CADBCD_Alloc: Allocation Function of Covariate Adjusted Doubly Biased Coin...

In caradpt: Covariate-Adjusted Response-Adaptive Designs for Clinical Trials

Allocation Function of Covariate Adjusted Doubly Biased Coin Design for Binary and Continuous Response

Description

Calculating the probability of assigning the upcoming patient to treatment A based on the patient's covariates and the previous patients' covariates and responses for Covariate Adjusted Doubly Biased Coin procedure proposed by Zhang and Hu.

Usage

CADBCD_Alloc(ptsb.cov, ptsb.t, ptsb.Y, ptnow.cov, v = 2, response, target)

Arguments

ptsb.cov	a n x k covariate matrix of previous patients.
ptsb.t	a treatment vector of previous patients with length n.
ptsb.Y	a response vector of previous patients with length n.
ptnow.cov	a covariate vector of the incoming patient with length k.
v	a non-negative integer that controls the randomness of CADBCD design. The default value is 2.
response	the type of the response. Options are "Binary" or "Cont".
target	the type of optimal allocation target. Options are "Neyman" or "RSIHR".

Details

To start, we let \boldsymbol{\theta}_0 be an initial estimate of \boldsymbol{\theta}, and assign m_0 subjects to each treatment using a restricted randomization. Assume that m (with m \geq 2 m_0) subjects have been assigned to treatments. Their responses \{\boldsymbol{Y}_j, j = 1, \ldots, m\} and the corresponding covariates \{\boldsymbol{\xi}_j, j = 1, \ldots, m\} are observed. We let \widehat{\boldsymbol{\theta}}_m = (\widehat{\boldsymbol{\theta}}_{m, 1}, \widehat{\boldsymbol{\theta}}_{m, 2}) be an estimate of \boldsymbol{\theta} = (\boldsymbol{\theta}_1, \boldsymbol{\theta}_2). For each k = 1, 2, the estimator \widehat{\boldsymbol{\theta}}_{m, k} = \widehat{\boldsymbol{\theta}}_{m, k}(Y_{j,k}, \boldsymbol{\xi}_j : X_{j,k} = 1, j = 1, \ldots, m) is based on the observed sample of size N_{m,k}, that is, \{(Y_{j,k}, \boldsymbol{\xi}_j) : X_{j,k} = 1, j = 1, \ldots, m\}.

Define \widehat{\rho}_m = \frac{1}{m} \sum_{i=1}^m \pi_1(\widehat{\boldsymbol{\theta}}_m, \boldsymbol{\xi}_i) and \widehat{\pi}_m = \pi_1(\widehat{\boldsymbol{\theta}}_m, \boldsymbol{\xi}_{m+1}). When the (m+1)-th subject is ready for randomization and the corresponding covariate \boldsymbol{\xi}_{m+1} is recorded, we assign the patient to treatment 1 with the probability:

\psi_{m+1,1} = \frac{\widehat{\pi}_m \left( \frac{\widehat{\rho}_m}{N_{m,1}/m} \right)^v} {\widehat{\pi}_m \left( \frac{\widehat{\rho}_m}{N_{m,1}/m} \right)^v+ (1 - \widehat{\pi}_m) \left( \frac{1 - \widehat{\rho}_m}{1 - N_{m,1}/m} \right)^v}

and to treatment 2 with probability \psi_{m+1,2} = 1 - \psi_{m+1,1}, where v \geq 0 is a constant controlling the degree of randomness—from the most random when v = 0 to the most deterministic when v \rightarrow \infty. See Zhang and Hu(2009) for more details.

Value

prob	Probability of assigning the upcoming patient to treatment A.

References

Zhang, L. X., Hu, F., Cheung, S. H., & Chan, W. S. (2007). Asymptotic properties of covariate-adjusted response-adaptive designs. Annals of Statistics, 35(3), 1166–1182.

Zhang, L. X., & Hu, F. F. (2009). A new family of covariate-adjusted response adaptive designs and their properties. Applied Mathematics—A Journal of Chinese Universities, 24(1), 1–13.

Examples

set.seed(123)

n_prev = 40
covariates = cbind(Z1 = rnorm(n_prev), Z2 = rnorm(n_prev))
treatment = sample(c(0, 1), n_prev, replace = TRUE)
response = rbinom(n_prev, size = 1, prob = 0.6)

# Simulate new incoming patient
new_patient_cov = c(Z1 = rnorm(1), Z2 = rnorm(1))

# Run allocation function
result = CADBCD_Alloc(
  ptsb.cov = covariates,
  ptsb.t = treatment,
  ptsb.Y = response,
  ptnow.cov = new_patient_cov,
  response = "Binary",
  target = "Neyman"
)
print(result$prob)

caradpt documentation built on Aug. 28, 2025, 9:09 a.m.