LMS approach

As of version 1.0.13, the modsem function supports the estimation of higher order interaction effects. A general implementation is however only available with the LMS approach. Currently, estimation of higher order interaction effects using the QML approach is not available, and only partially available for the product indicator approaches (i.e., modsem_pi()).

Interaction between two higher order latent variables

In modsem there are two datasets which are variants of the Theory of Planned Behaviour (TPB) dataset. The TPB_2SO contains two second order latent variables, INT (intention) which is a second order latent variable of ATT (attitude) and SN (subjective norm), and PBC (perceived behavioural control) which is a second order latent variable of PC (perceived control) and PB (perceived behaviour).

tpb_2so <- '
  # First order latent variables
  ATT =~ att1 + att2 + att3
  SN  =~ sn1 + sn2 + sn3
  PB =~ pb1 + pb2 + pb3
  PC =~ pc1 + pc2 + pc3
  BEH =~ b1 + b2

  # Higher order latent variables
  INT =~ ATT + SN
  PBC =~ PC + PB

  # Structural model
  BEH ~ PBC + INT + INT:PBC
'

est_lms_2so <- modsem(tpb_2so, data = TPB_2SO, method = "lms")
summary(est_lms_2so)

Interaction between a first order and a higher order latent variable

In the TPB_1SO dataset, the INT latent variable is a second order latent variable of ATT, SN and PBC. In this example, we will estimate the interaction effect between the INT (higher order latent variable) and PBC (first order latent variable).

tpb_1so <- '
  # First order latent variables
  ATT =~ att1 + att2 + att3
  SN  =~ sn1 + sn2 + sn3
  PBC =~ pbc1 + pbc2 + pbc3
  BEH =~ b1 + b2

  # Higher order latent variables
  INT =~ ATT + PBC + SN

  # Structural model
  BEH ~ PBC + INT + INT:PBC
'

est_lms_1so <- modsem(tpb_1so, data = TPB_1SO, method = "lms", nodes = 32)
summary(est_lms_1so)

Product Indicator Approaches

As of yet, the modsem package does not support using the interaction operator : between two higher order latent variables, when using one of the product indicator approaches (i.e., using modsem_pi()). However, you can still attempt to estimate the interaction effect between two higher order latent variables by specifying the interaction term in models using the product indicator approaches.

Interaction between two higher order latent variables

WARNING: Please note that the literature on higher order interactions in product indicator approaches is virtually non-existant, and you will likely need to experiment with different approaches to find one that works. As well as experiment with adding constraints to the model.

Here we see the same example as for the LMS approach, where ther is an interaction effect between two higher order latent variables.

tpb_2so <- '
  # First order latent variables
  ATT =~ att1 + att2 + att3
  SN  =~ sn1 + sn2 + sn3
  PB =~ pb1 + pb2 + pb3
  PC =~ pc1 + pc2 + pc3
  BEH =~ b1 + b2

  # Higher order latent variables
  INT =~ ATT + SN
  PBC =~ PC + PB

  # Higher order interaction
  INTxPBC =~ ATT:PC + ATT:PB + SN:PC + SN:PB

  # Structural model
  BEH ~ PBC + INT + INTxPBC
'

est_ca <- modsem(tpb_2so, data = TPB_2SO, method = "ca")
summary(est_ca)

est_dblcent <- modsem(tpb_2so, data = TPB_2SO, method = "dblcent")
summary(est_dblcent)

Interaction between a first order and a higher order latent variable

Here we see the same example as for the LMS approach, where ther is an interaction effect between a higher order latent variable, and a first order latent variable.

tpb_1so <- '
  # First order latent variables
  ATT =~ att1 + att2 + att3
  SN  =~ sn1 + sn2 + sn3
  PBC =~ pbc1 + pbc2 + pbc3
  BEH =~ b1 + b2

  # Higher order latent variables
  INT =~ ATT + PBC + SN

  # Higher order interaction
  INTxPBC =~ ATT:PBC + SN:PBC + PBC:PBC

  # Structural model
  BEH ~ PBC + INT + INTxPBC
'

est_ca <- modsem(tpb_1so, data = TPB_1SO, method = "ca")
summary(est_ca)

est_dblcent  <- modsem(tpb_1so, data = TPB_1SO, method = "dblcent")
summary(est_dblcent)