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

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

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)

Allocation Function of Covariate Adjusted Doubly Biased Coin Design for Survival 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 for survival trial.

Usage

CADBCD_Alloc_Surv(ptsb.cov, ptsb.t, ptsb.Y, ptsb.E, ptnow.cov, v = 2, target)

Arguments

Details

For the first m = 2m_0 patients, a restricted randomization procedure is used to allocate them equally to treatments A and B.

After the initial enrollment of 2m_0 patients, adaptive treatment assignment begins from patient 2m_0 + 1 onward. Importantly, this adaptive assignment does not require complete observation of outcomes from earlier patients. Instead, at each new patient arrival, the treatment effect parameters (\tilde{\boldsymbol{\beta}}_{A,m}, \tilde{\boldsymbol{\beta}}_{B,m}) are re-estimated via partial likelihood using the accrued survival data subject to dynamic administrative censoring. Specifically, each previously enrolled patient is censored at the current arrival time if their event has not yet occurred. These updated estimates are then used to guide covariate-adjusted allocation.

When the (m+1)-th patient enters the trial with covariate vector \boldsymbol{Z}_{m+1}, the probability of assigning this patient to treatment A is computed as:

\phi_{m+1} = g(\tilde{\boldsymbol{\beta}}_{A,m}, \tilde{\boldsymbol{\beta}}_{B,m}, \boldsymbol{Z}_{m+1}),

where 0 \leq g(\cdot) \leq 1 is an appropriately chosen allocation function that favors the better-performing treatment arm.

Following Zhang and Hu's CADBCD procedure, let N_A(m) and N_B(m) = m - N_A(m) denote the numbers of patients allocated to treatments A and B after m assignments. Let \tilde{\pi}_m = \pi_A(\tilde{\boldsymbol{\beta}}_{A,m}, \tilde{\boldsymbol{\beta}}_{B,m}, \boldsymbol{Z}_{m+1}) be the target allocation probability for the incoming patient. The average target allocation proportion among the first m patients is defined as:

\tilde{\rho}_m = \frac{1}{m} \sum_{i=1}^m \pi_A(\tilde{\boldsymbol{\beta}}_{A,m}, \tilde{\boldsymbol{\beta}}_{B,m}, \boldsymbol{Z}_i).

Under the CADBCD scheme, the probability of assigning treatment A to the (m+1)-th patient is given by:

\phi_{m+1} = \frac{\tilde{\pi}_m \left( \frac{\tilde{\rho}_m}{N_A(m)/m} \right)^v} {\tilde{\pi}_m \left( \frac{\tilde{\rho}_m}{N_A(m)/m} \right)^v + (1 - \tilde{\pi}_m) \left( \frac{1 - \tilde{\rho}_m}{N_B(m)/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. similiar procedure for survival responses are used in all other designs.

Value

References

Mukherjee, A., Jana, S., & Coad, S. (2024). Covariate-adjusted response-adaptive designs for semiparametric survival models. Statistical Methods in Medical Research, 09622802241287704.

Examples

set.seed(123)
n = 40
covariates = cbind(rexp(40),rexp(40))
treatment = sample(c(0, 1), n, replace = TRUE)
survival_time = rexp(n, rate = 1)
censoring = runif(n)
event = as.numeric(survival_time < censoring)
new_patient_cov = c(Z1 = 1, Z2 = 0.5)
result = CADBCD_Alloc_Surv(
 ptsb.cov = covariates,
 ptsb.t = treatment,
 ptsb.Y = survival_time,
 ptsb.E = event,
 ptnow.cov = new_patient_cov,
 v = 2,
 target = "Neyman"
)
print(result$prob)

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

Description

This function simulates a clinical trial using the Covariate Adjusted Doubly Biased Coin Design (CADBCD) with Binary or Continuous Responses.

