Package {grouprar}

grouprar-package: Group Response-Adaptive Randomization for Clinical Trials

Description

Implements group response-adaptive randomization procedures, which include standard (non-group) response-adaptive randomization methods as special cases. The package also handles delayed and missing responses, which broadens its use in real-world trials. It offers functions for simulating a variety of response-adaptive randomization procedures, to help guide the choice of design for a clinical trial.

Urn designs: RPWRule, WeiUrn, PolyaUrn, DLRule, GDLRule, Bai.Hu.Shen.Urn and BirthDeathUrn, with CRDesign (complete randomization) as a non-adaptive benchmark.

Doubly adaptive biased coin designs (DBCD) for binary and continuous responses, with immediate or delayed responses, for individual patients or groups of patients: DBCD_Bin, DBCD_Cont, dyldDBCD_Bin, dyldDBCD_Cont, Group.DBCD_Bin, Group.DBCD_Cont, Group.dyldDBCD_Bin and Group.dyldDBCD_Cont. These designs support any number of arms, the allocation function of Hu and Zhang (2004) or the multi-arm efficient randomized adaptive design (ERADE), several target allocations including the k-arm optimal allocations of Tymofyeyev, Rosenberger and Hu (2007) with a lower bound, and missing responses.

Further tools: nextAlloc computes the allocation probabilities of the next patient or group in an ongoing trial, and sqMonitor and sqBoundary add group sequential monitoring with alpha spending to two-arm designs (Zhu and Hu, 2010). Every design accepts a user-supplied test (test.fun), can estimate the type I error (typeI) and takes a seed. The results are objects of class "grouprar" with print and summary methods (print.grouprar), and contain the allocation sequences of all simulated trials and, for delayed designs, the trial duration.

For details of the methods and algorithms, see the following references: Wei, L. J. (1979) doi:10.1214/aos/1176344614; Wei, L. J. and Durham, S. (1978) doi:10.1080/01621459.1978.10480109; Durham, S. D., Flournoy, N. and Li, W. (1998) doi:10.2307/3315771; Ivanova, A., Rosenberger, W. F., Durham, S. D. and Flournoy, N. (2000) https://www.jstor.org/stable/25053121; Bai, Z. D., Hu, F. and Shen, L. (2002) doi:10.1006/jmva.2001.1987; Ivanova, A. (2003) doi:10.1007/s001840200220; Hu, F. and Zhang, L. X. (2004) doi:10.1214/aos/1079120137; Hu, F. and Rosenberger, W. F. (2006, ISBN:978-0-471-65396-7); Zhang, L. X., Chan, W. S., Cheung, S. H. and Hu, F. (2007) https://www.jstor.org/stable/26432528; Zhang, L. and Rosenberger, W. F. (2006) doi:10.1111/j.1541-0420.2005.00496.x; Tymofyeyev, Y., Rosenberger, W. F. and Hu, F. (2007) doi:10.1198/016214506000000906; Hu, F., Zhang, L. X., Cheung, S. H. and Chan, W. S. (2008) doi:10.1002/cjs.5550360404; Hu, F., Zhang, L. X. and He, X. (2009) doi:10.1214/08-AOS655; Zhu, H. and Hu, F. (2010) doi:10.1214/10-AOS796; Zhai, G., Li, Y., Zhang, L. and Hu, F. (2024) doi:10.1002/sim.10220; Alkhnefr, N., Hu, F. and Zhai, G. (2025) doi:10.1177/09622802251362644.

Author(s)

Guannan Zhai guannanzhai1996@gmail.com; Feifang Hu feifang@gwu.edu.

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.

Bai, Z. D., Hu, F. and Shen, L. (2002). An adaptive design for multi-arm clinical trials. Journal of Multivariate Analysis, 81(1), 1-18.

Durham, S. D., Flournoy, N. and Li, W. (1998). A sequential design for maximizing the probability of a favourable response. Canadian Journal of Statistics, 26(3), 479-495.

Hu, F. and Rosenberger, W. F. (2003). Optimality, variability, power: evaluating response-adaptive randomization procedures for treatment comparisons. Journal of the American Statistical Association, 98(463), 671-678.

