Estimate treatment effect and the corresponding variance estimation on the treatment arm using different covariate adjustment methods.

Description

Implements a unified framework for comparing covariate adjustment method for completely randomized experiments under randomization-based framework.

Usage

esti_mean_treat(X, Y, A, H = NULL)

Arguments

Value

A list with two named vectors:

point_est

Point estimates for all estimators:

• unadj: Unadjusted estimator

• db: Debiased estimator (Lu et al., 2023)

• adj2c: HOIF-inspired debiased estimator (Zhao et al., 2024), the same as db

• adj2: HOIF-motivated adjusted estimator (Zhao et al., 2024)

• adj3: Bias-free adjusted estimator based on adj2

• lin: Covariate-adjusted estimator (Lin, 2013)

• lin_db: Debiased estimator with population leverage scores (Lei, 2020)

var_est

Variance estimates corresponding to each estimator:

• unadj: Variance estimate for unadjusted estimator

• db: Variance estimate for debiased estimator (Lu et al., 2023)

• adj2c: Variance for adj2c, using formulas given in (Lu et al., 2023)

• adj2c_v2: Conservative variance for adj2c (Zhao et al., 2024)

• adj2: Variance for adj2, with formulas motivated by (Lu et al., 2023)

• adj2_v2: Conservative variance for adj2 (Zhao et al., 2024)

• adj3: Variance for adj3, with formulas motivated by (Lu et al., 2023)

• adj3_v2: Conservative variance for adj3 (Zhao et al., 2024)

• lin: HC3-type variance for Lin's (2013) estimator

• lin_db: HC3-type variance for Lei's (2020) estimator

References

Lin, W. (2013). Agnostic notes on regression adjustments to experimental data: Reexamining Freedman's critique. The Annals of Statistics, Vol. 7(1), 295–318, doi:10.1214/12-AOAS583.
Lei, L. and Ding, P. (2020) Regression adjustment in completely randomized experiments with a diverging number of covariates. Biometrika, Vol. 108(4), 815–828, doi:10.1093/biomet/asaa103.
Lu, X., Yang, F. and Wang, Y. (2023) Debiased regression adjustment in completely randomized experiments with moderately high-dimensional covariates. arXiv preprint, arXiv:2309.02073, doi:10.48550/arXiv.2309.02073.
Zhao, S., Wang, X., Liu, L. and Zhang, X. (2024) Covariate Adjustment in Randomized Experiments Motivated by Higher-Order Influence Functions. arXiv preprint, arXiv:2411.08491, doi:10.48550/arXiv.2411.08491.

Examples

set.seed(100)
n <- 500
p <- n * 0.3
beta <- runif(p, -1 / sqrt(p), 1 / sqrt(p))

X <- mvtnorm::rmvt(n, sigma = diag(1, p), df = 3)
Y1 <- as.numeric(X %*% beta)
Y0 <- rep(0, n)

pi1 <- 2/3
n1 <- ceiling(n * pi1)
ind <- sample(n, size = n1)
A <- rep(0, n)
A[ind] <- 1
Y <- Y1 * A + Y0 * (1 - A)

Xc_svd <- svd(X)
H <- Xc_svd$u %*% t(Xc_svd$u)

result_ls <- esti_mean_treat(X, Y, A, H)
point_est <- result_ls$point_est
var_est <- result_ls$var_est
print(paste0('True mean treat:', round(mean(Y1), digits = 3), '.'))
print('Absolute bias:')
print(abs(point_est - mean(Y1)))
print('Estimate variance:')
print(var_est)

Covariate-Adjusted Treatment Effect Estimation under the Randomization-based Framework

Description

Implements the (HOIF-inspired) debiased estimators for average treatment effect (ATE) or treatment effect on the treatment/control arm with variance estimation using asymptotic-variance. Designed for randomized experiments with moderately high-dimensional covariates.