Usage

CADBCD_Sim(n, thetaA, thetaB, m0 = 40, pts.cov, v = 2, response, target)

Arguments

Value

A list with the following elements:

Examples

set.seed(123)
results = CADBCD_Sim(n = 400,
                      pts.cov = cbind(rnorm(400), rnorm(400)),
                      thetaA = c(-1, 1, 1),
                      thetaB = c(3, 1, 1),
                      response = "Binary",
                      target = "Neyman")
results

## view the settings
results$method
results$sampleSize

## view the simulation results
results$parameter
results$assignments
results$proportion
results$responses
results$failureRate

Simulation For Covariate Adjusted Doubly Biased Coin Design for Survival Response

Description

This function simulates a clinical trial with time-to-event (survival) outcomes using the Covariate Adjusted Doubly Biased Coin Design (CADBCD). Patient responses are generated under the Cox proportional hazards model, assuming the proportional hazards assumption holds.

Usage

CADBCD_Sim_Surv(
  n,
  thetaA,
  thetaB,
  m0 = 40,
  pts.cov,
  v = 2,
  target,
  censor.time,
  arrival.rate
)

Arguments

Value

A list with the following elements:

Examples

set.seed(123)

## Run CADBCD simulation with survival response
results = CADBCD_Sim_Surv(
  thetaA = c(0.1, 0.1),
  thetaB = c(-1, 0.1),
  n = 400,
  pts.cov = cbind(sample(c(1, 0), 400, replace = TRUE), rnorm(400)),
  target = "RSIHR",
  censor.time = 2,
  arrival.rate = 150
)

Allocation Function of CARA Designs Based on Efficiency and Ethics 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 CARAEE procedure.

Usage

CARAEE_Alloc(ptsb.cov, ptsb.t, ptsb.Y, ptnow.cov, gamma, response)

Arguments

Details

Covariate-Adjusted Response-Adaptive with Ethics and Efficiency (CARAEE) Design: The CARAEE procedure balances both ethical considerations and statistical efficiency when assigning subjects to treatments.

As a start-up rule, m_0 subjects are assigned to each treatment using a balanced randomization scheme.

Assume that m \geq 2m_0 subjects have been assigned, and their responses \{\boldsymbol{X}_i, i = 1, \ldots, m\} and covariates \{\boldsymbol{Z}_i, i = 1, \ldots, m\} are observed. Let \hat{\boldsymbol{\theta}}(m) = \left( \hat{\theta}_1(m), \hat{\theta}_2(m) \right), where \hat{\theta}_k(m) is the maximum likelihood estimate of the treatment-specific parameter \theta_k based on the data for treatment group k.

For the incoming subject (m+1) with covariates \boldsymbol{Z}_{m+1}, we define the efficiency and ethics measures for each treatment as: \boldsymbol{d}(\boldsymbol{Z}, \boldsymbol{\theta}) = \left(d_1(\boldsymbol{Z}, \theta), d_2(\boldsymbol{Z}, \theta)\right), \quad \boldsymbol{e}(\boldsymbol{Z}, \boldsymbol{\theta}) = \left(e_1(\boldsymbol{Z}, \theta), e_2(\boldsymbol{Z}, \theta)\right).

The allocation probability of assigning subject (m+1) to treatment 1 is given by:

\phi_{m+1}(\boldsymbol{Z}_{m+1}, \hat{\boldsymbol{\theta}}(m)) = \frac{e_1(\boldsymbol{Z}_{m+1}, \hat{\boldsymbol{\theta}}(m)) \cdot d_1^\gamma(\boldsymbol{Z}_{m+1}, \hat{\boldsymbol{\theta}}(m))} {e_1(\boldsymbol{Z}_{m+1}, \hat{\boldsymbol{\theta}}(m)) \cdot d_1^\gamma(\boldsymbol{Z}_{m+1}, \hat{\boldsymbol{\theta}}(m)) + e_2(\boldsymbol{Z}_{m+1}, \hat{\boldsymbol{\theta}}(m)) \cdot d_2^\gamma(\boldsymbol{Z}_{m+1}, \hat{\boldsymbol{\theta}}(m))}.

This allocation rule is scale-invariant in both efficiency and ethics components due to its ratio-based form. The tuning parameter \gamma \geq 0 controls the trade-off between the two: when \gamma = 0, the assignment is based purely on ethical considerations; larger values of \gamma increase the emphasis on statistical efficiency. More details can be found in Hu, Zhu & Zhang(2015).