Hu, F. and Rosenberger, W. F. (2006). The Theory of Response-Adaptive Randomization in Clinical Trials. John Wiley & Sons.

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.

Hu, F., Zhang, L. X. and He, X. (2009). Efficient randomized-adaptive designs. The Annals of Statistics, 37(5A), 2543-2560.

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.

Ivanova, A. (2003). A play-the-winner-type urn design with reduced variability. Metrika, 58, 1-13.

Ivanova, A., Rosenberger, W. F., Durham, S. D. and Flournoy, N. (2000). A birth and death urn for randomized clinical trials: asymptotic methods. Sankhya, Series B, 62(1), 104-118.

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.

Wei, L. J. (1979). The generalized Polya's urn design for sequential medical trials. The Annals of Statistics, 7(2), 291-296.

Wei, L. J. and Durham, S. (1978). The randomized play-the-winner rule in medical trials. Journal of the American Statistical Association, 73(364), 840-843.

Zelen, M. (1969). Play the winner rule and the controlled clinical trial. Journal of the American Statistical Association, 64(325), 131-146.

Zhai, G., Li, Y., Zhang, L. and Hu, F. (2024). Group response-adaptive randomization with delayed and missing responses. Statistics in Medicine, 43(27), 5047-5059.

Zhang, L. and Rosenberger, W. F. (2006). Response-adaptive randomization for clinical trials with continuous outcomes. Biometrics, 62(2), 562-569.

Zhang, L. X., Chan, W. S., Cheung, S. H. and Hu, F. (2007). A generalized drop-the-loser urn for clinical trials with delayed responses. Statistica Sinica, 17(1), 387-409.

Zhu, H. and Hu, F. (2010). Sequential monitoring of response-adaptive randomized clinical trials. The Annals of Statistics, 38(4), 2218-2241.

Bai Hu Shen's Urn

Description

Bai, Hu, and Shen (2002) proposed an adaptive design for multi-arm clinical trials. The allocation probabilities adapt to the performance of the patients already treated: a success on a treatment increases the chance that the next patient is assigned to it, and a failure moves probability to the other treatments in proportion to their estimated success rates. This function simulates the Bai, Hu, and Shen urn with two-sided hypothesis testing in a clinical trial context.

Usage

Bai.Hu.Shen.Urn(k, p, ssn, Y0 = NULL, nsim = 2000, alpha = 0.05,
                test.fun = NULL, typeI = FALSE, seed = NULL)

Arguments

Details

Bai, Hu and Shen's urn can be described as follows. An urn initially contains balls of K types, where balls of types 1, 2, \ldots, K represent treatments 1, 2, \ldots, K. A ball is drawn at random from the urn and, if it is of type k, the next patient is assigned to treatment k. After the response is observed, the urn composition is updated. A success on treatment k adds one ball of type k to the urn. A failure on treatment k adds \hat p_j/(\hat M - \hat p_k) balls of each other type j \ne k, where \hat p_j = (S_j + 1)/(N_j + 1) is the current estimate of the success rate of treatment j (S_j successes among N_j patients) and \hat M = \hat p_1 + \cdots + \hat p_K. The estimates use the responses of the previous patients only. This is adaptive design 3 of Bai, Hu and Shen (2002), the design proposed in the paper. (Versions of grouprar before 0.2.0 used the true success rates instead, which is their design 2.)

Value

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

References

Bai, Z. D., Hu, F. and Shen, L. (2002). An adaptive design for multi-arm clinical trials. Journal of Multivariate Analysis, 81(1), 1-18. doi:10.1006/jmva.2001.1987

Examples

## a simple use
bhs.res <- Bai.Hu.Shen.Urn(k = 3,
                           p = c(0.7, 0.8, 0.6),
                           ssn = 200,
                           Y0 = NULL,
                           nsim = 100,
                           alpha = 0.05)

## view the output
bhs.res

  ## view all simulation settings
  bhs.res[["method"]]
  bhs.res[["parameter"]]

  ## view the simulation results
  bhs.res[["propotion"]]
  bhs.res[["failure rate"]]
  bhs.res[["power"]]
  bhs.res[["data: assignment"]]
  

Birth and Death Urn

Description

Simulating the birth and death urn procedure (number of arms \ge 2) with two-sided hypothesis testing in a clinical trial context.

