DTR.Boots.KernSmooth: Make inference about the linear treatment regime vector and the optimal value

Description

This function estimates the optimal treatment regime among all linear regimes with smoothed estimation methods and doubly robust correction, and construct element-wise 100(1-alpha)% confidence intervals for the optimal linear treatment regime vector, and the 100(1-alpha)% confidence interval for the optimal value if the population follows treatments recommended by the optimal linear regime. It outputs a 'DTR.Boots.KernSmooth' model object

Usage

DTR.Boots.KernSmooth(
  X,
  y,
  a,
  intercept = TRUE,
  prob = 0.5,
  B = 500,
  alpha = 0.05,
  m0 = mean(y[a == 0]),
  m1 = mean(y[a == 1]),
  kernel = "normal",
  phi0 = 1,
  gamma = 2,
  err_tol = 1e-04,
  iter_tol = 200
)

Value

An object of class "DTR.Boots.KernSmooth", which is a list containing the following components:

alpha: The confidence level of the confidence interval.
B: The number of repetitions in the inference procedure by weighted bootstrap.
smooth_est: The fitted "DTR.KernSmooth" object based on the input data and parameters.
Beta_CI: The 100(1-alpha)% confidence intervals for each element of the optimal treatment regime vector.
value_CI: The 100(1-alpha)% confidence interval for the optimal average response value among all linear treatment regimes.

Arguments

X: Input matrix, of dimension n_obs x n_vars; each row is an observation vector.
y: Response variable to be maximized on average if every subject follows the treatment recommended by the optimal regime.
a: Received treatments for n_obs subjects. Must be bivariate, and labeled as {0,1}.
intercept: Logical. TRUE (default) if the intercept is included in estimating the optimal treatment regime and FALSE if not.
prob: The propensity score for n_obs subjects, i.e., P(a=1|X). If NULL, it would be estimated by logistic regression a~X.
B: The number of repetitions in the inference procedure by weighted bootstrap. The default value is 500.
alpha: The confidence level of the confidence interval. The default value is 0.05.
m0: The estimated response values if the subjects receive treatment 0. The default is the average response value of all subjects who receive treatment 0.
m1: The estimated response values if the subjects receive treatment 1. The default is the average response value of all subjects who receive treatment 1.
kernel: The kernel function to be used in smoothed estimation. Should be one of "normal", "poly1" and "poly2". The default value is "normal". See more details in the "Details" section of DTR.KernSmooth.
phi0: The initial step size to be used in the Proximal Algorithm. The default value is 1.
gamma: The multiplier of the step sizes to be used in the Proximal Algorithm. Must be gamma > 1. The default value is 2.
err_tol: The desired accuracy in the estimation. The default value is 1e-4.
iter_tol: The maximum number of iterations in the estimation algorithm. The default value is 200.