Value

References

Hu, J., Zhu, H., & Hu, F. (2015). A unified family of covariate-adjusted response-adaptive designs based on efficiency and ethics. Journal of the American Statistical Association, 110(509), 357–367.

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 = CARAEE_Alloc(
 ptsb.cov = covariates,
 ptsb.t = treatment,
 ptsb.Y = response,
 ptnow.cov = new_patient_cov,
 response = "Binary",
 gamma=1
)
print(result$prob)

Allocation Function of CARA Designs Based on Efficiency and Ethics for Survival 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 using CARA Designs Based on Efficiency and Ethics for survival trial.

Usage

CARAEE_Alloc_Surv(
  ptsb.cov,
  ptsb.t,
  ptsb.Y,
  ptsb.E,
  ptnow.cov,
  gamma,
  event.prob
)

Arguments

Value

Examples

set.seed(123)
n = 40
covariates = cbind(rexp(40),rexp(40))
treatment = sample(c(0, 1), n, replace = TRUE)
survival_time = rexp(n, rate = 1)
censoring = runif(n)
event = as.numeric(survival_time < censoring)

new_patient_cov = c(Z1 = 1, Z2 = 0.5)

result = CARAEE_Alloc_Surv(
 ptsb.cov = covariates,
 ptsb.t = treatment,
 ptsb.Y = survival_time,
 ptsb.E = event,
 ptnow.cov = new_patient_cov,
 gamma=1,
 event.prob = c(0.5,0.7)
)
print(result$prob)

Simulation Function of of CARA Designs Based on Efficiency and Ethics for Binary and Continuous Response.

Description

This function simulates a clinical trial using CARA Designs Based on Efficiency and Ethics (CARAEE) with Binary or Continuous Responses.

Usage

CARAEE_Sim(n, thetaA, thetaB, m0 = 40, pts.cov, response, gamma)

Arguments

Value

A list with the following elements:

Examples

set.seed(123)
results = CARAEE_Sim(n = 400,
                     pts.cov = cbind(rnorm(400), rnorm(400)),
                     thetaA = c(-1, 1, 1),
                     thetaB = c(3, 1, 1),
                     response = "Binary",
                     gamma=1)

Simulation Function of of CARA Designs Based on Efficiency and Ethics for Survival Response.

Description

This function simulates a clinical trial with time-to-event (survival) outcomes using the CARA Designs Based on Efficiency and Ethics for Survival Response(CARAEE). Patient responses are generated under the Cox proportional hazards model, assuming the proportional hazards assumption holds.

Usage

CARAEE_Sim_Surv(
  n,
  thetaA,
  thetaB,
  m0 = 40,
  pts.cov,
  gamma,
  censor.time,
  arrival.rate
)

Arguments

Value

A list with the following elements:

Examples

set.seed(123)

results = CARAEE_Sim_Surv(
 thetaA = c(0.1, 0.1),
 thetaB = c(-1, 0.1),
 n = 400,
 pts.cov = cbind(sample(c(1, 0), 400, replace = TRUE), rnorm(400)),
 gamma=1,
 censor.time = 2,
 arrival.rate = 150
)

Allocation Function of Weighted Balance Ratio 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 Weighted Balance Ratio procedure for binary and continuous response.

Usage

WBR_Alloc(
  ptsb.X,
  ptsb.Z,
  ptsb.Y,
  ptsb.t,
  ptnow.X,
  ptnow.Z,
  weight,
  v = 2,
  response
)

Arguments

Details

This function implements a covariate-adjusted response-adaptive design using a weighted balancing ratio rule (WBR) combined with a DBCD-type allocation function.

The first two steps follow Zhao et al. (2022): the first 2K patients are randomized using restricted randomization with K patients in each treatment group. For patient n > 2K, suppose the prognostic covariates \boldsymbol{Z}_n fall into stratum (k_1^*, \ldots, k_J^*).