Usage

BirthDeathUrn(k, p, ssn, Y0 = NULL, nsim = 2000, alpha = 0.05,
              test.fun = NULL, typeI = FALSE, seed = NULL)

Arguments

Details

The birth and death urn works as follows. Initially the urn contains balls of K treatment types and an immigration ball. A ball is drawn at random with replacement. If it is the immigration ball, one ball of each treatment type is added to the urn, no patient is treated, and the next ball is drawn. This is repeated until a type i ball (i = 1, \ldots, K) is drawn, and then the patient is assigned to treatment i. After a success a type i ball is added to the urn, and after a failure a type i ball is removed (Hu and Rosenberger, 2006). More details can be found in Ivanova et al. (2000).

When the best success rate is at least 1/2 the urn concentrates on that arm, and the other arms can end with very few patients. The asymptotic tests can then be well above their nominal level under the null hypothesis, especially for K \ge 3. Use typeI = TRUE to check the type I error.

Value

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

References

Hu, F. and Rosenberger, W. F. (2006). The Theory of Response-Adaptive Randomization in Clinical Trials. John Wiley & Sons.

Ivanova, A., Rosenberger, W. F., Durham, S. D. and Flournoy, N. (2000). A birth and death urn for randomized clinical trials: asymptotic methods. Sankhya, Series B, 62(1), 104-118.

Examples

## a simple use
bd.res <- BirthDeathUrn(k = 3, p = c(0.6, 0.7, 0.6), ssn = 200, Y0 = NULL,
                        nsim = 100, alpha = 0.05)

## view the output
bd.res

  ## view all simulation settings
  bd.res[["method"]]
  bd.res[["parameter"]]

  ## view the simulation results
  bd.res[["propotion"]]
  bd.res[["failure rate"]]
  bd.res[["power"]]
  bd.res[["data: assignment"]]
  

Complete Randomization

Description

Simulating complete randomization with two-sided hypothesis testing in a clinical trial context.

Usage

CRDesign(k, p, ssn, nsim = 2000, alpha = 0.05, test.fun = NULL, typeI = FALSE,
         seed = NULL)

Arguments

Details

Complete randomization assigns each participant to one of the treatment groups with equal probability, independently of all other participants.

Value

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

Examples

## a simple use
CR.res <- CRDesign(k = 3, p = c(0.7, 0.8, 0.6), ssn = 400, nsim = 100)
## view the output
CR.res

  ## view all simulation settings
  CR.res[["method"]]
  CR.res[["parameter"]]
  ## view the simulation results
  CR.res[["propotion"]]
  CR.res[["failure rate"]]
  CR.res[["power"]]
  CR.res[["data: assignment"]]
  

## estimate the type I error as well, with a reproducible seed
res <- CRDesign(k = 2, p = c(0.6, 0.8), ssn = 100, nsim = 50, typeI = TRUE, seed = 1)
summary(res)

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

Description

Simulating Hu and Zhang's doubly biased coin design with 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

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

Arguments

Details

The objective of Hu and Zhang's doubly biased coin design is to allocate patients sequentially while closely approximating the desired allocation proportion, which is a function of certain unknown parameters related to the response variable under each treatment.

The process begins by assigning n0 patients to treatment groups using restricted randomization and collecting their responses. Initial parameter estimates for the response variable are then obtained for each treatment group. Based on these estimates, the desired allocation proportion is calculated. Then Hu and Zhang's allocation function is applied to determine the probabilities for the next patient to be assigned to each treatment group, which drive the allocation proportion towards the desired one. This process is repeated sequentially for each patient until the predetermined number of patients has been allocated.

This methodology was introduced by Hu and Zhang (2004) in their paper 'Asymptotic properties of doubly adaptive biased coin designs for multitreatment clinical trials'.

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.

