(fit_corr_incorrect)

# Case 4: Uncorrelated errors, but modeled as correlated
fit_uncorr_incorrect <- sem(model = model_corr, data = Data_uncorr)
results_uncorr_incorrect <- extract_estimates_b1_b2(fit_uncorr_incorrect)

# Combine all estimates for comparison
estimates_combined <- list(
    "Correlated (Correct)" = results_corr$estimates,
    "Uncorrelated (Correct)" = results_uncorr$estimates,
    "Correlated (Incorrect)" = results_corr_incorrect$estimates,
    "Uncorrelated (Incorrect)" = results_uncorr_incorrect$estimates
)

# Combine all into a single table
comparison_table <- do.call(rbind, lapply(names(estimates_combined), function(case) {
    df <- estimates_combined[[case]]
    df$Case <- case
    df
}))

# Merge with ground truth for final comparison
comparison_table <- merge(comparison_table, ground_truth, by = "Parameter")

# Display the comparison table
comparison_table
#>   Parameter       est          se                     Case GroundTruth
#> 1        b1 0.7002984 0.013870433     Correlated (Correct)        0.70
#> 2        b1 0.6973612 0.009859426   Uncorrelated (Correct)        0.70
#> 3        b1 0.7002984 0.010010367   Correlated (Incorrect)        0.70
#> 4        b1 0.6973612 0.009859634 Uncorrelated (Incorrect)        0.70
#> 5        b2 0.4871118 0.013805615     Correlated (Correct)        0.48
#> 6        b2 0.4868318 0.010009908   Uncorrelated (Correct)        0.48
#> 7        b2 0.4871118 0.009963588   Correlated (Incorrect)        0.48
#> 8        b2 0.4868318 0.010010119 Uncorrelated (Incorrect)        0.48

# Display model fit statistics for each case
fit_stats_combined <- list(
    "Correlated (Correct)" = results_corr$fit_stats,
    "Uncorrelated (Correct)" = results_uncorr$fit_stats,
    "Correlated (Incorrect)" = results_corr_incorrect$fit_stats,
    "Uncorrelated (Incorrect)" = results_uncorr_incorrect$fit_stats
)

fit_stats_combined
#> $`Correlated (Correct)`
#>      aic      bic    rmsea    chisq 
#> 77932.45 77997.34     0.00     0.00 
#> 
#> $`Uncorrelated (Correct)`
#>       aic       bic     rmsea     chisq 
#> 84664.312 84721.995     0.000     0.421 
#> 
#> $`Correlated (Incorrect)`
#>       aic       bic     rmsea     chisq 
#> 84453.208 84510.891     0.808  6522.762 
#> 
#> $`Uncorrelated (Incorrect)`
#>      aic      bic    rmsea    chisq 
#> 84665.89 84730.78     0.00     0.00

36.1.3.2 Multiple Treatments

(Hayes and Preacher 2014)

Code in Process

36.2 Causal Inference Approach

36.2.1 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)
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
model.M <- lm(M ~ X, 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
model.Y <- lm(Y ~ X + M, 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 (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       0.2119         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 = 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 Yamamoto (2010). More details of the package can be found here

2 types of Inference in this package:

  1. 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).

  2. 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.

36.3 Model-based causal mediation analysis

Other resources:

Fit 2 models

  • mediator model: conditional distribution of the mediators \(M_i | T_i, X_i\)

  • Outcome model: conditional distribution of \(Y_i | T_i, M_i, X_i\)

mediation can accommodate almost all types of model for both mediator model and outcome model except Censored mediator model.

The update here is that estimation of ACME does not rely on product or difference of coefficients (see 36.2.1 ,

which requires very strict assumption: (1) linear regression models of mediator and outcome, (2) \(T_i\) and \(M_i\) effects are additive and no interaction

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

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")
    )

# Quasi-Bayesian Monte Carlo 
med.out <-
    mediate(
        med.fit,
        out.fit,
        treat = "treat",
        mediator = "emo",
        robustSE = TRUE,
        sims = 100 # should be 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

Nonparametric bootstrap version

med.out <-
    mediate(
        med.fit,
        out.fit,
        boot = TRUE,
        treat = "treat",
        mediator = "emo",
        sims = 100, # should be 10000 in practice
        boot.ci.type = "bca" # bias-corrected and accelerated intervals
    )
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 theoretically understanding suggests that there is treatment and mediator interaction

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) # test treatment-mediator 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
plot(med.out)

mediation can be used in conjunction with any of your imputation packages.

And it can also handle mediated moderation or non-binary treatment variables, or multi-level data

Sensitivity Analysis for sequential ignorability

  • test for unobserved pre-treatment covariates

  • \(\rho\) = correlation between the residuals of the mediator and outcome regressions.

  • If \(\rho\) is significant, we have evidence for violation of sequential ignorability (i.e., there is unobserved pre-treatment confounders).

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
    )
sens.out <-
    medsens(med.out,
            rho.by = 0.1, # \rho varies from -0.9 to 0.9 by 0.1
            effect.type = "indirect", # sensitivity on ACME
            # effect.type = "direct", # sensitivity on ADE
            # effect.type = "both", # sensitivity on ACME and ADE
            sims = 100)
summary(sens.out)
#> 
#> Mediation Sensitivity Analysis: Average Mediation Effect
#> 
#> Sensitivity Region: ACME for Control Group
#> 
#>      Rho ACME(control) 95% CI Lower 95% CI Upper R^2_M*R^2_Y* R^2_M~R^2_Y~
#> [1,] 0.3        0.0062      -0.0073       0.0188         0.09       0.0493
#> [2,] 0.4       -0.0084      -0.0238       0.0017         0.16       0.0877
#> 
#> Rho at which ACME for Control Group = 0: 0.3
#> R^2_M*R^2_Y* at which ACME for Control Group = 0: 0.09
#> R^2_M~R^2_Y~ at which ACME for Control Group = 0: 0.0493 
#> 
#> 
#> Sensitivity Region: ACME for Treatment Group
#> 
#>      Rho ACME(treated) 95% CI Lower 95% CI Upper R^2_M*R^2_Y* R^2_M~R^2_Y~
#> [1,] 0.3        0.0071      -0.0092       0.0213         0.09       0.0493
#> [2,] 0.4       -0.0101      -0.0295       0.0023         0.16       0.0877
#> 
#> Rho at which ACME for Treatment Group = 0: 0.3
#> R^2_M*R^2_Y* at which ACME for Treatment Group = 0: 0.09
#> R^2_M~R^2_Y~ at which ACME for Treatment Group = 0: 0.0493
plot(sens.out, sens.par = "rho", main = "Anxiety", ylim = c(-0.2, 0.2))

ACME confidence intervals contains 0 when \(\rho \in (0.3,0.4)\)

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")