Define the weighted imbalance ratio \boldsymbol{r}_{n-1}(\boldsymbol{X}_n) for patient n as:

\boldsymbol{r}_{n-1}(\boldsymbol{X}_n) = w_o \frac{\boldsymbol{N}_{n-1}(\boldsymbol{X}_n)}{N_{n-1}(\boldsymbol{X}_n)} + \sum_{j=1}^{J} w_{m,j} \frac{\boldsymbol{N}_{(j; k_j^*), n-1}(\boldsymbol{X}_n)}{N_{(j; k_j^*), n-1}(\boldsymbol{X}_n)} + w_s \frac{\boldsymbol{N}_{(k_1^*, \ldots, k_J^*), n-1}(\boldsymbol{X}_n)}{N_{(k_1^*, \ldots, k_J^*), n-1}(\boldsymbol{X}_n)},

where w_o, w_{m,1}, ..., w_{m,J}, w_s are non-negative weights that sum to 1.

Based on the patient’s covariates, the target allocation probability \hat{\rho}_{n-1}(\boldsymbol{X}_n) is estimated from previous data.

The final probability of assigning patient n to treatment k is computed using a CARA-type allocation function from Hu and Zhang (2009):

\phi_{n,k}(\boldsymbol{X}_n) = g_k(\boldsymbol{r}_{n-1}(\boldsymbol{X}_n), \hat{\rho}_{n-1}(\boldsymbol{X}_n)) = \frac{ \hat{\rho}_{n-1} \left( \frac{\hat{\rho}_{n-1}}{r_{n-1}} \right)^v } { \hat{\rho}_{n-1} \left( \frac{\hat{\rho}_{n-1}}{r_{n-1}} \right)^v + (1 - \hat{\rho}_{n-1}) \left( \frac{1 - \hat{\rho}_{n-1}}{1 - r_{n-1}} \right)^v },

where v \geq 0 is a tuning parameter controlling the degree of randomness. A value of v = 2 is commonly recommended.

This approach combines stratified covariate balancing with covariate-adjusted optimal targeting. More details can be found in Yu(2025).

Value

References

Yu, J. (2025). A New Family of Covariate-Adjusted Response-Adaptive Randomization Procedures for Precision Medicine (Doctoral dissertation, The George Washington University).

Examples

set.seed(123)

# Generate historical data for 400 patients
ptsb.X = sample(c(1, -1), 400, replace = TRUE)  # predictive covariate
ptsb.Z = cbind(
  sample(c(1, -1), 400, replace = TRUE),         # prognostic covariate 1
  sample(c(1, -1), 400, replace = TRUE)          # prognostic covariate 2
)
ptsb.Y = sample(c(1, 0), 400, replace = TRUE)   # binary responses
ptsb.t = sample(c(1, 0), 400, replace = TRUE)   # treatment assignments

# Incoming patient covariates
ptnow.X = 1
ptnow.Z = c(1, -1)

# Calculate allocation probability using WBR method
result = WBR_Alloc(
  ptsb.X = ptsb.X,
  ptsb.Z = ptsb.Z,
  ptsb.Y = ptsb.Y,
  ptsb.t = ptsb.t,
  ptnow.X = ptnow.X,
  ptnow.Z = ptnow.Z,
  weight = rep(0.25, 4),
  response = "Binary"
)

# View probability of assigning to treatment A
result$prob

Allocation Function of Weighted Balance Ratio Design for Survival 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 Weighted Balance Ratio procedure for survival response.

Usage

WBR_Alloc_Surv(
  ptsb.X,
  ptsb.Z,
  ptsb.Y,
  ptsb.t,
  ptsb.E,
  ptnow.X,
  ptnow.Z,
  v = 2,
  weight
)

Arguments

Value

Examples

set.seed(123)

# Generate historical data for 400 patients
ptsb.X = sample(c(1, -1), 400, replace = TRUE)  # predictive covariate
ptsb.Z = cbind(
  sample(c(1, -1), 400, replace = TRUE),         # prognostic covariate 1
  sample(c(1, -1), 400, replace = TRUE)          # prognostic covariate 2
)
ptsb.Y = rexp(400, rate = 1)                    # observed time
ptsb.E = sample(c(1, 0), 400, replace = TRUE)   # event indicator (1=event, 0=censored)
ptsb.t = sample(c(1, 0), 400, replace = TRUE)   # treatment assignments