Sequential monitoring. With monitor = sqMonitor(t, spend) (two arms only) the trial is analysed at the information times t. Looks are taken after ceiling(t * ssn) patients. At each look the Wald statistic Z = (\hat\theta_1 - \hat\theta_2)/\sqrt{\hat v_1/n_1 + \hat v_2/n_2} is computed from all patients enrolled so far, with \hat p_k(1 - \hat p_k) as variance estimates (missing responses are excluded), and the trial stops and rejects the null hypothesis as soon as |Z| reaches the boundary of sqBoundary. Zhu and Hu (2010) showed that under the DBCD the sequential statistics are asymptotically a Brownian motion in the information time, so alpha spending boundaries keep the type I error. Their theory covers two arms and the DBCD, and the same boundaries are used with ERADE. The result then also contains the stopping probability at each look and the expected sample size.

Value

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

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. 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. 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. doi:10.1214/08-AOS655

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. 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. doi:10.1198/016214506000000906

Zhu, H. and Hu, F. (2010). Sequential monitoring of response-adaptive randomized clinical trials. The Annals of Statistics, 38(4), 2218-2241. doi:10.1214/10-AOS796

Examples

res <- DBCD_Bin(n0 = 20, p = c(0.7, 0.8), k = 2, ssn = 300, theta0 = NULL,
                target.alloc = "RPW", r = 2, nsim = 50, mRate = NULL, alpha = 0.05)
res

## ERADE instead of the DBCD allocation function
res.erade <- DBCD_Bin(n0 = 20, p = c(0.7, 0.8), k = 2, ssn = 200, target.alloc = "RSIHR",
                      nsim = 50, allocation = "ERADE", erade.alpha = 0.5, seed = 1)
res.erade

## three arms with the optimal RSIHR target of Tymofyeyev, Rosenberger and Hu (2007)
res.opt <- DBCD_Bin(n0 = 30, p = c(0.5, 0.7, 0.8), k = 3, ssn = 200,
                    target.alloc = "OptimalRSIHR", lower.bound = 0.1, nsim = 20, seed = 1)
res.opt[["propotion"]]

## group sequential monitoring with two interim looks (O'Brien-Fleming-type spending)
res.sq <- DBCD_Bin(n0 = 20, p = c(0.6, 0.8), k = 2, ssn = 200, target.alloc = "RSIHR",
                   nsim = 50, monitor = sqMonitor(c(1/3, 2/3)), seed = 1)
res.sq[["stopping probability"]]
res.sq[["expected sample size"]]

## a user-supplied test: Fisher's exact test
fisher <- function(outcome, assignment)
  fisher.test(table(factor(assignment, 1:2), factor(outcome, 0:1)))$p.value
res.f <- DBCD_Bin(n0 = 20, p = c(0.6, 0.8), k = 2, ssn = 100, nsim = 20,
                  test.fun = fisher, seed = 1)
res.f[["power"]]

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

Description

Simulating Hu and Zhang's doubly biased coin design with 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

DBCD_Cont(n0 = 20, theta, k, ssn, theta0 = NULL, target.alloc = "Neyman",
          r = 2, nsim = 2000, alpha = 0.05, allocation = "DBCD",
          erade.alpha = 0.5, lower.bound = 0, monitor = NULL, test.fun = NULL,
          typeI = FALSE, seed = NULL)

Arguments

Details

The objective of Hu and Zhang's doubly biased coin design is to allocate patients sequentially while closely approximating the desired allocation proportion, which is a function of certain unknown parameters related to the response variable under each treatment.

The process begins by assigning n0 patients to treatment groups using restricted randomization and collecting their responses. Initial parameter estimates for the response variable are then obtained for each treatment group. Based on these estimates, the desired allocation proportion is calculated. Then Hu and Zhang's allocation function is applied to determine the probabilities for the next patient to be assigned to each treatment group, which drive the allocation proportion towards the desired one. This process is repeated sequentially for each patient until the predetermined number of patients has been allocated.

This methodology was introduced by Hu and Zhang (2004) in their paper 'Asymptotic properties of doubly adaptive biased coin designs for multitreatment clinical trials'.

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.

