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Generate Synthetic Data

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 (C1,C2,...,C6), 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