Non-centered interaction terms

Using the LMS and QML approaches it is possible to estimate interaction terms where the means of the latent variables are not centered (i.e., they have non-zero means).

Here we can see an example using the TPB dataset:

tpb <- '
# Outer Model (Based on Hagger et al., 2007)
  ATT =~ att1 + att2 + att3 + att4 + att5
  SN =~ sn1 + sn2
  PBC =~ pbc1 + pbc2 + pbc3
  INT =~ int1 + int2 + int3
  BEH =~ b1 + b2

# Inner Model (Based on Steinmetz et al., 2011)
  INT ~ ATT + SN + PBC
  BEH ~ INT + PBC
  BEH ~ INT:PBC

# Adding Latent Intercepts
  INT ~ 1
  BEH ~ 1
  PBC ~ 1
  SN  ~ 1
  ATT ~ 1
'

est <- modsem(tpb, TPB, method = "lms", nodes = 32)
summary(est)

Comparing this to the estimates we get when PBC and INT have zero means, we see that the coefficients BEH~PBC and BEH~INT are drastically changed. This is not a bug, and is a function of the interaction effect rescaling the coefficients, when not centered at zero. When using the standardized_estimates function, or summary(est, standardized = TRUE) the interaction effect is centered, and we can see that the coefficients BEH~PBC and BEH~INT are rescaled once again.

summary(est, standardized = TRUE, centered = TRUE)

It is also possible to get the centered solution using the centered_estimates() function. Note, that centered_estimates() removes the mean structure of the model all together.

centered_estimates(est)