# observed variables in the LMS- and QML approach

library(modsem)

# The Latent Moderated Structural Equations (LMS) and the Quasi Maximum Likelihood (QML) approach

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.

## The LMS approach

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 QML approach

### If you are using the latest GitHub version

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)

### If you are using the CRAN version

If you are using the latest CRAN version, there is a slight caveat, in that all endogenous variables have to have atleast two indicators. This is due to a transformation, and the approximation of the distribution of the indicators in the endogenous variables. This problem will likely be fixed in a later update, but as of now, latent endogenous variable need at least two indicators. If a latent variable in the QML approach can be expressed without using an interaction term, you can in some cases use the ‘cov.syntax’ argument as a workaround. If this is the case, see the vignette on interaction effects between two endogenous variable in the LMS- and QML approach (vignette("interaction_two_etas"))

m4 <- '
# Outer Model
X =~ x1 # X is observed
Z =~ z1 # Z is observed
Y =~ y1 + y2 # Y needs to be latent, needing atleast two indicators

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

qml4 <- modsem(m3, oneInt, method = "qml")
summary(qml4)