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

In grouprar: Group Response Adaptive Randomization for Clinical Trials

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

Description

Simulating Hu and Zhang's doubly biased coin design with delayed continuous 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_Cont(n0 = 20, theta, k, ssn, ent.param, rspT.dist, rspT.param,
              target.alloc = "Neyman", 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.
theta	A numerical vector of length 2k giving the true mean and variance of each treatment, used to generate data for the simulations. For example, with k = 2 use theta = c(13, 4.0^2, 15, 2.5^2).
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 distribution of each treatment. For "exponential" give the k means. For "normal" give k (mean, sd) pairs and for "uniform" give k (min, max) pairs, that is 2k values in total. Normal times below 0 are set to 0. For example, with 3 treatments and normal times with (mean, sd) pairs (3, 2), (2, 1) and (4, 1), use rspT.param = c(3, 2, 2, 1, 4, 1).
target.alloc	Desired allocation proportion. One of "Neyman", "ZR", "OptimalNeyman" or "DaOptimal". The default is "Neyman". 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 "ZR" and "OptimalNeyman". 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 the mean of every arm set to the average of the means in theta and the variance of each arm kept, 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 continuous 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. "Neyman" is proportional to \sigma_k. "ZR" is the allocation of Zhang and Rosenberger (2006), which minimizes the total expected response (smaller responses are better, and all means must be positive). For two arms it is their rule (7): \rho_1 = \sigma_1\sqrt{\mu_2}/(\sigma_1\sqrt{\mu_2} + \sigma_2\sqrt{\mu_1}) when this assigns more patients to the arm with the smaller mean, and 1/2 otherwise. For three or more arms it minimizes the total expected response for a fixed noncentrality parameter of the chi-squared test of equal means, with every proportion at least lower.bound, the continuous analogue of Tymofyeyev, Rosenberger and Hu (2007). "OptimalNeyman" minimizes the total sample size under the same constraints and equals "Neyman" for two arms with lower.bound = 0. For three or more arms a positive lower.bound such as 0.1 is recommended with "ZR" and "OptimalNeyman". "DaOptimal" is proportional to \sigma_k^{4/3}. While the target cannot be estimated (for example with fewer than two responses in an arm) equal allocation is used.

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 means and variances used in the simulations, named muA, sigma2A, muB, ...
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 response over the simulations (for continuous responses this element holds the mean response, not a failure rate).
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 means. 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.

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")}

Zhang, L. and Rosenberger, W. F. (2006). Response-adaptive randomization for clinical trials with continuous outcomes. Biometrics, 62(2), 562-569. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/j.1541-0420.2005.00496.x")}

Examples

# a simple use
## Arguments for generating the simulated data
### For response simulation
theta = c(13, 4.0^2, 15, 2.5^2)
k = 2
ssn = 88

### for entry time and response time simulation
ent.param = 5
rspT.param = rep(10, 2)
rspT.dist = "exponential"

## Arguments for the design
target.alloc = "Neyman"

res <- dyldDBCD_Cont(n0 = 10, theta = theta, k = k, ssn = ssn, ent.param = ent.param,
                     rspT.dist = rspT.dist, rspT.param = rspT.param,
                     target.alloc = target.alloc, r = 2, nsim = 100,
                     mRate = 0.2, alpha = 0.05)

# View the output (a list of all results)
res

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