---
title: "LCSM Parameters"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{lcsm-parameters}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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

```{r setup}
library(lcsm)
```


## Overview of estimated LCSM parameters

### Univariate LCSM

Depending on the specified model, the following parameters can be estimated for **univariate** LCSMs: 

|Parameter   |Description                                                             |
|:-----------|:-----------------------------------------------------------------------|
|gamma_lx1   |Mean of latent true scores x (Intercept)                                |
|sigma2_lx1  |Variance of latent true scores x                                        |
|sigma2_ux   |Variance of observed scores x                                           |
|alpha_g2    |Mean of change factor (g2)                                              |
|alpha_g3    |Mean of change factor (g3)                                              |
|sigma2_g2   |Variance of change factor (g2)                                          |
|sigma2_g3   |Variance of change factor (g3)                                          |
|sigma_g2lx1 |Covariance of change factor (g2) with the initial true score x          |
|sigma_g3lx1 |Covariance of change factor (g3) with the initial true score x          |
|sigma_g2g3  |Covariance of change factors within construct x                         |
|beta\_x     | Proportional change x                                                  |
|phi_x       |Autoregression of change scores x                                       |

### Bivariate LCSMs

For bivariate LCSMs, estimated parameters can be categorised into (1) **Construct X**, (2) **Construct Y**, and (3) **Coupling between X and Y**.

|Parameter    |Description                                                             |
|:------------|:-----------------------------------------------------------------------|
|**Construct X**                                                                       |
|gamma_lx1    |Mean of latent true scores x (Intercept)                                |
|sigma2_lx1   |Variance of latent true scores x                                        |
|sigma2_ux    |Variance of observed scores x                                           |
|alpha_g2     |Mean of change factor (g2)                                              |
|alpha_g3     |Mean of change factor (g3)                                              |
|sigma2_g2    |Variance of change factor (g2)                                          |
|sigma2_g3    |Variance of change factor (g3)                                          |
| beta\_x     | Proportional change x                                                  |
|sigma_g2lx1  |Covariance of change factor (g2) with the initial true score x (lx1)    |
|sigma_g3lx1  |Covariance of change factor (g3) with the initial true score x (lx1)    |
|sigma_g2g3   |Covariance of change factors within construct x                         |
|phi_x        |Autoregression of change scores x                                       |
|**Construct Y**                                                                       |
|gamma_ly1    |Mean of latent true scores y (Intercept)                                |
|sigma2_ly1   |Variance of latent true scores y                                        |
|sigma2_uy    |Variance of observed scores y                                           |
|alpha_j2     |Mean of change factor (j2)                                              |
|alpha_j3     |Mean of change factor (j3)                                              |
|sigma2_j2    |Variance of change factor (j2)                                          |
|sigma2_j3    |Variance of change factor (j3)                                          |
| beta\_y     | Proportional change y                                                  |
|sigma_j2ly1  |Covariance of change factor (j2) with the initial true score y (ly1)    |
|sigma_j3ly1  |Covariance of change factor (j3) with the initial true score y (ly1)    |
|sigma_j2j3   |Covariance of change factors within construct y                         |
|phi_y        |Autoregression of change scores y                                       |
|**Coupeling X & Y**                                                                   |
|sigma_su     |Covariance of residuals x and y                                         |
|sigma_ly1lx1 |Covariance of intercepts x and y                                        |
|sigma_g2ly1  |Covariance of change factor x (g2) with the initial true score y (ly1)  |
|sigma_g3ly1  |Covariance of change factor x (g3) with the initial true score y (ly1)  |
|sigma_j2lx1  |Covariance of change factor y (j2) with the initial true score x (lx1)  |
|sigma_j3lx1  |Covariance of change factor y (j3) with the initial true score x (lx1)  |
|sigma_j2g2   |Covariance of change factors y (j2) and x (g2)                          |
|sigma_j2g3   | Covariance of change factors y (j2) and x (g3)                         |
|sigma_j3g2   |Covariance of change factors y (j3) and x (g2)                          |
|delta_con_xy |Change score x (t) determined by true score y (t)                       |
|delta_con_yx |Change score y (t) determined by true score x (t)                       |
|delta_lag_xy |Change score x (t) determined by true score y (t-1)                     |
|delta_lag_yx |Change score y (t) determined by true score x (t-1)                     |
|xi_con_xy    |Change score x (t) determined by change score y (t)                     |
|xi_con_yx    |Change score y (t) determined by change score x (t)                     |
|xi_lag_xy    |Change score x (t) determined by change score y (t-1)                   |
|xi_lag_yx    |Change score y (t) determined by change score x (t-1)                   |