library(modsem)In contrast to the other approaches, the LMS and QML approaches are
designed to handle latent variables only. Thus observed variables cannot
be as easily used, as in the other approaches. One way of getting 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. Then, the only difference between the
latent variable and its indicator, is that (assuming that it is an
exogenous variable) it has a zero-mean. This will work for both the LMS-
and QML approach in most cases, except for two exceptions.
For the LMS approach you can use the above mentioned approach in
almost all cases, except in the case where you wish to use an observed
variable as a moderating variable. In the LMS approach, you will usually
select one variable in an interaction term as a moderator. The
interaction effect is then estimated via numerical integration, at
n quadrature nodes of the moderating variable. This process
however, requires that the moderating variable has an error-term, as the
distribution of the moderating variable is modelled as \(X \sim N(Az, \varepsilon)\), where \(Az\) is the expected value of \(X\) at quadrature point k, and
\(\varepsilon\) 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. Thus if only
one of the variables are latent, you should put the latent variable
first in the interaction term. If both are observed, you have to 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(m1, oneInt, method = "lms")
summary(lms2)The estimation of the QML approach is different from the LMS approach, and you do not have the same issue as in the LMS approach. Thus you don’t have to specify a measurement error for moderating variables.
m3 <- '
# Outer Model
X =~ x1 # X is observed
Z =~ z1 # Z is observed
Y =~ y1 # Y is observed
# Inner model
Y ~ X + Z
Y ~ X:Z
'
qml3 <- modsem(m3, oneInt, method = "qml")
summary(qml3)