# Incoming patient covariates
ptnow.X = 1
ptnow.Z = c(1, -1)

# Calculate allocation probability using WBR for survival response
result = WBR_Alloc_Surv(
  ptsb.X = ptsb.X,
  ptsb.Z = ptsb.Z,
  ptsb.Y = ptsb.Y,
  ptsb.E = ptsb.E,
  ptsb.t = ptsb.t,
  ptnow.X = ptnow.X,
  ptnow.Z = ptnow.Z,
  weight = rep(0.25, 4)
)

# View probability of assigning to treatment A
result$prob

Simulation Function of Weighted Balance Ratio Design for Binary and Continuous Response

Description

This function simulates a trial using Weighted Balance Ratio design for binary and continuous responses.

Usage

WBR_Sim(n, mu, beta, gamma, m0 = 40, pts.X, pts.Z, response, weight, v = 2)

Arguments

Value

#'

Examples

set.seed(123)

# Simulation settings
n = 400                              # total number of patients
mu = c(0.8, 0.8)                     # treatment effects (muA, muB)
beta = c(0.8, -0.8)                  # predictive effect and interaction
gamma = c(0.8, 0.8)                  # prognostic covariate effects
weight = rep(0.25, 4)               # weights for imbalance components

# Generate patient covariates
pts.X = sample(c(1, -1), n, replace = TRUE)  # predictive covariate
pts.Z = cbind(
  sample(c(1, -1), n, replace = TRUE),        # prognostic Z1
  sample(c(1, -1), n, replace = TRUE)         # prognostic Z2
)

# Run simulation for continuous response
result = WBR_Sim(
  n = n,
  mu = mu,
  beta = beta,
  gamma = gamma,
  pts.X = pts.X,
  pts.Z = pts.Z,
  response = "Cont",
  weight = weight
)

Simulation Function of Weighted Balance Ratio Design for Survival Response

Description

This function simulates a trial using Weighted Balance Ratio design for survival responses.

Usage

WBR_Sim_Surv(
  n,
  mu,
  beta,
  gamma,
  m0 = 40,
  pts.X,
  pts.Z,
  censor.time,
  arrival.rate,
  weight,
  v = 2
)

Arguments

Value

A list with the following elements:

Examples

set.seed(123)

# Simulation settings
n = 400                            # total number of patients
mu = 0.5                           # treatment effect (log hazard ratio)
beta = c(0.5, -0.5)                # predictive effect and interaction
gamma = c(0.5, 0.5)                # prognostic covariate effects
censor.time = 2                   # maximum censoring time (years)
arrival.rate = 1.5                # arrival rate per year
weight = rep(0.25, 4)             # imbalance weights for overall, margins, and stratum

# Generate patient covariates
pts.X = sample(c(1, -1), n, replace = TRUE)  # predictive covariate
pts.Z = cbind(
  sample(c(1, -1), n, replace = TRUE),        # prognostic Z1
  sample(c(1, -1), n, replace = TRUE)         # prognostic Z2
)

# Run simulation for survival outcome
result = WBR_Sim_Surv(
  n = n,
  mu = mu,
  beta = beta,
  gamma = gamma,
  pts.X = pts.X,
  pts.Z = pts.Z,
  censor.time = censor.time,
  arrival.rate = arrival.rate,
  weight = weight

)

Allocation Function of Zhao's New Design for Binary and Continuous Response

Description

Calculating the probability of assigning the upcoming patient to treatment A based on the patient's predictive covariates and the previous patients' predictive covariates and responses for Zhao's New procedure.

Usage

ZhaoNew_Alloc(
  ptsb.X,
  ptsb.Z,
  ptsb.t,
  ptsb.Y,
  ptnow.X,
  ptnow.Z,
  response,
  omega,
  p = 0.8
)

Arguments

Details

This function implements a stratified covariate-adjusted response-adaptive design that balances treatment allocation within and across strata defined by prognostic covariates.