Usage

fit.adj2.adj2c.Random(Y, X, A, pi1 = NULL, target = "ATE")

Arguments

Value

A list containing three named vectors, including point estimates and variance estimates:

tau_vec

Point estimates:

• adj2: Point estimation of the HOIF-inspired debiased estimator given by Zhao et al.(2024).

• adj2c: Point estimation of the debiased estimator given by Lu et al. (2023), which is also the HOIF-inspired debiased estimator given by Zhao et al.(2024).

var_vec_v1

Variance estimates for adj2 and adj2c, with formulas inspired by Lu et al. (2023).:

• adj2: Variance for adj2.

• adj2c: Variance for adj2c.

var_vec_v2

Variance estimates for adj2 and adj2c, with formulas given in Zhao et al. (2024), which is more conservative.

• adj2: Variance for adj2.

• adj2c: Variance for adj2c.

References

Lu, X., Yang, F. and Wang, Y. (2023) Debiased regression adjustment in completely randomized experiments with moderately high-dimensional covariates. arXiv preprint, arXiv:2309.02073, doi:10.48550/arXiv.2309.02073.
Zhao, S., Wang, X., Liu, L. and Zhang, X. (2024) Covariate Adjustment in Randomized Experiments Motivated by Higher-Order Influence Functions. arXiv preprint, arXiv:2411.08491, doi:10.48550/arXiv.2411.08491.

Examples

set.seed(100)
n <- 500
p <- n * 0.3
beta <- runif(p, -1 / sqrt(p), 1 / sqrt(p))

X <- mvtnorm::rmvt(n, sigma = diag(1, p), df = 3)
Y1 <- as.numeric(X %*% beta)
Y0 <- as.numeric(X %*% beta - 1)

pi1 <- 2/3
n1 <- ceiling(n * pi1)
ind <- sample(n, size = n1)
A <- rep(0, n)
A[ind] <- 1
Y <- Y1 * A + Y0 * (1 - A)

Xc <- cbind(1, scale(X, scale = FALSE))
result.adj2.adj2c.random.ate.ls <- fit.adj2.adj2c.Random(Y, Xc, A, target = 'ATE')
result.adj2.adj2c.random.ate.ls

result.adj2.adj2c.random.treat.ls <- fit.adj2.adj2c.Random(Y, Xc, A, target = 'EY1')
result.adj2.adj2c.random.treat.ls

result.adj2.adj2c.random.control.ls <- fit.adj2.adj2c.Random(Y, Xc, A, target = 'EY0')
result.adj2.adj2c.random.control.ls

Covariate-Adjusted Treatment Effect Estimation under the Super-Population Framework

Description

Implements HOIF-inspired debiased estimators for average treatment effect (ATE) or treatment effect on the treatment/control arm with variance estimation using influence function-based and asymptotic-variance. Designed for randomized experiments with moderately high-dimensional covariates.

Usage

fit.adj2.adj2c.Super(
  Y,
  X,
  A,
  intercept = TRUE,
  pi1 = NULL,
  target = "ATE",
  lc = FALSE
)

Arguments

Value

A list containing three named vectors, including point estimates and variance estimates:

tau_vec

Point estimates:

• adj2: Point estimation of the HOIF-inspired debiased estimator (Zhao et al., 2024).

• adj2c: Point estimation of the the HOIF-inspired debiased estimator (Zhao et al., 2024), which is also the debiased estimator given by Lu et al. (2023).

var_infl_vec

Influence function-based variance estimates:

• adj2: Variance for adj2 via the sample variance of its influence function formula.

• adj2c: Variance for adj2c via the sample variance of its influence function formula.

var_rb_vec

Variance estimates inspired by Bannick et al. (2025):

• adj2: Variance for adj2 following the asymptotic variance given by Bannick et al. (2025).

• adj2c: Variance for adj2c following the asymptotic variance given by Bannick et al. (2025).

