| DTR.Boots.KernSmooth {DTRKernSmooth} | R Documentation |
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
)
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. |
prob |
The propensity score for n_obs subjects, i.e., P(a=1|X). If |
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 |
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. |
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.
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. |
Author(s)
Yunan Wu and Lan Wang
Maintainer:
Yunan Wu <yunan.wu@utdallas.edu>
References
Wu, Y. and Wang, L. (2021), Resampling-based Confidence Intervals for Model-free Robust Inference on Optimal Treatment Regimes, Biometrics, 77: 465– 476, doi:10.1111/biom.13337.
See Also
predict.DTR.Boots.KernSmooth, obj_value,
DTR.KernSmooth
Examples
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
## Not run:
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
## End(Not run)