The Basic Syntax

modsem introduces a new feature to the lavaan syntax—the semicolon operator (:). The semicolon operator works the same way as in the lm() function. To specify an interaction effect between two variables, you join them by Var1:Var2.

Models can be estimated using one of the product indicator approaches ("ca", "rca", "dblcent", "pind") or by using the latent moderated structural equations approach ("lms") or the quasi maximum likelihood approach ("qml"). The product indicator approaches are estimated via lavaan, while the lms and qml approaches are estimated via modsem itself.

A Simple Example

Here is a simple example of how to specify an interaction effect between two latent variables in lavaan.

m1 <- '
  # Outer Model
  X =~ x1 + x2 + x3
  Y =~ y1 + y2 + y3
  Z =~ z1 + z2 + z3
  
  # Inner Model
  Y ~ X + Z + X:Z 
'

est1 <- modsem(m1, oneInt)
summary(est1)

By default, the model is estimated using the "dblcent" method. If you want to use another method, you can change it using the method argument.

est1 <- modsem(m1, oneInt, method = "lms")
summary(est1)

Interactions Between Two Observed Variables

modsem allows you to estimate interactions between not only latent variables but also observed variables. Below, we first run a regression with only observed variables, where there is an interaction between x1 and z2, and then run an equivalent model using modsem().

Using a Regression

reg1 <- lm(y1 ~ x1*z1, oneInt)
summary(reg1)

Using `modsem`

modsem can also be used to to estimate interaction effects between observed variables. Interaction effects with observed variables are not supported by the LMS and QML approaches. In most cases, you can define a latent variable with a single indicator to estimate the interaction effect between two observed variables in the LMS and QML approaches.

est2 <- modsem('y1 ~ x1 + z1 + x1:z1', data = oneInt, method = "dblcent")
summary(est2)

Interactions Between Latent and Observed Variables

modsem also allows you to estimate interaction effects between latent and observed variables. To do so, simply join a latent and an observed variable with a colon (e.g., 'latent:observed'). In most of the product indicator approaches the residuals of product indicators with common variables (e.g., x1z1 and x1z2) are allowed to covary freely. This is problematic when there is an interaction term between an observed variable (or a latent variable with a single indicator) and a latent variables (with multiple indicators), since all of the product indicators share a common element.

In the example below, all the generated product indicators (x1z1, x2z1 and x3z1) share z1. If all the indicator residuals of a latent variabel are allowed to covary freely, the model will (in general) be unidenfified. The simplest way to fix this issue, is to constrain the residual covariances to be zero, by using the res.cov.method argument.

m3 <- '
  # Outer Model
  X =~ x1 + x2 + x3
  Y =~ y1 + y2 + y3
  
  # Inner Model
  Y ~ X + z1 + X:z1 
'

est3 <- modsem(m3, oneInt, method = "dblcent", res.cov.method = "none")
summary(est3)

Quadratic Effects

Quadratic effects are essentially a special case of interaction effects. Thus, modsem can also be used to estimate quadratic effects.

m4 <- '
# Outer Model
X =~ x1 + x2 + x3
Y =~ y1 + y2 + y3
Z =~ z1 + z2 + z3

# Inner Model
Y ~ X + Z + Z:X + X:X
'

est4 <- modsem(m4, oneInt, method = "qml")
summary(est4)

More Complicated Examples

Here is a more complex example using the theory of planned behavior (TPB) model.

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 + INT:PBC  
'

# The double-centering approach
est_tpb <- modsem(tpb, TPB)

# Using the LMS approach
est_tpb_lms <- modsem(tpb, TPB, method = "lms")
summary(est_tpb_lms)

Here is an example that includes two quadratic effects and one interaction effect, using the jordan dataset. The dataset is a subset of the PISA 2006 dataset.

m2 <- '
ENJ =~ enjoy1 + enjoy2 + enjoy3 + enjoy4 + enjoy5
CAREER =~ career1 + career2 + career3 + career4
SC =~ academic1 + academic2 + academic3 + academic4 + academic5 + academic6
CAREER ~ ENJ + SC + ENJ:ENJ + SC:SC + ENJ:SC
'

est_jordan <- modsem(m2, data = jordan)
est_jordan_qml <- modsem(m2, data = jordan, method = "qml")
summary(est_jordan_qml)

modsem