# Nonparametric bootstrap inference on the derivative effect via our localized derivative estimator. This function implements the nonparametric bootstrap inference on the derivative of a dose-response curve via our localized derivative estimator. ```r DerivEffectBoot( Y, X, t_eval = NULL, boot_num = 500, alpha = 0.95, h_bar = NULL, kernT_bar = "gaussian", h = NULL, b = NULL, C_h = 7, C_b = 3, print_bw = TRUE, degree = 2, deriv_ord = 1, kernT = "epanechnikov", kernS = "epanechnikov", parallel = TRUE, cores = 6 ) ``` ## Arguments - `Y`: The input n-dimensional outcome variable vector. - `X`: The input n*(d+1) matrix. The first column of X stores the treatment/exposure variables, while the other d columns are confounding variables. - `t_eval`: The m-dimensional vector for evaluating the derivative. (Default: t_eval = NULL. Then, t_eval = `X[,1]`, which consists of the observed treatment variables.) - `boot_num`: The number of bootstrapping times. (Default: boot_num = 500.) - `alpha`: The confidence level of both the uniform confidence band and pointwise confidence interval. (Default: alpha = 0.95.) - `h_bar`: The bandwidth parameter for the Nadaraya-Watson conditional CDF estimator. (Default: h_bar = NULL. Then, the Silverman's rule of thumb is applied. See Chen et al. (2016) for details.) - `kernT_bar`: The name of the kernel function for the Nadaraya-Watson conditional CDF estimator. (Default: kernT_bar = "gaussian".) - `h, b`: The bandwidth parameters for the treatment/exposure variable and confounding variables in the local polynomial regression. (Default: h = NULL, b = NULL. Then, the rule-of-thumb bandwidth selector in Eq. (A1) of Yang and Tschernig (1999) is used with additional scaling factors C_h and C_b, respectively.) - `C_h, C_b`: The scaling factors for the rule-of-thumb bandwidth parameters. - `print_bw`: The indicator of whether the current bandwidth parameters should be printed to the console. (Default: print_bw = TRUE.) - `degree`: Degree of local polynomials. (Default: degree = 2.) - `deriv_ord`: The order of the estimated derivative of the conditional mean outcome function. (Default: deriv_ord = 1. It shouldn't be changed in most cases.) - `kernT, kernS`: The names of kernel functions for the treatment/exposure variable and confounding variables. (Default: kernT = "epanechnikov", kernS = "epanechnikov".) - `parallel`: The indicator of whether the function should be parallel executed. (Default: parallel = TRUE.) - `cores`: The number of cores for parallel execution. (Default: cores = 6.) ## Returns A list that contains four elements. - **theta_est**: The estimated derivative of the dose-response curve evaluated at points `t_eval`. - **theta_est_boot**: The estimated derivative of the dose-response curve evaluated at points `t_eval` for all the bootstrap samples. - **theta_alpha**: The width of the uniform confidence band. - **theta_alpha_var**: The widths of the pointwise confidence bands at evaluation points `t_eval`. ## Examples ```r set.seed(123) n <- 300 S2 <- cbind(2 * runif(n) - 1, 2 * runif(n) - 1) Z2 <- 4 * S2[, 1] + S2[, 2] E2 <- 0.2 * runif(n) - 0.1 T2 <- cos(pi * Z2^3) + Z2 / 4 + E2 Y2 <- T2^2 + T2 + 10 * Z2 + rnorm(n, mean = 0, sd = 1) X2 <- cbind(T2, S2) t_qry2 = seq(min(T2) + 0.01, max(T2) - 0.01, length.out = 100) chk <- Sys.getenv("_R_CHECK_LIMIT_CORES_", "") if (nzchar(chk) && chk == "TRUE") { # use 2 cores in CRAN/Travis/AppVeyor num_workers <- 2L } else { # use all cores in devtools::test() num_workers <- parallel::detectCores() } # Increase bootstrap times "boot_num" to a larger integer in practice theta_boot2 = DerivEffectBoot(Y2, X2, t_eval = t_qry2, boot_num = 3, alpha = 0.95, h_bar = NULL, kernT_bar = "gaussian", h = NULL, b = NULL, C_h = 7, C_b = 3, print_bw = FALSE, degree = 2, deriv_ord = 1, kernT = "epanechnikov", kernS = "epanechnikov", parallel = FALSE, cores = num_workers) ``` ## References Zhang, Y., Chen, Y.-C., and Giessing, A. (2024) **Nonparametric Inference on Dose-Response Curves Without the Positivity Condition.** [https://arxiv.org/abs/2405.09003](https://arxiv.org/abs/2405.09003). ## Author(s) Yikun Zhang, yikunzhang@foxmail.com

DerivEffectBoot function