---
title: "Specifying Copula Models through Correlation"
author: "Chase Mathis"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Specifying Copula Models through Correlation}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---
```{r load causl}
library(causl)
```

The standard way we choose to parametrize copula dependence structure is 
similar to the way we parametrize the other causal structures. We specify
a variable is a link function of a linear combination of other variables. 

For instance, we return 
to the following example: \begin{align*}
Z &\sim N(0, \; 1)\\
X \mid Z=z &\sim N(0.5\times z, \; 1)\\
Y \mid \operatorname{do}(X=x) &\sim N(0.3\times x, \; 1)
\end{align*}

The formulas and family variables are familiar. 
```{r forms and fams}
formulas=list(z~1, x~z, y~x, ~1)
family=rep(1,4)
```

When defining the `pars` variable
we replace the familiar `beta` in the cop entry to a `tau`. If the user specifies
a $\tau$, the program will convert the specified Kendall's $\tau$ to the correct
parameter so that the copula between the outcome and confounder has a 
Kendall correlation of $\tau$. This is a simpler way to encode the dependence information
as $\tau \in [-1,1]$ and easier to interpret. The code looks like.

```{r pars}
pars <- list(z = list(beta = 0, phi=1),
             x = list(beta = c(0,0.5), phi=1),
             y = list(beta = c(0,0.3), phi=1),
             cop = list(y= list(z = list(tau = 0.25)))) # NOTE tau argument
```

We note that we define the dependence between y and z to be uncoditional 
on the treatment x. This assumption is necessary for specifying Kendall's $\tau$.
(Is it?)

```{r}
cm <- causl_model(formulas = formulas, family = family, pars = pars)
dat <- rfrugal(n = 10000, causl_model = cm)
```