Author

Yunan Wu and Lan Wang
Maintainer: Yunan Wu <yunan.wu@utdallas.edu>

Details

This function constructs confidence intervals for the optimal linear treatment regime vector by wild bootstrap procedures. The bootstrapped estimate of the smoothed robust estimator is defined as the vector $\widehat{\bm{\beta}}^*$ that maximizes $$n^{-1}\sum_{i=1}^n \frac{\left[A_i K\left(\frac{\bm{x}_i^T\bm{\beta}}{h_n}\right)+(1-A_i)\left\{1-K\left(\frac{\bm{x}_i^T\bm{\beta}}{h_n}\right)\right\}\right]r_iY_i}{\pi_a(\bm{x}_i, \widehat{\bm{\xi}})}- n^{-1}\sum_{i=1}^n \frac{\left[A_i-\pi_a(\bm{x}_i, \widehat{\bm{\xi}})\right]r_i\widehat{m}_1(\bm{x}_i)K\left(\frac{\bm{x}_i^T\bm{\beta}}{h_n}\right)+\left[1-A_i-\pi_a(\bm{x}_i, \widehat{\bm{\xi}})\right]r_i \widehat{m}_0(\bm{x}_i) \left\{1-K\left(\frac{\bm{x}_i^T\bm{\beta}}{h_n}\right)\right\}} {\pi_a(\bm{x}_i, \widehat{\bm{\xi}})},$$ with the second term as the doubly correction, where $r_i$'s are i.i.d N(1,1).
Let $\xi_j^{\circ(\alpha/2)}$ and $\xi_j^{\circ(1-\alpha/2)}$ be the $(\alpha/2)$-th and $(1-\alpha/2)$-th quantile of the bootstrap distribution of $(nh_n)^{1/2}(\widehat{\beta}_j^*-\widehat{\beta}_j)$, respectively, where $\alpha$ is a small positive number. We can estimate $\xi_j^{\circ(\alpha/2)}$ and $\xi_j^{\circ(1-\alpha/2)}$ from a large number of bootstrap samples. An asymptotic $100(1-\alpha)\%$ bootstrap confidence interval for $\beta_{0j}$, is given by $$\left\{\widehat{\beta}_j-(nh_n)^{-1/2}\xi_j^{\circ(1-\alpha/2)}, \widehat{\beta}_j-(nh_n)^{-1/2}\xi_j^{\circ(\alpha/2)}\right\}.$$ To construct confidence intervals for the optimal value $V(\bm{\beta}_0)$, we define $$V_n^*(\widehat{\bm{\beta}}) = n^{-1}\sum_{i=1}^n \frac{\left[A_i I\left(\bm{x}_i^T\widehat{\bm{\beta}}>0\right)+(1-A_i)I\left(\bm{x}_i^T\widehat{\bm{\beta}}\leq 0 \right) \right]r_iY_i}{\pi_a(\bm{x}_i, \widehat{\bm{\xi}})}- n^{-1}\sum_{i=1}^n \frac{\left[A_i-\pi_a(\bm{x}_i, \widehat{\bm{\xi}})\right]r_i \widehat{m}_1(\bm{x}_i)I\left(\bm{x}_i^T\bm{\beta}>0\right)+\left[1-A_i-\pi_a(\bm{x}_i, \widehat{\bm{\xi}})\right]r_i \widehat{m}_0(\bm{x}_i)I\left(\bm{x}_i^T\bm{\beta}\leq 0\right)} {\pi_a(\bm{x}_i, \widehat{\bm{\xi}})},$$ where $r_i$'s are i.i.d N(1,1). Let $d^{\circ(\alpha/2)}$ and $d^{\circ(1-\alpha/2)}$ be the $(\alpha/2)$-th and $(1-\alpha/2)$-th quantile of the bootstrap distribution of $n^{1/2}\{V_n^*(\widehat{\bm{\beta}})-V_n(\widehat{\bm{\beta}})\}$, respectively. An asymptotic $100(1-\alpha)\%$ bootstrap confidence interval for $V(\bm{\beta}_0)$ is $$\left\{V_n(\widehat{\bm{\beta}})-n^{-1/2}d^{\circ(1-\alpha/2)}, V_n(\widehat{\bm{\beta}})-n^{-1/2}d^{\circ(\alpha/2)}\right\}.$$ See more details in the reference.

References

Wu, Y. and Wang, L. (2021), Resampling-based Confidence Intervals for Model-free Robust Inference on Optimal Treatment Regimes, Biometrics, 77: 465– 476, tools:::Rd_expr_doi("10.1111/biom.13337").

Examples

Run this code

n <- 500; p <- 3
beta <- c(0.2,1,-0.5,-0.8)*0.7
beta1 <- c(1,-0.5,-0.5,0.5)

set.seed(12345)
X <- matrix(rnorm(n*p),n)
a <- rbinom(n,1,0.7)
mean1 <- exp(cbind(1,X) %*% beta1)
mean2 <- 8/(1 + exp(-cbind(1,X) %*% beta)) - 4
y <- mean1 + a * mean2 + rnorm(n)

boots_smooth_model_ci <- DTR.Boots.KernSmooth(X, y, a, prob = 0.4*a+0.3, B = 100)
boots_smooth_model_ci$Beta_CI
boots_smooth_model_ci$value_CI

if (FALSE) {
boots_smooth_model_ic <- DTR.Boots.KernSmooth(X, y, a, B = 100, m0 = mean1,
                                              m1 = mean1 + mean2)
boots_smooth_model_ic$Beta_CI
boots_smooth_model_ic$value_CI

boots_smooth_model_cc <- DTR.Boots.KernSmooth(X, y, a, prob = 0.4*a+0.3, B = 100,
                                              m0 = mean1, m1 = mean1 + mean2)
boots_smooth_model_cc$Beta_CI
boots_smooth_model_cc$value_CI

boots_smooth_model_ii <- DTR.Boots.KernSmooth(X, y, a, B = 100)
boots_smooth_model_ii$Beta_CI
boots_smooth_model_ii$value_CI
}

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