Mediation – How to Interpret Output from ‘Mediation’ Package: Can Full Mediation Be Claimed?

causalitymediationtreatmenttreatment-effect

I have an outcome binary variable Y, a continuous mediator M, a binary treatment T and some covariates C.

I have a linear regression model for the mediator:

$ M = \alpha + \beta T + \delta C $

and a logistic model for the outcome:

$ \text{logit}(\text{Pr}(Y=1)) = \alpha +\beta T + \gamma M + \delta C $

Using these two models I have run the mediation package obtaining the following output:

Causal Mediation Analysis 

Nonparametric Bootstrap Confidence Intervals with the Percentile Method

                          Estimate 95% CI Lower 95% CI Upper p-value    
ACME (control)           -0.000498    -0.000545         0.00  <2e-16 ***
ACME (treated)           -0.000472    -0.000541         0.00  <2e-16 ***
ADE (control)            -0.000254    -0.000828         0.00   0.442    
ADE (treated)            -0.000228    -0.000745         0.00   0.442    
Total Effect             -0.000726    -0.001270         0.00   0.016 *  
Prop. Mediated (control)  0.685899     0.391613         2.87   0.016 *  
Prop. Mediated (treated)  0.650018     0.322575         3.08   0.016 *  
ACME (average)           -0.000485    -0.000536         0.00  <2e-16 ***
ADE (average)            -0.000241    -0.000786         0.00   0.442    
Prop. Mediated (average)  0.667959     0.358519         2.97   0.016 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Sample Size Used: 283515 


Simulations: 1000

My questions are:

Can I claim full mediation with this output, given that the ACME is statistically significant and the ADE is not?
If so, how to explain that the proportion mediated of the effect is 66.79% and not 100%?

Best Answer

To claim full mediation is to claim that there is no direct effect. The tests for ADE test the null hypothesis that there is no direct effect, but failing to reject the null hypothesis is not the same as accepting the null hypothesis. Indeed, the confidence intervals for the ADE indicate that it is possible the direct effect is larger in magnitude than the indirect effect. The proportion mediated is a point estimate, but you can see that the confidence intervals include 100% mediated (as well as high values, i.e., that the direct and indirect effects are in opposite directions) and 50% mediated (i.e., that the direct and indirect effects are of equal size and direction).

You don't have to use a term like "full mediation" to describe the results in a meaningful and useful way. "We found evidence of mediation, but not of a direct effect." This explains the situation very clearly without using that specific term.

Related Solutions

Solved – Interpreting Mediation Output when Direct Effect is not Stat. Sig but ACME and Total are

This does suggest a "full" mediation, in which all of the IV's influence is mediated. The ACME being significant shows that the mediating process appears to be present. On the other hand, you don't have evidence that there is an ADE (insignificant result). The reason the total effect is larger than the ACME alone is that the estimate still includes the estimate for the ADE, however uncertain that estimate is for the ADE. The total effect has all the certainty of the ACME and the uncertainty of the ADE, but is still statistically significant because of the apparent strength of the ACME observation.

With that said, I would be careful in discussing these results since this analysis isn't designed to prove that there is no direct effect, but rather show that there is one. Your insignificant estimate for the ADE reflects a lack of evidence, not contrary evidence per se.

Solved – Interpreting Mediation Output when ACME is stat. sig but ADE and Total are not

This should not happen if you use only one mediator in the model. If there are other mediators for a negative hidden away, they should reflect in the direct effect.

But there are certain conditions under which it would happen. Most likely, the IV weakly relates or has no relation to the DV in these conditions.

# first, some syntax for Sobel's test
sobel <- function(mm, my) {
  a <- coef(mm)["x"]
  ase <- coef(summary(mm))["x", 2]

  b <- coef(my)["m"]
  bse <- coef(summary(my))["m", 2]

  abse <- sqrt(a ^ 2 * bse ^ 2 + b ^ 2 * ase ^ 2)
  # return ACME and z-test
  c(acme = a * b, z = a * b / abse, p = (1 - pnorm(a * b / abse)) * 2)
}

set.seed(899398)
n <- 100
x <- rnorm(n)
m <- rnorm(1) * x + rnorm(n)
y <- rnorm(1) * m + rnorm(1) * x + rnorm(n)

coef(summary(lm(y ~ x)))["x", ] # total effect not stat sig
  Estimate Std. Error    t value   Pr(>|t|) 
0.2125229  0.1470346  1.4453944  0.1515374 

coef(summary(mm <- lm(m ~ x)))["x", ] # IV affects MV
  Estimate Std. Error    t value   Pr(>|t|) 
0.25667463 0.10464021 2.45292550 0.01593713 

coef(summary(my <- lm(y ~ m + x))) # Only MV significant for DV
              Estimate Std. Error   t value     Pr(>|t|)
(Intercept) -0.06147867 0.09632753 -0.6382253 5.248309e-01
m            1.08186228 0.09104230 11.8830723 1.241992e-20
x           -0.06516366 0.09716138 -0.6706745 5.040217e-01

sobel(mm, my) # returns ACME, z and z-test of ACME
    acme.x        z.x        p.x 
0.27768660 2.40227887 0.01629328

The conditions OP described are all present here. This is one example of how it could happen. In reality, the relationships are probably way more complicated but it definitely is possible. If you have a not statistically significant total effect and the mediated effect is statistically significant in a one mediator model, I'd doubt any mediation was going on.

The way I found this example was by using a search for the right seed:

find <- FALSE
tab <- 1:1e6
while (find == FALSE) {
  seed <- sample(tab, 1)
  set.seed(seed)
  n <- 100
  x <- rnorm(n)
  m <- rnorm(1) * x + rnorm(n)
  y <- rnorm(1) * m + rnorm(1) * x + rnorm(n)
  te <- coef(summary(lm(y ~ x)))["x", 4]
  mm <- lm(m ~ x)
  dme <- coef(summary(my <- lm(y ~ m + x)))[c("m", "x"), 4]
  me <- dme[1]; de <- dme[2]
  ie <- sobel(mm, my)[3]
  if (te > .15 & de > .15 & ie < .04) {
    find <- TRUE
  }
}

It is possible to expand the search function to store all such conditions and study the patterns ACME, ADE and ATE that produce the OP's situation.

Best Answer

Related Solutions

Solved – Interpreting Mediation Output when Direct Effect is not Stat. Sig but ACME and Total are

Solved – Interpreting Mediation Output when ACME is stat. sig but ADE and Total are not

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