Sequential monitoring. With monitor = sqMonitor(t, spend) (two arms only) the trial is analysed at the information times t. Looks are taken after ceiling(t * ssn) patients. At each look the Wald statistic Z = (\hat\theta_1 - \hat\theta_2)/\sqrt{\hat v_1/n_1 + \hat v_2/n_2} is computed from all patients enrolled so far, with the sample variances as variance estimates (missing responses are excluded), and the trial stops and rejects the null hypothesis as soon as |Z| reaches the boundary of sqBoundary. Zhu and Hu (2010) showed that under the DBCD the sequential statistics are asymptotically a Brownian motion in the information time, so alpha spending boundaries keep the type I error. Their theory covers two arms and the DBCD, and the same boundaries are used with ERADE. The result then also contains the stopping probability at each look and the expected sample size.

Value

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

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. 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. 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. 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.

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. 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. doi:10.1111/j.1541-0420.2005.00496.x

Zhu, H. and Hu, F. (2010). Sequential monitoring of response-adaptive randomized clinical trials. The Annals of Statistics, 38(4), 2218-2241. doi:10.1214/10-AOS796

See Also

See DBCD_Bin for simulations of Hu and Zhang's doubly biased coin design with binary response.

See dyldDBCD_Cont for simulations of Hu and Zhang's doubly biased coin design with delayed continuous response.

Examples

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

res <- DBCD_Cont(n0 = 20, theta = theta, k = k, ssn = ssn, theta0 = NULL,
                 target.alloc = "Neyman", r = 2, nsim = 100, alpha = 0.05)

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

## three arms with the k-arm ZR target and ERADE
res3 <- DBCD_Cont(n0 = 30, theta = c(13, 16, 15, 6.25, 14, 9), k = 3, ssn = 150,
                  target.alloc = "ZR", lower.bound = 0.1, nsim = 30,
                  allocation = "ERADE", seed = 1)
summary(res3)

Drop the loser rule

Description

Simulating the drop-the-loser rule (number of arms \ge 2) with two-sided hypothesis testing in a clinical trial context.

Usage

DLRule(k, p, ssn, Y0 = NULL, nsim = 2000, alpha = 0.05, test.fun = NULL,
       typeI = FALSE, seed = NULL)

Arguments

Details

The drop-the-loser rule can be described as follows. An urn initially contains balls of K treatment types and an immigration ball (type 0). A ball is drawn at random. If a treatment ball of type k is drawn, treatment k is assigned to the subject and the response is observed. If the response is a failure, the ball is not replaced, otherwise it is replaced. If the immigration ball is drawn, no treatment is assigned, and the ball is returned to the urn together with one ball of each treatment type. With two treatments A and B this is the rule of Ivanova (2003).

Value

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

References

Ivanova, A. (2003). A play-the-winner-type urn design with reduced variability. Metrika, 58, 1-13. doi:10.1007/s001840200220

Examples

## a simple use
dl.res <- DLRule(k = 2, p = c(0.7, 0.8), ssn = 200, Y0 = NULL, nsim = 100, alpha = 0.05)

## view the output
dl.res


  ## view all simulation settings
  dl.res[["method"]]
  dl.res[["parameter"]]

  ## view the simulation results
  dl.res[["propotion"]]
  dl.res[["failure rate"]]
  dl.res[["power"]]
  dl.res[["data: assignment"]]
  

Generalized drop-the-loser rule

Description

Simulating the generalized drop-the-loser rule (number of arms \ge 2) with two-sided hypothesis testing in a clinical trial context.

Usage

GDLRule(k, p, ssn, aK, Y0 = NULL, nsim = 2000, alpha = 0.05, test.fun = NULL,
        typeI = FALSE, seed = NULL)

Arguments

Details

Consider an urn containing balls of K+1 types. Balls of types 1, \ldots, K represent treatments, and balls of type 0 are called immigration balls. When a subject arrives for randomization, a ball is drawn at random. If the ball is of type 0 (an immigration ball), no subject is treated, and the ball is returned to the urn together with A=a_1+\cdots+a_K additional balls, a_k of treatment type k, k=1, \ldots, K. If a treatment ball is drawn (say, of type k, for some k=1, \ldots, K), the subject is given treatment k. If the observed response of this subject is a success, the ball is replaced, otherwise it is not replaced.

