---
title: "Generate Synthetic Data"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{generate_synthetic_data}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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

We provide **`gen_syn_data`** to generate synthetic data for CausalGPS package

## Usage

Input parameters:

**`sample_size`** Number of data samples

**`seed`** The seed of R's random number generator

**`outcome_sd`** Standard deviation used to generate the outcome

**`gps_spec`** A numerical value (1-7) that indicates the GPS model used to generate synthetic data. See the following section for more details.

**`cova_spec`** A numerical value (1-2) to modify the covariates. See the code for more details.

## Technical Details for Data Generating Process
We generate six confounders $(C_1,C_2,...,C_6)$, which include a combination of continuous and categorical variables,
\begin{align*}
    C_1,\ldots,C_4 \sim N(0,\boldsymbol{I}_4), C_5 \sim U\{-2,2\}, C_6 \sim U(-3,3),
\end{align*}
and generate $W$ using six specifications of the generalized propensity score model,

1)  $W = 9 \{-0.8+ (0.1,0.1,-0.1,0.2,0.1,0.1) \boldsymbol{C}\} +17  + N(0,5)$

2)  $W = 15\{-0.8+ (0.1,0.1,-0.1,0.2,0.1,0.1) \boldsymbol{C}\} + 22 + T(2)$

3)  $W = 9 \{-0.8+ (0.1,0.1,-0.1,0.2,0.1,0.1) \boldsymbol{C}\} + 3/2 C_3^2 + 15 + N(0,5)$

4)  $W = \frac{49 \exp(\{-0.8+ (0.1,0.1,-0.1,0.2,0.1,0.1) \boldsymbol{C}\})}{1+ \exp(\{-0.8+ (0.1,0.1,-0.1,0.2,0.1,0.1) \boldsymbol{C}\})} -6 + N(0,5)$

5)  $W = \frac{42}{1+ \exp(\{-0.8+ (0.1,0.1,-0.1,0.2,0.1,0.1) \boldsymbol{C}\})} - 18 + N(0,5)$

6)  $W = 7 \text{log} ( \{-0.8+ (0.1,0.1,-0.1,0.2,0.1,0.1) \boldsymbol{C}\}) + 13 + N(0,4)$

We generate $Y$ from an outcome model which is assumed to be a cubical function of $W$ with additive terms for the confounders and interactions between $W$ and confounders $\mathbf{C}$,

$$Y | W, \mathbf{C} \sim N\{\mu(W, \mathbf{C}),\text{sd}^2\}$$

$$\mu(W, \mathbf{C}) = -10 - (2, 2, 3, -1,2,2)\mathbf{C}  - W(0.1 - 0.1C_1 + 0.1C_4 + 0.1C_5 + 0.1C_3^2)  + 0.13^2W^3$$