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

The mediation package in R returns results in which:

• The Average Causal Mediated Effect (ACME) (the effect of the mediator alone) is positive and statistically significant
• Average Direct Effect (ADE) (the unmediated effect) and the Total Effect (ADE+ACME) is not statistically significant

Does this indicate that the IV positively influences my DV via my mediator, however there are other mediators through which the IV has a negative effect on the DV (and as a result, the total effect can't be differentiated from 0?).

Thank you for any help.

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; de <- dme
ie <- sobel(mm, my)
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.

• I don't get why this should not happen. – FeldO Oct 3 '18 at 16:54
• @FeldO I try to do that through the text. Also see the comment about what is the most realistic explanation at the end. For one, total always equal direct plus indirect. – Heteroskedastic Jim Oct 3 '18 at 16:56
• Yes of course, ADE+ACME= Total Effect. This is the case. However, I was refering to the uncertainty concerning the corresponding estimates. When I follow your arguments, you don't say why it is not possible that the ACME is insignificant, while the ADE and Total Effect are not. – FeldO Oct 3 '18 at 17:03
• @FeldO I rewrote my answer, with an example where I replicate what you claimed. – Heteroskedastic Jim Oct 4 '18 at 14:13

@heteroskedastic Jim "Final most realistic alternative is your IV really has no relationship to the DV or the MV, but the MV is so strongly related to the DV that almost anything you multiply by it will work out as a mediator. The numbers will still add up though."

How should one interpret the results then? Is this evidence of mediation or not?

• Put this in the comments section which is just below @heteroskedastic Jim's answer – prashanth Oct 4 '18 at 10:31
• If the IV has no relationship to the outcome, then what is the mediator mediating? – Heteroskedastic Jim Oct 4 '18 at 13:26