---
title: "higher order interactions"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{higher order interactions}
  %\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)
```

# LMS approach
As of version `1.0.13`, the `modsem` function supports the estimation of higher order interaction effects.
A general implementation is however only available with the LMS approach. Currently,
estimation of higher order interaction effects using the QML approach is not available,
and only partially available for the product indicator approaches (i.e., `modsem_pi()`).

## Interaction between two higher order latent variables
In `modsem` there are two datasets which are variants of the Theory of Planned Behaviour (`TPB`) dataset.
The `TPB_2SO` contains two second order latent variables,
`INT` (intention) which is a second order latent variable of `ATT` (attitude) and `SN` (subjective norm),
and `PBC` (perceived behavioural control) which is a second order latent variable of
`PC` (perceived control) and `PB` (perceived behaviour).

```{r}
tpb_2so <- '
  # First order latent variables
  ATT =~ att1 + att2 + att3
  SN  =~ sn1 + sn2 + sn3
  PB =~ pb1 + pb2 + pb3
  PC =~ pc1 + pc2 + pc3
  BEH =~ b1 + b2

  # Higher order latent variables
  INT =~ ATT + SN
  PBC =~ PC + PB

  # Structural model
  BEH ~ PBC + INT + INT:PBC
'

est_lms_2so <- modsem(tpb_2so, data = TPB_2SO, method = "lms")
summary(est_lms_2so)
```

## Interaction between a first order and a higher order latent variable 
In the `TPB_1SO` dataset, the `INT` latent variable is a second order latent variable of `ATT`, `SN` and `PBC`.
In this example, we will estimate the interaction effect between the `INT` (higher order latent variable) and `PBC` (first order latent variable).

```{r}
tpb_1so <- '
  # First order latent variables
  ATT =~ att1 + att2 + att3
  SN  =~ sn1 + sn2 + sn3
  PBC =~ pbc1 + pbc2 + pbc3
  BEH =~ b1 + b2

  # Higher order latent variables
  INT =~ ATT + PBC + SN

  # Structural model
  BEH ~ PBC + INT + INT:PBC
'

est_lms_1so <- modsem(tpb_1so, data = TPB_1SO, method = "lms", nodes = 32)
summary(est_lms_1so)
```


# Product Indicator Approaches
As of yet, the `modsem` package does not support using the interaction operator `:` between two higher order latent variables,
when using one of the product indicator approaches (i.e., using `modsem_pi()`).
However, you can still attempt to estimate the interaction effect between two higher order latent variables by specifying the interaction term in
models using the product indicator approaches.

## Interaction between two higher order latent variables
**WARNING:** Please note that the literature on higher order
interactions in product indicator approaches is virtually non-existant, and you
will likely need to experiment with different approaches to find one that works.
As well as experiment with adding constraints to the model.

Here we see the same example as for the LMS approach, where ther is an interaction
effect between two higher order latent variables.

```{r}
tpb_2so <- '
  # First order latent variables
  ATT =~ att1 + att2 + att3
  SN  =~ sn1 + sn2 + sn3
  PB =~ pb1 + pb2 + pb3
  PC =~ pc1 + pc2 + pc3
  BEH =~ b1 + b2

  # Higher order latent variables
  INT =~ ATT + SN
  PBC =~ PC + PB

  # Higher order interaction
  INTxPBC =~ ATT:PC + ATT:PB + SN:PC + SN:PB

  # Structural model
  BEH ~ PBC + INT + INTxPBC
'

est_ca <- modsem(tpb_2so, data = TPB_2SO, method = "ca")
summary(est_ca)

est_dblcent <- modsem(tpb_2so, data = TPB_2SO, method = "dblcent")
summary(est_dblcent)
```

## Interaction between a first order and a higher order latent variable 
Here we see the same example as for the LMS approach, where ther is an interaction
effect between a higher order latent variable, and a first order latent variable.

```{r}
tpb_1so <- '
  # First order latent variables
  ATT =~ att1 + att2 + att3
  SN  =~ sn1 + sn2 + sn3
  PBC =~ pbc1 + pbc2 + pbc3
  BEH =~ b1 + b2

  # Higher order latent variables
  INT =~ ATT + PBC + SN

  # Higher order interaction
  INTxPBC =~ ATT:PBC + SN:PBC + PBC:PBC

  # Structural model
  BEH ~ PBC + INT + INTxPBC
'

est_ca <- modsem(tpb_1so, data = TPB_1SO, method = "ca")
summary(est_ca)

est_dblcent  <- modsem(tpb_1so, data = TPB_1SO, method = "dblcent")
summary(est_dblcent)
```