References

Bannick, M. S., Shao, J., Liu, J., Du, Y., Yi, Y. and Ye, T. (2025) A General Form of Covariate Adjustment in Clinical Trials under Covariate-Adaptive Randomization. Biometrika, Vol. xx(x), 1-xx, doi:10.1093/biomet/asaf029.
Lu, X., Yang, F. and Wang, Y. (2023) Debiased regression adjustment in completely randomized experiments with moderately high-dimensional covariates. arXiv preprint, arXiv:2309.02073, doi:10.48550/arXiv.2309.02073.
Zhao, S., Wang, X., Liu, L. and Zhang, X. (2024) Covariate Adjustment in Randomized Experiments Motivated by Higher-Order Influence Functions. arXiv preprint, arXiv:2411.08491, doi:10.48550/arXiv.2411.08491.

Examples

set.seed(120)
alpha0 <- 0.1;
n <- 400;

p0 <- ceiling(n * alpha0)
beta0_full <- 1 / (1:p0) ^ (1 / 2) * (-1) ^ c(1:p0)
beta <- beta0_full / norm(beta0_full,type='2')

Sigma_true <- matrix(0, nrow = p0, ncol = p0)
for (i in 1:p0) {
  for (j in 1:p0) {
    Sigma_true[i, j] <- 0.1 ** (abs(i - j))
  }
}

X <- mvtnorm::rmvt(n, sigma = Sigma_true, df = 3)

lp0 <- X %*% beta
delta_X <- 1  -  1/4 * X[, 2] -  1/8 * X[, 3]
lp1 <- lp0 + delta_X

Y0 <- lp0 + rnorm(n)
Y1 <- lp1 + rnorm(n)


pi1 <- 1 / 2
A <- rbinom(n, size = 1, prob = pi1)
Y <- A * Y1 + (1 - A) * Y0

Xc <- cbind(1, scale(X, scale = FALSE))
result.adj2.adj2c.sp.ate.ls <- fit.adj2.adj2c.Super(Y, Xc, A, intercept = TRUE,
                                                    target = 'ATE', lc = TRUE)
result.adj2.adj2c.sp.ate.ls
result.adj2.adj2c.sp.treat.ls <- fit.adj2.adj2c.Super(Y, Xc, A, intercept = TRUE,
                                                      target = 'EY1', lc = TRUE)
result.adj2.adj2c.sp.treat.ls
result.adj2.adj2c.sp.control.ls <- fit.adj2.adj2c.Super(Y, Xc, A, intercept = TRUE,
                                                        target = 'EY0', lc = TRUE)
result.adj2.adj2c.sp.control.ls

Covariate-Adjusted Treatment Effect Estimation under the Randomization-based Framework

Description

Implements the (HOIF-inspired) debiased estimators for average treatment effect (ATE) with variance estimation using asymptotic-variance. Designed for randomized experiments with moderately high-dimensional covariates.

Usage

fit.ate.adj2.adj2c.Random(Y, X, A, pi1_hat = NULL)

Arguments

Value

A list containing three named vectors, including point estimates and variance estimates:

tau_vec

Point estimates:

• adj2: Point estimation of the HOIF-inspired debiased estimator given by Zhao et al.(2024).

• adj2c: Point estimation of the debiased estimator given by Lu et al. (2023), which is also the HOIF-inspired debiased estimator given by Zhao et al.(2024).

var_vec_v1

Variance estimates for adj2 and adj2c, with formulas inspired by Lu et al. (2023).:

• adj2: Variance for adj2.

• adj2c: Variance for adj2c.

var_vec_v2

Variance estimates for adj2 and adj2c, with formulas given in Zhao et al. (2024), which is more conservative.

• adj2: Variance for adj2.

• adj2c: Variance for adj2c.

References

Lu, X., Yang, F. and Wang, Y. (2023) Debiased regression adjustment in completely randomized experiments with moderately high-dimensional covariates. arXiv preprint, arXiv:2309.02073, doi:10.48550/arXiv.2309.02073.
Zhao, S., Wang, X., Liu, L. and Zhang, X. (2024) Covariate Adjustment in Randomized Experiments Motivated by Higher-Order Influence Functions. arXiv preprint, arXiv:2411.08491, doi:10.48550/arXiv.2411.08491.