The first 2K patients are randomized using a restricted randomization procedure, with K patients assigned to each treatment. For patient n > 2K, let \mathbf{X}_n denote predictive covariates, and \mathbf{Z}_n denote stratification covariates, placing the patient into stratum (k_1^*, \ldots, k_I^*).

Based on the covariate profiles and responses of the first n-1 patients, we estimate the target allocation probability \widehat{\rho}(\mathbf{X}_n) for the current patient.

If assigned to treatment 1, we compute imbalance measures between actual and target allocation at three levels:

• Overall imbalance: D_n^{(1)}(\mathbf{X}_n),

• Marginal imbalance: D_n^{(1)}(i; k_i^*; \mathbf{X}_n),

• Within-stratum imbalance: D_n^{(1)}(k_1^*, \ldots, k_I^*; \mathbf{X}_n).

These are combined into a weighted imbalance function:

\operatorname{Imb}_n^{(1)}(\mathbf{X}_n) = w_o [D_n^{(1)}(\mathbf{X}_n)]^2 + \sum_{i=1}^I w_{m,i} [D_n^{(1)}(i; k_i^*; \mathbf{X}_n)]^2 + w_s [D_n^{(1)}(k_1^*, \ldots, k_I^*; \mathbf{X}_n)]^2.

A similar imbalance \operatorname{Imb}_n^{(2)}(\mathbf{X}_n) is defined for treatment 2. The patient is then assigned to treatment 1 with probability:

\phi_n = g\left( \operatorname{Imb}_n^{(1)}(\mathbf{X}_n) - \operatorname{Imb}_n^{(2)}(\mathbf{X}_n) \right),

where g(x) is a biasing function satisfying g(-x) = 1 - g(x) and g(x) < 0.5 for x \geq 0. One common choice is Efron's biased coin function:

g(x) = \begin{cases} q, & \text{if } x > 0 \\ 0.5, & \text{if } x = 0 \\ p, & \text{if } x < 0 \end{cases}

with p > 0.5 and q = 1 - p.

This design unifies covariate-adjusted response-adaptive randomization and marginal/stratified balance. It reduces to Hu & Hu's design when \mathbf{X}_n is excluded, and to CARA designs when \mathbf{Z}_n is ignored. More detail can be found in Zhao et al.(2022).

Value

References

Zhao, W., Ma, W., Wang, F., & Hu, F. (2022). Incorporating covariates information in adaptive clinical trials for precision medicine. Pharmaceutical Statistics, 21(1), 176–195.

Examples

set.seed(123)
ptsb.X = sample(c(1, -1), 400, replace = TRUE)
ptsb.Z = cbind(
 sample(c(1, -1), 400, replace = TRUE),
 sample(c(1, -1), 400, replace = TRUE)
)
ptsb.Y = sample(c(1, 0), 400, replace = TRUE)
ptsb.t = sample(c(1, 0), 400, replace = TRUE)

## Incoming patient
ptnow.X = 1
ptnow.Z = c(1, -1)

## Run allocation probability calculation
prob = ZhaoNew_Alloc(
 ptsb.X = ptsb.X,
 ptsb.Z = ptsb.Z,
 ptsb.Y = ptsb.Y,
 ptsb.t = ptsb.t,
 ptnow.X = ptnow.X,
 ptnow.Z = ptnow.Z,
 response = "Binary",
 omega = rep(0.25, 4),
 p = 0.8
)

## View allocation probability for treatment A
prob

Allocation Function of Zhao's Design for Survival 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 Zhao's New procedure for survival trials.

