Data Generating Process with a Binary Covariate

This function generates potential outcomes from a data generating process similar to the one described in Parast et al. (2024) , but with the addition of a binary covariate X. It creates a dataset of potential outcomes $$P = (Y_1, S_1, Y_0, S_0)$$ and observed outcomes $$P_{observed} = (Y, S)$$ based on a random treatment assignment $Z$.

Usage

DGP_X_binary(n, p, q, mu_0, mu_1, Sigma_0, Sigma_1)

Arguments

n: Integer. Total sample size.
p: Numeric. Probability of being assigned to the treatment group $(Z=1)$.
q: Numeric. Probability of the binary covariate X being 1.
mu_0: Numeric vector. Mean vector for $P$ when $X=0$.
mu_1: Numeric vector. Mean vector for $P$ when $X=1$.
Sigma_0: Matrix. Covariance matrix for $P$ when $X=0$.
Sigma_1: Matrix. Covariance matrix for $P$ when $X=1$.

Value

A list containing:

X: The binary covariate vector.
n_X1: Number of units with $X=1$.
n_X0: Number of units with $X=0$.
Z: Treatment assignment vector.
n1: Number of treated units.
n0: Number of control units.
P: Full matrix of potential outcomes.
P_observed: Observed outcomes $(Y, S)$ corresponding to the assigned treatment $Z$.
P_unobserved: Counterfactual outcomes under the opposite treatment.

This function is useful for generating synthetic data to test or explore the method, for instance to verify the behavior of BSET_X under known simulation settings.

Details

The potential outcomes are generated from multivariate normal distributions with different mean vectors and covariance matrices depending on the value of $X$. Specifically: $$P \mid X = 0 \sim \mathcal{N}_{4}(\mu^{0}, \Sigma^{0}), \\ P \mid X = 1 \sim \mathcal{N}_{4}(\mu^{1}, \Sigma^{1}).$$

References

Carlotti P, Parast L (2026). “A Bayesian Critique of Rank-Based Methods for Surrogate Marker Evaluation.” arXiv preprint arXiv:2603.14381.

Examples

set.seed(123)
data <- DGP_X_binary(
  n = 100,
  p = 0.5,
  q = 0.5,
  mu_0 = c(5, 5, 0, 0),
  mu_1 = c(5, -5, 0, -10),
  Sigma_0 = kronecker(diag(2), matrix(c(1, 1, 1, 2), 2, 2)),
  Sigma_1 = kronecker(diag(2), matrix(c(1, 1, 1, 2), 2, 2))
)