Value

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

References

Zhang, L. X., Chan, W. S., Cheung, S. H. and Hu, F. (2007). A generalized drop-the-loser urn for clinical trials with delayed responses. Statistica Sinica, 17(1), 387-409.

Examples

## a simple use
gdl.res <- GDLRule(k = 3, p = c(0.6, 0.7, 0.6),
                   ssn = 200, aK = c(1, 1, 1), Y0 = NULL, nsim = 100, alpha = 0.05)

## view the output
gdl.res


  ## view all simulation settings
  gdl.res[["method"]]
  gdl.res[["parameter"]]

  ## view the simulation results
  gdl.res[["propotion"]]
  gdl.res[["failure rate"]]
  gdl.res[["power"]]
  gdl.res[["data: assignment"]]
  

Group Doubly Biased Coin Design with Binary Response Type

Description

Simulating the group doubly biased coin design with 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

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

Arguments

Details

Hu and Zhang's doubly biased coin design (DBCD) adjusts the probability of assigning each patient to a specific treatment group in a clinical trial, based on the responses of all previous patients. The group DBCD is a more practical version of this approach. It updates the allocation probabilities for the patients in each group based on the responses of all preceding groups, either when the data become available or at fixed time intervals (for example weekly or biweekly).

The process begins by assigning the first n0 patients (possibly the first few groups) to treatment groups using restricted randomization and collecting their responses. Initial parameter estimates for the response variable are then obtained for each treatment group. Based on these estimates, the estimated desired allocation proportion is calculated. Then Hu and Zhang's allocation function is applied to determine the probabilities for the next group of patients to be assigned to each treatment group, which drive the allocation proportion towards the desired one. This process is repeated sequentially for each group until the predetermined number of patients has been allocated.

This methodology was introduced by Zhai, Li, Zhang and Hu (2024) in their paper 'Group response-adaptive randomization with delayed and missing responses'.

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.

Sequential monitoring. With monitor = sqMonitor(t, spend) (two arms only) the trial is analysed at the information times t. Looks are taken at the end of the first group in which at least ceiling(t * ssn) patients are enrolled. If one group passes several planned looks, only the last of them is analysed and the alpha of the skipped looks is not spent, which makes the test slightly conservative. At each look the Wald statistic Z = (\hat\theta_1 - \hat\theta_2)/\sqrt{\hat v_1/n_1 + \hat v_2/n_2} is computed from all patients enrolled so far, with \hat p_k(1 - \hat p_k) as variance estimates (missing responses are excluded), and the trial stops and rejects the null hypothesis as soon as |Z| reaches the boundary of sqBoundary. Zhu and Hu (2010) showed that under the DBCD the sequential statistics are asymptotically a Brownian motion in the information time, so alpha spending boundaries keep the type I error. Their theory covers two arms and the DBCD, and the same boundaries are used with ERADE. The result then also contains the stopping probability at each look and the expected sample size.

Value

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

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. 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. 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. doi:10.1214/08-AOS655

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. 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. doi:10.1198/016214506000000906

Zhai, G., Li, Y., Zhang, L. and Hu, F. (2024). Group response-adaptive randomization with delayed and missing responses. Statistics in Medicine, 43(27), 5047-5059. doi:10.1002/sim.10220

Zhu, H. and Hu, F. (2010). Sequential monitoring of response-adaptive randomized clinical trials. The Annals of Statistics, 38(4), 2218-2241. doi:10.1214/10-AOS796

Examples

res <- Group.DBCD_Bin(n0 = 20, p = c(0.65, 0.8), k = 2, gsize.param = 5,
                      ssn = 300, theta0 = NULL, target.alloc = "RPW",
                      r = 2, nsim = 100, mRate = NULL, alpha = 0.05)
res

## monitored group design with Pocock-type spending
res.sq <- Group.DBCD_Bin(n0 = 20, p = c(0.6, 0.8), k = 2, gsize.param = 5, ssn = 200,
                         target.alloc = "RSIHR", nsim = 50,
                         monitor = sqMonitor(c(0.5), spend = "Pocock"), seed = 1)
res.sq

