---
title: "Specifying Formulas in a DAG"
output: rmarkdown::html_vignette
author: "Robin Denz"
vignette: >
  %\VignetteIndexEntry{Specifying Formulas in a DAG}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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

# Introduction

In this small vignette, we give more detailed examples on how best to use the `formula` argument in the `node()` and `node_td()` functions. This argument allows users to directly specify the full structural equation that should be used to generate the respective node in a clear and easy way, that does not directly rely on the `parents`, `betas` and associated arguments. Note that the `formula` argument may only be used with certain node types, as mentioned in the documentation.

# A simple example

We will start with a very simple example. Suppose we want to generate some data from a simple DAG with no time-varying variables. Consider the following DAG:

```{r}
library(simDAG)

dag <- empty_dag() +
  node("A", type="rnorm", mean=0, sd=1) +
  node("B", type="rbernoulli", p=0.5, output="numeric") +
  node("C", type="rcategorical", probs=c(0.3, 0.2, 0.5),
       output="factor", labels=c("low", "medium", "high"))
```

This DAG contains only three root nodes of different types. $A$ is normally distributed, $B$ is Bernoulli distributed and $C$ is a simple categorical variable with the levels "low", "medium" and "high". If we generate data from this DAG alone, it would look like this:

```{r}
set.seed(23143)

dat <- sim_from_dag(dag, n_sim=10)
head(dat)
```

Suppose we now want to generate an additional child node called $D$ which should be based on a linear regression model of the form:

$$D \sim -8 + A \cdot 0.4 + B \cdot -2 + N(0, 1.5).$$

We could do this using the `node()` function, by supplying appropriate values to the `parents`, `betas`, `intercept` and `error` arguments. The following code could be used:

```{r}
dag_without_formula <- dag +
  node("D", type="gaussian", parents=c("A", "B"), betas=c(0.4, -2),
       intercept=-8, error=1.5)
```

This does work just fine, but it may be a little cumbersome to specify the DAG in this way. Since we want to use a linear regression model, we could instead use the `formula` argument like this:

```{r}
dag_with_formula <- dag +
  node("D", type="gaussian", formula= ~ -8 + A*0.4 + B*-2, error=1.5)
```

Given the same random number generator seed, the same output will be produced from both DAGs, as shown below:

```{r}
set.seed(34)
dat1 <- sim_from_dag(dag_without_formula, n_sim=100)

set.seed(34)
dat2 <- sim_from_dag(dag_with_formula, n_sim=100)

all.equal(dat1, dat2)
```

Formulas should always start with a `~` sign and have nothing else on the left hand side. All parts of the formula should be connected by `+` signs, never `-` signs. The name of the respective variable should always be connected to the associated coefficient by a `*` sign. It does not matter whether the name of the term or the coefficient go first, but it has to be consistent in a formula. For example, `~ 1 + A*2 + B*3` works, and `~ 1 + 2*A + 3*B` also works, but `~ 1 + 2*A + B*2` will produce an error. The formula may also be supplied as a string and will produce the same output.

Apart from being easier to read, this also allows the user a lot more options. Through the use of formulas it is possible to specify nodes that have categorical parents. It is also possible to include any order of interaction effects and cubic terms using formulas, as shown below.

# Using a Categorical Parent Variable

Suppose that $D$ should additionally depend on $C$, a categorical variable. For example, suppose this is the regression model we want to generate data from:

$$D \sim -8 + A \cdot 0.4 + B \cdot -2 + Cmedium \cdot -1 + Chigh \cdot -3 + N(0, 1.5).$$

In this model, the "low" category is used as a reference category. If this is what we want to do, using the simple `parents`, `betas`, `intercept` approach no longer works. We have to use a formula. Fortunately, this is really simple to do using the following code:

```{r}
dag2 <- dag +
  node("D", type="gaussian", error=1.5,
       formula=~ -8 + A*0.4 + B*-2 + Cmedium*-1 + Chigh*-3,
       parents=c("A", "B", "C"))
```

Essentially, all we have to do is use the name of the categorical variable immediately followed by the category name. Note that if a different reference category should be used, the user needs to re-define the factor levels of the categorical variable accordingly first.

Note that we also defined the `parents` argument in this case. This is not strictly necessary to generate the data in this case, but it is recommended whenever categorical variables are used in a `formula` for two reasons:

* **1.)** If `parents` is not specified, the `sim_from_dag()` function will not know that $C$ is a parent of $D$. If `sort_dag=TRUE` and/or the nodes are not specified in a correctly topologically sorted order, this may lead to errors when trying to generate the data.
* **2.)** If `parents` is not specified, other functions that take DAG objects as input (such as the `plot.DAG()` function) may produce incorrect output, because they won't know that $C$ is a parent of $D$. 

# Using Interaction Effects

Interactions of any sort may also be added to the DAG. Suppose we want to generate data from the following regression model:

$$D \sim -8 + A \cdot 0.4 + B \cdot -2 + A*B \cdot -5 + N(0, 1.5),$$

where $A*B$ indicates the interaction between $A$ and $B$. This can be specified in the `formula` argument using the `:` sign:

```{r}
dag3 <- dag +
  node("D", type="gaussian", formula= ~ -8 + A*0.4 + B*-2 + A:B*-5, error=1.5)
```

Since both $A$ and $B$ are coded as numeric variables here, this works fine. If we instead want to include an interaction which includes a categorical variable, we again have to use the name with the respective category appended to it. For example, the following DAG includes an interaction between $A$ and $C$:

```{r}
dag4 <- dag +
  node("D", type="gaussian", error=1.5,
       formula=~ -8 + A*0.4 + B*-2 + Cmedium*-1 + Chigh*-3 + A:Cmedium*0.3 + 
         A:Chigh*10,
       parents=c("A", "B", "C"))
```

Higher order interactions may be specified in exactly the same way, just using more `:` symbols. It may not always be obvious in which order the variables for the interaction need to be specified. If the "wrong" order was used, the `sim_from_dag()` function will return a helpful error message explaining which ones should be used instead. For example, if we had used "Cmedium:A" instead of "A:Cmedium", this would not work because internally only the latter is recognized as a valid column. Note that because $C$ is categorical, we also specified the `parents` argument here just to be safe.

# Using Cubic Terms

Sometimes we also want to include non-linear relationships between a continuous variable and the outcome in a data generation process. This can be done by including cubic terms of that variable in a formula. Suppose the regression model that we want to use has the following form:

$$D \sim -8 + A \cdot 0.4 + A^2 \cdot 0.02 + B \cdot -2 + N(0, 1.5).$$

The following code may be used to define such as node:

```{r}
dag_with_formula <- dag +
  node("D", type="gaussian", formula= ~ -8 + A*0.4 + I(A^2)*0.02 + B*-2,
       error=1.5)
```

Users may of course use as many cubic terms as they like.

# Using Functions in formula

There is also limited support for including functions in the formula as well. For example, it is allowed to call any function on the beta coefficients, which is useful to specify betas on a different scale (for example using Odds-Ratios instead of betas). For example:

```{r}
dag_with_fun <- dag +
  node("D", type="binomial", formula= ~ -3 + A*log(0.5) + B*0.2)
```

is valid syntax. Any function can be used in the place of `log()`, as long as it is a single function that is called on a beta-coefficient.