Examples

set.seed(100)
n <- 500
p <- n * 0.3
beta <- runif(p, -1 / sqrt(p), 1 / sqrt(p))

X <- mvtnorm::rmvt(n, sigma = diag(1, p), df = 3)
Y1 <- as.numeric(X %*% beta)
Y0 <- as.numeric(X %*% beta - 1)

pi1 <- 2/3
n1 <- ceiling(n * pi1)
ind <- sample(n, size = n1)
A <- rep(0, n)
A[ind] <- 1
Y <- Y1 * A + Y0 * (1 - A)

Xc <- cbind(1, scale(X, scale = FALSE))
result.adj2.adj2c.random.ls <- fit.ate.adj2.adj2c.Random(Y, Xc, A)
point_est <- result.adj2.adj2c.random.ls$tau_vec
var_est_v1 <- result.adj2.adj2c.random.ls$var_vec_v1
var_est_v2 <- result.adj2.adj2c.random.ls$var_vec_v2
point_est
var_est_v1
var_est_v2

Covariate-Adjusted Treatment Effect Estimation under the Randomization-based Framework

Description

Implements the (HOIF-inspired) debiased estimators for treatment effect on the treatment/control arm with variance estimation using asymptotic-variance. Designed for randomized experiments with moderately high-dimensional covariates.

Usage

fit.treat.adj2.adj2c.Random(Y, X, A, pi1_hat = NULL)

Arguments

Value

A list containing three named vectors, including point estimates and variance estimates:

tau_vec

Point estimates:

• adj2: Point estimation of the HOIF-inspired debiased estimator given by Zhao et al.(2024).

• adj2c: Point estimation of the debiased estimator given by Lu et al. (2023), which is also the HOIF-inspired debiased estimator given by Zhao et al.(2024).

var_vec_v1

Variance estimates for adj2 and adj2c, with formulas inspired by Lu et al. (2023).:

• adj2: Variance for adj2.

• adj2c: Variance for adj2c.

var_vec_v2

Variance estimates for adj2 and adj2c, with formulas given in Zhao et al. (2024), which is more conservative.

• adj2: Variance for adj2.

• adj2c: Variance for adj2c.

References

Lu, X., Yang, F. and Wang, Y. (2023) Debiased regression adjustment in completely randomized experiments with moderately high-dimensional covariates. arXiv preprint, arXiv:2309.02073, doi:10.48550/arXiv.2309.02073.
Zhao, S., Wang, X., Liu, L. and Zhang, X. (2024) Covariate Adjustment in Randomized Experiments Motivated by Higher-Order Influence Functions. arXiv preprint, arXiv:2411.08491, doi:10.48550/arXiv.2411.08491.

Examples

set.seed(100)
n <- 500
p <- n * 0.3
beta <- runif(p, -1 / sqrt(p), 1 / sqrt(p))

X <- mvtnorm::rmvt(n, sigma = diag(1, p), df = 3)
Y1 <- as.numeric(X %*% beta)
Y0 <- as.numeric(X %*% beta - 1)

pi1 <- 2/3
n1 <- ceiling(n * pi1)
ind <- sample(n, size = n1)
A <- rep(0, n)
A[ind] <- 1
Y <- Y1 * A + Y0 * (1 - A)

Xc <- cbind(1, scale(X, scale = FALSE))
result.adj2.adj2c.random.ls <- fit.treat.adj2.adj2c.Random(Y, Xc, A)
point_est <- result.adj2.adj2c.random.ls$tau_vec
var_est_v1 <- result.adj2.adj2c.random.ls$var_vec_v1
var_est_v2 <- result.adj2.adj2c.random.ls$var_vec_v2
point_est
var_est_v1
var_est_v2

Estimate the oracle bias, the exact variance and approximated variance of the debiased estimator tau_adj2c inspired by HOIF (Zhao et al.(2024)).

Description

Estimate the oracle bias, the exact variance and approximated variance of the debiased estimator tau_adj2c inspired by HOIF (Zhao et al.(2024)).

