set.seed(1234)

# define set of data with 3 covariates for testing
n <- 20
X3 <- matrix(rnorm(n * 3), nrow = n, ncol = 3)
X3 <- cbind(matrix(1, nrow = n, ncol = 1), X3) # add intercept
theta3_truth <- matrix(c(1, 0, 1, 0), ncol = 1)
theta3_truth_bool <- as.logical(theta3_truth)
theta3_truth_idx <- which(theta3_truth_bool)
y3 <- X3 %*% theta3_truth + rnorm(n)

n <- 200
X6 <- matrix(rnorm(n * 6), nrow = n, ncol = 6)
X6 <- cbind(matrix(1, nrow = n, ncol = 1), X6) # add intercept
theta6_truth <- matrix(c(0, 0, 1, 1, 0, 1, 1), ncol = 1)
theta6_truth_bool <- as.logical(theta6_truth)
theta6_truth_idx <- which(theta6_truth_bool)
y6 <- X6 %*% theta6_truth + rnorm(n)

n <- 150
X9 <- matrix(rnorm(n * 9), nrow = n, ncol = 9)
X9 <- cbind(matrix(1, nrow = n, ncol = 1), X9) # add intercept
theta9_truth <- matrix(c(0, 1, 1, 0, 1, 1, 1, 0, 0, 0), ncol = 1)
theta9_truth_bool <- as.logical(theta9_truth)
theta9_truth_idx <- which(theta9_truth_bool)
y9 <- X9 %*% theta9_truth + rnorm(n)
groups9 <- c(1,2,3,4,5,5,5,6,6,6)