Group Doubly Biased Coin Design with Continuous Response Type

Description

Simulating the group doubly biased coin design with 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

Group.DBCD_Cont(n0 = 20, theta, k, gsize.param, ssn, theta0 = NULL,
                target.alloc = "Neyman", r = 2, nsim = 2000, mRate = NULL,
                alpha = 0.05, allocation = "DBCD", erade.alpha = 0.5,
                lower.bound = 0, monitor = NULL, test.fun = NULL,
                typeI = FALSE, seed = NULL)

Arguments

Details

Hu and Zhang's doubly biased coin design (DBCD) adjusts the probability of assigning each patient to a specific treatment group in a clinical trial, based on the responses of all previous patients. The group DBCD is a more practical version of this approach. It updates the allocation probabilities for the patients in each group based on the responses of all preceding groups, either when the data become available or at fixed time intervals (for example weekly or biweekly).

The process begins by assigning the first n0 patients (possibly the first few groups) to treatment groups using restricted randomization and collecting their responses. Initial parameter estimates for the response variable are then obtained for each treatment group. Based on these estimates, the estimated desired allocation proportion is calculated. Then Hu and Zhang's allocation function is applied to determine the probabilities for the next group of patients to be assigned to each treatment group, which drive the allocation proportion towards the desired one. This process is repeated sequentially for each group until the predetermined number of patients has been allocated.

This methodology was introduced by Zhai, Li, Zhang and Hu (2024) in their paper 'Group response-adaptive randomization with delayed and missing responses'.

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.

Sequential monitoring. With monitor = sqMonitor(t, spend) (two arms only) the trial is analysed at the information times t. Looks are taken at the end of the first group in which at least ceiling(t * ssn) patients are enrolled. If one group passes several planned looks, only the last of them is analysed and the alpha of the skipped looks is not spent, which makes the test slightly conservative. At each look the Wald statistic Z = (\hat\theta_1 - \hat\theta_2)/\sqrt{\hat v_1/n_1 + \hat v_2/n_2} is computed from all patients enrolled so far, with the sample variances as variance estimates (missing responses are excluded), and the trial stops and rejects the null hypothesis as soon as |Z| reaches the boundary of sqBoundary. Zhu and Hu (2010) showed that under the DBCD the sequential statistics are asymptotically a Brownian motion in the information time, so alpha spending boundaries keep the type I error. Their theory covers two arms and the DBCD, and the same boundaries are used with ERADE. The result then also contains the stopping probability at each look and the expected sample size.

Value

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

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. 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. 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. doi:10.1214/08-AOS655

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. doi:10.1198/016214506000000906

Zhai, G., Li, Y., Zhang, L. and Hu, F. (2024). Group response-adaptive randomization with delayed and missing responses. Statistics in Medicine, 43(27), 5047-5059. doi:10.1002/sim.10220

Zhang, L. and Rosenberger, W. F. (2006). Response-adaptive randomization for clinical trials with continuous outcomes. Biometrics, 62(2), 562-569. doi:10.1111/j.1541-0420.2005.00496.x

Zhu, H. and Hu, F. (2010). Sequential monitoring of response-adaptive randomized clinical trials. The Annals of Statistics, 38(4), 2218-2241. doi:10.1214/10-AOS796

Examples

theta = c(13, 4.0^2, 15, 2.5^2)
k = 2
gsize.param = 5
ssn = 120
res <- Group.DBCD_Cont(n0 = 20, theta = theta, k = k, gsize.param = gsize.param,
                       ssn = ssn, target.alloc = "Neyman", r = 2, nsim = 100,
                       mRate = NULL, alpha = 0.05)
res

Group Doubly Biased Coin Design with Delayed Binary Response

Description

Simulating the group 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

Group.dyldDBCD_Bin(n0 = 20, p, k, ssn, gsize.param, rspT.dist, rspT.param,
                   theta0 = NULL, target.alloc = "RPW", r = 2, nsim = 2000,
                   eTime = 7, mRate = NULL, alpha = 0.05, allocation = "DBCD",
                   erade.alpha = 0.5, lower.bound = 0, test.fun = NULL,
                   typeI = FALSE, seed = NULL)

Arguments

Details

Hu and Zhang's doubly biased coin design (DBCD) adjusts the probability of assigning each patient to a specific treatment group in a clinical trial, based on the responses of all previous patients. The group DBCD is a more practical version of this approach. It updates the allocation probabilities for the patients in each group based on the available responses of all preceding groups, either when the data become available or at fixed time intervals (for example weekly or biweekly). This function implements the group doubly biased coin design for delayed binary responses.

