Lavaan mediation + moderation + 2 X's

Question

I'm trying to build a SEM that looks like the picture shown below:

Where:

• $X$ and $X^2$ are the independent variables
• $Y$ is the dependent variable
• $M_1$ and $M_2$ are the mediators
• $W$ is a moderator

None of the default mediation packages that I've tried so far support such a structure so I'm using lavaan SEM instead. I've been reading up on Hayes and have figured out the model without the moderator:

  # direct effect
  Y ~ cprime1*X + cprime2*X2 + controls

  # mediator 1 paths
  M1 ~ a1*X+a2*X2  + controls          
  Y ~ b1*M1

  # mediator 2 paths
  M2 ~ a1prime*X + a2prime*X2 + d*M1 + controls
  Y ~ b2*M2
       
  # indirect effect (a*b) using the Hayes2010 formula (Table 1) 
  ab :=  (a1 + 2*a2*X)*b1 + (a1prime + 2*a2prime + (a1+2*a2*X)*d ) * b2
  
  # total effect
  cprime := (cprime1+2*cprime2)
  total := cprime + (ab)

I'm aware of how to moderate with 1 input variable, for instance for M1 you'd have something like:

M1 ~ a1*X1 + a2*W + a3*X1:W + controls

But it's unclear to me how one would add the moderators for X2 in these and the other parts of the model.

Answer 1

I simulated some irrelevant data:

library(lavaan)

dat <- as.data.frame(mvrnorm(5e2, rep(0, 6), matrix(.25, 6, 6) + .75 * diag(6)))
colnames(dat) <- c("X", "X2", "M1", "M2", "W", "Y")
dat$X2 <- dat$X ^ 2
head(dat)
#            X         X2         M1         M2          W          Y
# 1 -1.2556613 1.57668526  0.9558917 -0.6155703 -0.4540778 -0.3747208
# 2 -0.8463471 0.71630340  0.2066086 -0.1680590 -0.9181445 -0.2819832
# 3  0.1081230 0.01169059 -0.1934604  1.2662057  0.4817797 -0.5342619
# 4  0.6180336 0.38196555  0.3301925 -0.1277026  1.3222274  0.3626635

Given these data, this is the model code you want:

summary(sem(
  "M1 ~ a11 * X + a12 * X2 + w10 * W + w11 * X:W + w12 * X2:W
  M2 ~ a21 * X + a22 * X2 + w20 * W + w21 * X:W + w22 * X2:W + d * M1
  Y ~ c1 * X + c2 * X2 + w30 * W + w31 * X:W + w32 * X2:W + b1 * M1 + b2 * M2
  a11db2 := a11 * d * b2
  a12db2 := a12 * d * b2
  a11b1 := a11 * b1
  a12b1 := a12 * b1
  a21b2 := a21 * b2
  a22b2 := a22 * b2",
  dat), rsquare = TRUE)

With this, you can work out the exact combinations that get you the total indirect, direct and total effects. The nature of W makes things a bit difficult. Right now, the computed effects at the bottom are for when W = 0. You might want to play around with the computed effects depending on the type of variable W is.

Answer 2

Moderation is basically the product of the IVs ($X_i$) with the moderator ($W$). Since you have two IVs, you have two products $X_1W$ and $W_2W$, you just have to do it once for each predictor $X_i$ on each outcome ($M_1,M_2,Y$), since you want the moderator on each of them.

I used the notation in the graph for the code,

Y ~ cprime1*X + cprime2*X2 + w3_1*X1:W + w3_2*X2:W + controls    
M2 ~ aprime1*X1 + aprime2*W + w2_1*X1:W + w2_2*X2:W + + d*M1 + controls
M1 ~ a1*X1 + a2*W + w2_1*X1:W + w2_2*X2:W + controls

plus your indirect effects.

You can also consider a triple interaction X1:X2:W. You would have to add the double interaction of $X_1X_2$ and the triple interaction $X_1X_2W$. At this point you would have an IV and two moderators (conceptually speaking; statistically, it is arbitrary because there is no difference between the variables). I don't think it is what you want though.