---
title: "observed variables in the LMS- and QML approach"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{observed variables in the LMS- and QML approach}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
EVAL_DEFAULT <- FALSE
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  eval = EVAL_DEFAULT
)
```

```{r setup}
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.

```{r}
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
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. 

```{r} 
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)
``` 
