---
title: "fit measures for LMS and QML"
output: rmarkdown::html_vignette
vignette: |
  %\VignetteIndexEntry{fit measures for LMS and QML}
  %\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)
```

# Introduction

This vignette demonstrates how to evaluate and compare model fit for latent interaction models estimated via

- **LMS** (Latent Moderated Structural Equations)
- **QML** (Quasi-Maximum Likelihood)

using the **modsem** package (v≥1.0.8). Because standard Chi-square statistics are not available under LMS/QML, we assess fit by:

1. Examining fit indices for the baseline (no-interaction) model.
2. Conducting a likelihood-ratio difference test to compare the baseline and interaction models (Klein & Moosbrugger, 2000; Klein & Múthen, 2007).

If the baseline model shows acceptable fit and adding the latent interaction significantly improves fit, the interaction model can also be deemed well-fitting.

# Example

We define a model with three latent variables (X, Y, Z) and their interaction (X:Z):

```{r}
m1 <- "
# Outer (measurement) model
X =~ x1 + x2 + x3
Y =~ y1 + y2 + y3
Z =~ z1 + z2 + z3

# Inner (structural) model
Y ~ X + Z + X:Z
"

# Estimate the full (H1) model via LMS
est_h1 <- modsem(m1, oneInt, method = "lms")

# Estimate the baseline (H0) model without interaction
est_h0 <- estimate_h0(est_h1, calc.se = FALSE) # std.errors are not needed
```
# Fit measures baseline model

To get fit measures for the baseline model you can use the `fit_modsem_da()` function.
```{r}
fit_modsem_da(est_h0)
```
It can also be used to get fit measures for the full model, but should be pared with 
`chisq = FALSE` to avoid the Chi-square test. If it is set to `TRUE` it will 
calculate the Chi-square test while ignoring the interaction terms in the model.

```{r}
fit_modsem_da(est_h1, chisq = FALSE)
```

# Difference Test of Fit

Compare H0 vs. H1 using a log-likelihood ratio test:

```{r}
compare_fit(est_h0, est_h1)
```

A **significant** p-value indicates the latent interaction term significantly improves model fit.

# Inspecting Fit Indices
For convenience, you can also use the `modsem_inspect()` function with 
`what = "fit"` to get fit indices for both models, and comparative fit in one go.

```{r}
modsem_inspect(est_h1, what = "fit")
```

# References
    Klein, A., & Moosbrugger, H. (2000). 
      <doi:10.1007/BF02296338>.
      "Maximum likelihood estimation of latent interaction effects with the LMS method."
    Klein, A. G., & Muthén, B. O. (2007). 
      <doi:10.1080/00273170701710205>.
      "Quasi-maximum likelihood estimation of structural equation models with multiple interaction and quadratic effects."