Quadratic Effects and Interaction Effects

Quadratic effects are essentially a special case of interaction effects—where a variable interacts with itself. As such, all of the methods in modsem can also be used to estimate quadratic effects.

Below is a simple example using the LMS approach.

library(modsem)
m1 <- '
# Outer Model
X =~ x1 + x2 + x3
Y =~ y1 + y2 + y3
Z =~ z1 + z2 + z3

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

est1_lms <- modsem(m1, data = oneInt, method = "lms")
summary(est1_lms)

In this example, we have a simple model with two quadratic effects and one interaction effect. We estimate the model using both the QML and double-centering approaches, with data from 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
'

est2_dca <- modsem(m2, data = jordan)
est2_qml <- modsem(m2, data = jordan, method = "qml")
summary(est2_qml)

NOTE: We can also use the LMS approach to estimate this model, but it will be a lot slower, since we have to integrate along both ENJ and SC. In the first example it is sufficient to only integrate along X, but the addition of the SC:SC term means that we have to explicitly model SC as a moderator. This means that we (by default) have to integrate along 24^2=576 nodes. This both affects the the optimization process, but also dramatically affects the computation time of the standard errors. To make the estimation process it is possible to reduce the number of quadrature nodes, and calculate standard errors using the outer product of the score function, instead of the negative of the hessian matrix. Additionally, we can also pass mean.observed = FALSE, constraining the intercepts of the indicators to zero.

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
'

est2_lms <- modsem(m2, data = jordan, method = "lms", 
                   nodes = 15, OFIM.hessian = FALSE, 
                   mean.observed = FALSE)
summary(est2_lms)