How to perform simplest Moderation in lavaan and measuring its effect to other vars?

Question

How to perform simplest Moderation in lavaan and measuring its effect to other vars?

You have a model in r, in lavaan - how you add an moderation term?:

1. Does by adding a ~b + c make "c" a moderator term?
2. What is the difference between interaction term and moderation?
3. lavaan project do not offer a such example, and many ask similar questions without a proper and clear answer.

Not a clear answer i found on web, in terms of simple words and example.

model_sem <- 
  '
a=~ x1 + x2
b=~ z1 + z2
a ~ b
'

Answer 1

Moderation = Testing if Interaction term is Significant or not (in Lavaan). This simple thing is not so clearly written to web with some exceptions. see here

When you have two observed variables (those that containing data) then the things are very simple

Let say you have X regressed onto (->) Y (X~Y) and M = Moderation variable

Note: that a simple lm syntax with interaction term is valid too. eg.

summary( lm( Y ~ X + X*Y ) )

In Lavaan see here and here:

library(psych)
library(dplyr)
library(lavaan)
library(semTools)

# Note that Centering variables is essential 

# Creating interaction term prior of sem running
myData <- myData %>% mutate(Χ_x_Μ = myData$X * myData$M)

moderation_model <- '
  # Hayes Process model 1
  # regressions
  Y ~ b1*X
  Y ~ b2*M
  Y ~ b3*Χ_x_Μ
  
  # define mean parameter label for centered math for use in simple slopes
  M ~ M.mean*1
  
  # define variance parameter label for centered math for use in simple slopes
  M ~~ M.var*M
  
  # simple slopes for condition effect
  SD.below := b1 + b3*(M.mean - sqrt(M.var))
  mean := b1 + b3*(M.mean)
  SD.above := b1 + b3*(M.mean + sqrt(M.var))
  '

model <- moderation_model 
sem_fit <- lavaan::sem( model, std.lv = TRUE, data = myData, fixed.x = FALSE   )
summary( sem_fit, standardized = TRUE,  rsquare = TRUE )
standardizedSolution( sem_fit, type = "std.all" )

However, things can go weird when you would like to interact 1st order or even 2nd order latent variables (ps. in some cases Factors). Because, Lavaan does not support this, in an easy way.

Steps:

# Create observed Factors by using their items

    df$x1_o  = (df$f11 + df$f12 + df$f13 ) /3
    df$x2_o  = (df$f21 + df$f22 + df$f33 ) /3
    
    df$M_o   = (df$m1 + df$m2 + df$m3 ) /3
    df$Y_o   = (df$y1 + df$y2 ) /3
    df$X_o   = (df$x1_o + df$x2_o ) /3


# Mean Center all your SEM variables

    cn                        = colnames( df )
    df_center                 = df - data.frame( matrix( rep( colMeans( df ), nrow( df ) ), ncol = ncol( df ), byrow = T))  # mean centering
    colnames( df_center )     = cn      

# Make to interact each item of Moderator (M) with the X Variable to test its effect on the Y variable

    prod = df_center$x1_o   * df_center$m1;   df_center$prod11 = prod - mean(prod)
    prod = df_center$x1_o   * df_center$m2;   df_center$prod12 = prod - mean(prod)
    prod = df_center$x1_o   * df_center$m3;   df_center$prod13 = prod - mean(prod)
    prod = df_center$x2_o   * df_center$m1;   df_center$prod11 = prod - mean(prod)
    prod = df_center$x2_o   * df_center$m2;   df_center$prod12 = prod - mean(prod)
    prod = df_center$x2_o   * df_center$m3;   df_center$prod13 = prod - mean(prod)
    head(df_center, 2)

# Make your Lavaan moderated model 
# Note that in defining mean and variance of (M) Moderator, 
# we used the observed factored Moderator (M_o)
# and not the Latent (without values) construct (M).

# This happened cause Lavaan was saying that M.var has no free parameters. 
# So, I did that. If you think something else, please make a comment or a 
# post.  

    moderation_model  <- 
      '
    x1  =  f11 + f12 + 13 
    x2  =  f21 + f22 + f33 
    M   =   m1 + m2 + m3 
    Y   =   y1 + y2 
    X   =   x1 + x2
    
    INT   =~ prod11 + prod12 + prod13 + prod21 + prod22 + prod23 
    
    Y ~  b1 * X
    Y ~  b2 * M
    Y ~  b3 * INT
    
    
     # define mean parameter label for centered math for use in simple slopes
     M_o  ~ M_o.mean*1
      
     # define variance parameter label for centered math for use in simple slopes
     M_o  ~~ M_o.var*M_o 
      
     # simple slopes for condition effect
      SD.below := b1 + b3*(M_o.mean - sqrt(M_o.var))
      mean     := b1 + b3*(M_o.mean)
      SD.above := b1 + b3*(M_o.mean + sqrt(M_o.var))
      
    ' 

# ps. If Lavaan says that there is non-definite matrix cause is negative, 
# please, remove some items from the relevant variable from moderation model. 


# Check if lavaan sem can run it: 


    model <- moderation_model 
    sem_fit <- lavaan::sem( model, std.lv = TRUE, data = df_center, fixed.x = FALSE   )
    summary( sem_fit, standardized = TRUE,  rsquare = TRUE )
    standardizedSolution( sem_fit, type = "std.all" )