---
title: "create_parameter_table"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{create_parameter_table}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
  %\DeclareUnicodeCharacter{2194}{$\leftrightarrow$}
  %\DeclareUnicodeCharacter{2192}{$\rightarrow$}
---

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

```{r setup, include = FALSE}
library(mxsem)
```

The core function of **mxsem** is to create a parameter table, where all loadings,
regressions, and (co-)variances are specified. This parameter table is then used
to set up an **mxModel** with the `mxPath`-function. It can be useful to
visually inspect the parameter table created by **mxsem**. To this end, set
the `return_parameter_table`-argument to `TRUE`:

```{r}
library(mxsem)

model <- '
  # latent variable definitions
     ind60 =~ x1 + x2 + x3
     dem60 =~ y1 + a1*y2 + b*y3 + c1*y4
     dem65 =~ y5 + a2*y6 + b*y7 + c2*y8

  # regressions
    dem60 ~ g1*ind60
    dem65 ~ g2*ind60 + g3*dem60

  # residual correlations
    y1 ~~ y5
    y2 ~~ y4 + y6
    y3 ~~ y7
    y4 ~~ y8
    y6 ~~ y8
    
! delta_a
! g1g3
a2   := a1 + delta_a
g1g3 := g1*g3
'

model_list <- mxsem(model = model,
                    data  = OpenMx::Bollen,
                    return_parameter_table = TRUE)

print(model_list$parameter_table)
```

The element `parameter_table$parameter_table` specifies all loadings (`op` is `=~`),
regressions (`op` is `~`), and (co-)variances (`op` is `~~`). The `modifier` specifies
parameter labels, `lbound` is the lower bound and `ubound` is the upper bound for
parameters. Finally, `free` specifies if a parameter is estimated (`TRUE`) or 
fixed (`FALSE`).

If there are algebras, these are listed in the `parameter_table$algebras` data.frame.
Note that the new parameters `delta_a` and `g1g3` used in these algebras are listed in 
`parameter_table$new_parameters`, while `parameter_table$new_parameters_free`
specifies for each of these new parameters if they are free or fixed. In this case
`g1g3` is fixed because it is the product of two other parameters.

The `variables` specify which of the variables are manifest (observed) and which
are latent (unobserved). Each manifest variable must also be found in the data set.