Usage

ZhaoNew_Alloc_Surv(
  ptsb.X,
  ptsb.Z,
  ptsb.t,
  ptsb.Y,
  ptsb.E,
  ptnow.X,
  ptnow.Z,
  omega,
  p = 0.8
)

Arguments

Value

Examples

set.seed(123)

# Generate historical data for 400 patients
ptsb.X = sample(c(1, -1), 400, replace = TRUE)  # predictive covariate
ptsb.Z = cbind(
 sample(c(1, -1), 400, replace = TRUE),         # prognostic covariate 1
 sample(c(1, -1), 400, replace = TRUE)          # prognostic covariate 2
)
ptsb.Y = rexp(400, rate = 1)                    # survival time (response)
ptsb.E = sample(c(1, 0), 400, replace = TRUE)   # event indicator (1 = event, 0 = censored)
ptsb.t = sample(c(1, 0), 400, replace = TRUE)   # treatment assignment

# Incoming patient covariates
ptnow.X = 1
ptnow.Z = c(1, -1)

# Allocation probability calculation
prob = ZhaoNew_Alloc_Surv(
 ptsb.X = ptsb.X,
 ptsb.Z = ptsb.Z,
 ptsb.Y = ptsb.Y,
 ptsb.E = ptsb.E,
 ptsb.t = ptsb.t,
 ptnow.X = ptnow.X,
 ptnow.Z = ptnow.Z,
 omega = rep(0.25, 4),
 p = 0.8
)

# View the allocation probability for treatment A
prob

Simulation Function of Zhao's New Design for Binary and Continuous Response

Description

This function simulates a trial using Zhao's new design for binary and continuous responses.

Usage

ZhaoNew_Sim(
  n,
  mu,
  beta,
  gamma,
  m0 = 40,
  pts.X,
  pts.Z,
  response,
  omega,
  p = 0.8
)

Arguments

Value

Examples

set.seed(123)

# Simulation settings
n = 400                    # total number of patients
m0 = 40                    # initial burn-in sample size
mu = c(0.5, 0.8)           # potential means (for continuous or logistic link)
beta = c(1, 1)             # treatment effect and predictive covariate effect
gamma = c(0.1, 0.5)        # prognostic covariate effects
omega = rep(0.25, 4)       # imbalance weights
p = 0.8                    # biased coin probability

# Generate patient covariates
pts.X = sample(c(1, -1), n, replace = TRUE)  # predictive covariate
pts.Z = cbind(
  sample(c(1, -1), n, replace = TRUE),        # prognostic Z1
  sample(c(1, -1), n, replace = TRUE)         # prognostic Z2
)

# Run the simulation (binary response setting)
result = ZhaoNew_Sim(
  n = n,
  mu = mu,
  beta = beta,
  gamma = gamma,
  m0 = m0,
  pts.X = pts.X,
  pts.Z = pts.Z,
  response = "Binary",
  omega = omega,
  p = p
)

Simulation Function for Zhao's New Design for Survival Trial

Description

This function simulates a clinical trial using Zhao's New design for survival responses.

Usage

ZhaoNew_Sim_Surv(
  n,
  mu,
  beta,
  gamma,
  m0 = 40,
  pts.X,
  pts.Z,
  censor.time,
  arrival.rate,
  omega,
  p = 0.8
)

Arguments

Value

A list with the following elements:

Examples

set.seed(123)

# Simulation settings
n = 400                    # total number of patients
m0 = 40                    # initial burn-in sample size
mu = 0.5           # potential means (for continuous or logistic link)
beta = c(1, 1)             # treatment effect and predictive covariate effect
gamma = c(0.1, 0.5)        # prognostic covariate effects
omega = rep(0.25, 4)       # imbalance weights
p = 0.8                    # biased coin probability
censor.time = 2            # maximum censoring time
arrival.rate = 150         # arrival rate of patients
# Generate patient covariates
pts.X = sample(c(1, -1), n, replace = TRUE)  # predictive covariate
pts.Z = cbind(
  sample(c(1, -1), n, replace = TRUE),        # prognostic Z1
  sample(c(1, -1), n, replace = TRUE)         # prognostic Z2
)

# Run the simulation (binary response setting)
result = ZhaoNew_Sim_Surv(
  n = n,
  mu = mu,
  beta = beta,
  gamma = gamma,
  m0 = m0,
  pts.X = pts.X,
  pts.Z = pts.Z,
  omega = omega,
  p = p,
  censor.time = censor.time,
  arrival.rate = arrival.rate
)