Usage

get_oracle_bias_var_adj2c(X, Y1, n1 = NULL)

Arguments

Value

A list of oracle bias and variance of the debised adjusted estimator tau_adj2c.

References

Zhao, S., Wang, X., Liu, L., & Zhang, X. (2024). Covariate adjustment in randomized experiments motivated by higher-order influence functions. arXiv preprint. https://arxiv.org/abs/2411.08491.

Examples

NULL

Estimate the oracle bias, the exact variance and approximated variance of the debiased estimator and the bias-free estimator motivated by HOIF (Zhao et al.(2024)).

Description

Estimate the oracle bias, the exact variance and approximated variance of the debiased estimator and the bias-free estimator motivated by HOIF (Zhao et al.(2024)).

Usage

get_oracle_bias_var_adj_2_3(X, Y1, n1 = NULL)

Arguments

Value

A list of oracle bias and variance of the adjusted estimator motivated by HOIF and the bias-free estimator.

References

Zhao, S., Wang, X., Liu, L., & Zhang, X. (2024). Covariate adjustment in randomized experiments motivated by higher-order influence functions. arXiv preprint. https://arxiv.org/abs/2411.08491

Examples

# Linear setting
set.seed(100)
n <- 500
p <- 50
beta <- rt(p,3)

X <- mvtnorm::rmvt(n, sigma = diag(1, p), df = 3)
Y1 <- as.numeric(X %*% beta)
pi1 <- 0.50
n1 <- ceiling(n*pi1)

result_adj_db <- get_oracle_bias_var_adj_db(X = X,Y1=Y1,n1=n1)
result_adj2c <- get_oracle_bias_var_adj2c(X = X,Y1=Y1,n1=n1)
result_adj2_3 <- get_oracle_bias_var_adj_2_3(X = X,Y1=Y1,n1=n1)
unlist(result_adj_db)
unlist(result_adj2c)
unlist(result_adj2_3)



# Nonlinear setting
n <- 500;
alpha <- 0.2;
set.seed(1000)
p <- ceiling(n*alpha)
Sigma_true <- matrix(0,nrow=p,ncol=p)
for(i in 1:p){
  for(j in 1:p){
    Sigma_true[i,j] <- 0.1**(abs(i-j))
  }
}

X <- mvtnorm::rmvt(n, sigma = Sigma_true, df = 3)
beta <- rt(p,3)
or_baseline <- sign(X %*% beta) * abs(X %*% beta)^(1/2) + sin(X %*% beta)
epsilon1 <- epsilon0 <- rt(n,3)
Y1 <- 1 + as.numeric(or_baseline) + epsilon1


pi1 <- 0.50
n1 <- ceiling(n*pi1)

result_adj_db <- get_oracle_bias_var_adj_db(X = X,Y1=Y1,n1=n1) # from LYW paper
result_adj2c <- get_oracle_bias_var_adj2c(X = X,Y1=Y1,n1=n1)
result_adj2_3 <- get_oracle_bias_var_adj_2_3(X = X,Y1=Y1,n1=n1)
unlist(result_adj_db)
unlist(result_adj2c)
unlist(result_adj2_3)

Estimate the oracle bias, the oracle variance of the unadjusted estimator, the adjusted estimator by Lei’s (2020) and the debiased estimator tau_db by Lu et al.(2023).

Description

Estimate the oracle bias, the oracle variance of the unadjusted estimator, the adjusted estimator by Lei’s (2020) and the debiased estimator tau_db by Lu et al.(2023).

Usage

get_oracle_bias_var_adj_db(X, Y1, n1 = NULL)

Arguments

Value

A list of the oracle bias and variance of .

References

Lihua Lei, Peng Ding. Regression adjustment in completely randomized experiments with a diverging number of covariates. Biometrika, 815–828, 2020.

Xin Lu, Fan Yang, and Yuhao Wang. Debiased regression adjustment in completely randomized experiments with moderately high-dimensional covariates. arXiv preprint arXiv:2309.02073, 2023.

Examples

NULL