dyldDBCD_Bin: Hu and Zhang's Doubly Biased Coin Design with Delayed Binary...

In grouprar: Group Response Adaptive Randomization for Clinical Trials

Hu and Zhang's Doubly Biased Coin Design with Delayed Binary Response

Description

Simulating Hu and Zhang's doubly biased coin design with delayed binary response (number of arms \ge 2) in a clinical trial context. (Inference: two-sided t-test for two arms or chi-square test for more than two arms.)

Usage

dyldDBCD_Bin(n0 = 20, p, k, ssn, ent.param, rspT.dist, rspT.param,
             theta0 = NULL, target.alloc = "RPW", r = 2, nsim = 2000,
             mRate = NULL, alpha = 0.05, allocation = "DBCD",
             erade.alpha = 0.5, lower.bound = 0, test.fun = NULL,
             typeI = FALSE, seed = NULL)

Arguments

n0	A positive integer and a multiple of k. The number of initial patients assigned by restricted randomization for initial parameter estimation.
p	A vector of length k with values between 0 and 1. The true success rates of the treatments, used to generate data for the simulations.
k	A positive integer. The number of treatment groups in the trial (k \ge 2).
ssn	A positive integer. The total number of participants in each simulated trial.
ent.param	A positive number. The mean time between consecutive patient arrivals. Inter-arrival times are drawn from an exponential distribution with this mean.
rspT.dist	The distribution of the time from enrollment until the response is observed. One of "exponential", "normal" or "uniform".
rspT.param	A numeric vector of parameters for the response-time distributions, one distribution per treatment and response, ordered as (treatment 1 failure, treatment 1 success, treatment 2 failure, treatment 2 success, ...). For "exponential" give the 2k means. For "normal" give 2k (mean, sd) pairs and for "uniform" give 2k (min, max) pairs, that is 4k values in total. Normal times below 0 are set to 0. For example, with k = 2 and exponential times, rspT.param = c(3, 2, 4, 1) gives mean times 3 and 2 for failures and successes on treatment 1, and 4 and 1 on treatment 2.
theta0	A vector of length k used to smooth the success-rate estimates, \hat p_k = (S_k + \theta_{0k})/(N_k + 1), where S_k and N_k are the number of observed successes and observed responses on treatment k. If NULL (default), all values are 0.5.
target.alloc	Desired allocation proportion. One of "Neyman", "RSIHR", "RPW", "WeisUrn", "OptimalNeyman" or "OptimalRSIHR". The default is "RPW". See Details.
r	A non-negative number. The tuning parameter of Hu and Zhang's allocation function, used when allocation = "DBCD". Values between 2 and 4 are common. The default value is 2.
nsim	A positive integer. The number of simulated trials, with a default value of 2000.
mRate	A number between 0 and 1 giving the probability that a response is missing. Missingness is simulated completely at random (MCAR), and missing responses are never observed and are excluded from estimation and testing. The default NULL means no missing responses. As in Zhai et al. (2024), missing responses are excluded from the estimates, the failure rate and the test, while the allocation proportions (both those used by the allocation function and the reported ones) count all enrolled patients.
alpha	A number between 0 and 1. The significance level of the two-sided test, with a default value of 0.05.
allocation	The allocation function. "DBCD" (default) uses the allocation function of Hu and Zhang (2004) with parameter r. "ERADE" uses the efficient randomized adaptive design of Hu, Zhang and He (2009), in its multi-arm version (Alkhnefr, Hu and Zhai, 2025), with parameter erade.alpha. See Details.
erade.alpha	A number between 0 and 1. The degree of randomization of ERADE, used when allocation = "ERADE". Smaller values push the allocation more strongly towards the target, and values of 1/2 or 2/3 are recommended. The default is 0.5.
lower.bound	A number between 0 and 1/k. The smallest allowed target proportion of each arm for the targets "OptimalNeyman" and "OptimalRSIHR". The default is 0.
test.fun	An optional function function(outcome, assignment) that returns the p-value of a test for one simulated trial, where outcome holds the observed responses and assignment the treatment labels (integers 1 to k); missing responses are removed first. The null hypothesis is rejected when the p-value is at most alpha. The default NULL uses the built-in two-sided tests: the t-test for two arms and the Wald chi-squared test of equal means for more than two arms (for binary responses, if some arms have no variability, the variances are computed from (S_k + 1)/(n_k + 2) as in Agresti and Caffo, 2000).
typeI	Logical. If TRUE, the simulation is repeated under the null hypothesis, with every arm's success rate set to the average of p, and the rejection rate is reported as type I error. The default is FALSE.
seed	An optional integer passed to set.seed before simulating, so that the results can be reproduced. The default NULL leaves the random number generator unchanged.

Details

Hu and Zhang's doubly biased coin design with delayed binary response uses the following allocation scheme.

(a) Initially, due to limited information about treatment efficacy, the first n0 patients are assigned to the K treatments using restricted randomization (as described by Rosenberger and Lachin, 2002).

(b) For m \ge n_0, patient (m+1) is allocated to treatment k with probability p_{m+1, k}, which depends on the responses available at that time and on the estimated target allocation through the allocation function g_k proposed by Hu and Zhang (2004).

For a more comprehensive description of the procedure, please refer to the paper 'Doubly adaptive biased coin designs with delayed responses' by Hu et al. (2008).

