In contrast to the other approaches, the LMS
and
QML
approaches are designed to handle latent variables
only. Thus, observed variables cannot be used as easily as in the other
approaches. One way to get around this is by specifying your observed
variable as a latent variable with a single indicator.
modsem()
will, by default, constrain the factor loading to
1
and the residual variance of the indicator to
0
. The only difference between the latent variable and its
indicator, assuming it is an exogenous variable, is that it has a
zero-mean. This approach works for both the LMS
and
QML
methods in most cases, with two exceptions.
For the LMS
approach, you can use the above-mentioned
method in almost all cases, except when using an observed variable as a
moderating variable. In the LMS
approach, you typically
select one variable in an interaction term as the moderator. The
interaction effect is then estimated via numerical integration at
n
quadrature nodes of the moderating variable. However,
this process requires that the moderating variable has an error term, as
the distribution of the moderating variable is modeled as X ∼ N(Az, ε),
where Az is the
expected value of X at
quadrature point k
, and ε is the error term. If the error
term is zero, the probability of observing a given value of X will not be computable.
In most instances, the first variable in the interaction term is
chosen as the moderator. For example, if the interaction term is
"X:Z"
, "X"
will usually be chosen as the
moderator. Therefore, if only one of the variables is latent, you should
place the latent variable first in the interaction term. If both
variables are observed, you must specify a measurement error (e.g.,
"x1 ~~ 0.1 * x1"
) for the indicator of the first variable
in the interaction term.
library(modsem)
# Interaction effect between a latent and an observed variable
m1 <- '
# Outer Model
X =~ x1 # X is observed
Z =~ z1 + z2 # Z is latent
Y =~ y1 # Y is observed
# Inner model
Y ~ X + Z
Y ~ Z:X
'
lms1 <- modsem(m1, oneInt, method = "lms")
# Interaction effect between two observed variables
m2 <- '
# Outer Model
X =~ x1 # X is observed
Z =~ z1 # Z is observed
Y =~ y1 # Y is observed
x1 ~~ 0.1 * x1 # Specify a variance for the measurement error
# Inner model
Y ~ X + Z
Y ~ X:Z
'
lms2 <- modsem(m2, oneInt, method = "lms")
summary(lms2)
The estimation process for the QML
approach differs from
the LMS
approach, and you do not encounter the same issue
as in the LMS
approach. Therefore, you don’t need to
specify a measurement error for moderating variables.