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
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.