Target allocations. With q_k = 1 - p_k, "Neyman" and "RSIHR" are proportional to \sqrt{p_k q_k} and \sqrt{p_k}. For two arms these are the Neyman allocation and the optimal allocation of Rosenberger et al. (2001), and for more arms they are simple generalizations. "RPW" and "WeisUrn" are proportional to 1/q_k, the limiting allocation of the randomized play-the-winner rule and of Wei's urn. "OptimalNeyman" and "OptimalRSIHR" are the k-arm optimal allocations of Tymofyeyev, Rosenberger and Hu (2007). They minimize the total sample size and the expected number of failures, respectively, for a fixed noncentrality parameter of the chi-squared test of equal success rates, with every proportion at least lower.bound. For two arms and lower.bound = 0 they equal "Neyman" and "RSIHR". For three or more arms the optimum without a lower bound can assign no patients to the middle arms, so a positive lower.bound such as 0.1 is recommended.

Allocation functions. With allocation = "ERADE" and \alpha = erade.alpha, arm k receives the next patient (in group designs, each patient of the next group) with probability \alpha\hat\rho_k if it is over-allocated (N_k/m > \hat\rho_k), \hat\rho_k if it is on target, and \hat\rho_k + (1-\alpha)\sum_{j \in S}\hat\rho_j/|T| if it is under-allocated, where S and T are the sets of over- and under-allocated arms, N_k/m is the current allocation proportion and \hat\rho is the estimated target (Alkhnefr, Hu and Zhai, 2025, eq. (1)). For two arms this is the ERADE of Hu, Zhang and He (2009). ERADE attains the lower bound of the asymptotic variance of the allocation proportions, so its allocation is less variable than that of the DBCD.

Value

An object of class "grouprar", a list that is printed as a short summary (see print.grouprar), with the following elements.

method	The name of the procedure.
sample size	The total sample size (Total) and the expected number of patients with observed responses (Effective Size).
parameter	The true success rates used in the simulations, named pA, pB, ...
propotion	The mean allocation proportion of each arm over the simulations, named treatment A, treatment B, ...
sd of propotion	The standard deviation of the allocation proportion of each arm over the simulations.
failure rate	The mean failure rate over the simulations.
sd of failure rate	The standard deviation of the failure rate (or mean response) over the simulations.
power	The proportion of simulated trials that reject the null hypothesis of equal success rates. Simulations in which the test cannot be computed are dropped.
data: failureRate	The failure rate (or mean response) of each simulated trial.
data: test	The test decision of each simulated trial (1 = reject).
data: assignment	The treatment assignments of the last simulated trial.
data: propotion	A data frame with the allocation proportions of each simulated trial.
data: allocation	An nsim by ssn matrix with the treatment assignments of every simulated trial.
type I error	Only if typeI = TRUE. The rejection rate under the null hypothesis.
duration	The mean time from the first enrollment to the last observed response. Trials without any observed response have duration NA and are left out of the mean.
sd of duration	The standard deviation of the duration over the simulations.
enrollment duration	The mean time from the first to the last enrollment.
data: duration	The duration of each simulated trial.
data: enrollment	The enrollment duration of each simulated trial.

References

Alkhnefr, N., Hu, F. and Zhai, G. (2025). Efficient randomized adaptive designs for multi-arm clinical trials. Statistical Methods in Medical Research, 34(9), 1886-1898. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1177/09622802251362644")}

Hu, F. and Zhang, L. X. (2004). Asymptotic properties of doubly adaptive biased coin designs for multitreatment clinical trials. The Annals of Statistics, 32(1), 268-301. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1214/aos/1079120137")}

Hu, F., Zhang, L. X. and He, X. (2009). Efficient randomized-adaptive designs. The Annals of Statistics, 37(5A), 2543-2560. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1214/08-AOS655")}

Hu, F., Zhang, L. X., Cheung, S. H. and Chan, W. S. (2008). Doubly adaptive biased coin designs with delayed responses. Canadian Journal of Statistics, 36(4), 541-559. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/cjs.5550360404")}

Rosenberger, W. F. and Lachin, J. M. (2002). Randomization in Clinical Trials: Theory and Practice. John Wiley & Sons.

Rosenberger, W. F., Stallard, N., Ivanova, A., Harper, C. N. and Ricks, M. L. (2001). Optimal adaptive designs for binary response trials. Biometrics, 57(3), 909-913. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/j.0006-341X.2001.00909.x")}

Tymofyeyev, Y., Rosenberger, W. F. and Hu, F. (2007). Implementing optimal allocation in sequential binary response experiments. Journal of the American Statistical Association, 102(477), 224-234. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1198/016214506000000906")}

Examples

# a simple use
## Arguments for generating the simulated data
### For response simulation
p = c(0.6, 0.8)
k = 2
ssn = 100
### for entry time and response time simulation
ent.param = 0.7
rspT.dist = "exponential"
rspT.param = c(1, 1, 3, 1)

## Arguments for the design
n0 = 20
target.alloc = "RSIHR"

res <- dyldDBCD_Bin(n0 = n0, p = p, k = k, ssn = ssn, ent.param = ent.param,
                    rspT.dist = rspT.dist, rspT.param = rspT.param, theta0 = NULL,
                    target.alloc = target.alloc, r = 2, nsim = 100,
                    mRate = NULL, alpha = 0.05)
res

grouprar documentation built on Oct. 9, 2026, 9:07 a.m.