---
title: "quadratic effects"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{quadratic effects}
  %\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)
```
# 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.

```{r}
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
'

est1Lms <- modsem(m1, data = oneInt, method = "lms")
summary(est1Lms)
```

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.

```{r}
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
'

est2Dblcent <- modsem(m2, data = jordan)
est2Qml <- modsem(m2, data = jordan, method = "qml")
summary(est2Qml)
```

**Note**: The other approaches (e.g., `LMS` and constrained methods) can also be used but may be slower depending on the number of interaction effects, especially for the `LMS` and constrained approaches.