The process begins by assigning the first n0 patients (possibly the first few groups) to treatment groups using restricted randomization and collecting their responses. Initial parameter estimates for the response variable are then obtained for each treatment group. Based on these estimates, the estimated desired allocation proportion is calculated. Then Hu and Zhang's allocation function is applied to determine the probabilities for the next group of patients to be assigned to each treatment group, which drive the allocation proportion towards the desired one. This process is repeated sequentially for each group until the predetermined number of patients has been allocated.

This methodology was introduced by Zhai, Li, Zhang and Hu (2024) in their paper 'Group response-adaptive randomization with delayed and missing responses'.

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.

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. 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. 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. doi:10.1214/08-AOS655

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. 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. doi:10.1198/016214506000000906

Zhai, G., Li, Y., Zhang, L. and Hu, F. (2024). Group response-adaptive randomization with delayed and missing responses. Statistics in Medicine, 43(27), 5047-5059. doi:10.1002/sim.10220

Examples

# a simple use
## Arguments for generating the simulated data
### For response simulation
p = c(0.7, 0.5)
k = 2
ssn = 200

### for entry time and response time simulation
eTime = 7
rspT.param = rep(10, 4)
rspT.dist = "exponential"
gsize.param = 5

## Arguments for the design
n0 = 10
target.alloc = "RPW"

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

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

## three arms, ERADE and the optimal RSIHR target, with the trial duration
res3 <- Group.dyldDBCD_Bin(n0 = 21, p = c(0.6, 0.7, 0.8), k = 3, ssn = 120,
                           gsize.param = 5, rspT.dist = "exponential",
                           rspT.param = rep(5, 6), target.alloc = "OptimalRSIHR",
                           lower.bound = 0.1, nsim = 30, allocation = "ERADE", seed = 1)
res3[["duration"]]

Group Doubly Biased Coin Design with Delayed Continuous Response

Description

Simulating the group 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

Group.dyldDBCD_Cont(n0 = 20, theta, k, ssn, gsize.param, rspT.dist,
                    rspT.param, target.alloc = "Neyman", r = 2, nsim = 2000,
                    eTime = 7, mRate = NULL, alpha = 0.05,
                    allocation = "DBCD", erade.alpha = 0.5, lower.bound = 0,
                    test.fun = NULL, typeI = FALSE, seed = NULL)

Arguments

Details

Hu and Zhang's doubly biased coin design (DBCD) adjusts the probability of assigning each patient to a specific treatment group in a clinical trial, based on the responses of all previous patients. The group DBCD is a more practical version of this approach. It updates the allocation probabilities for the patients in each group based on the available responses of all preceding groups, either when the data become available or at fixed time intervals (for example weekly or biweekly). This function implements the group doubly biased coin design for delayed continuous responses.

The process begins by assigning the first n0 patients (possibly the first few groups) to treatment groups using restricted randomization and collecting their responses. Initial parameter estimates for the response variable are then obtained for each treatment group. Based on these estimates, the estimated desired allocation proportion is calculated. Then Hu and Zhang's allocation function is applied to determine the probabilities for the next group of patients to be assigned to each treatment group, which drive the allocation proportion towards the desired one. This process is repeated sequentially for each group until the predetermined number of patients has been allocated.

This methodology was introduced by Zhai, Li, Zhang and Hu (2024) in their paper 'Group response-adaptive randomization with delayed and missing responses'.

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.

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. 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. 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. doi:10.1214/08-AOS655

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. doi:10.1198/016214506000000906

Zhai, G., Li, Y., Zhang, L. and Hu, F. (2024). Group response-adaptive randomization with delayed and missing responses. Statistics in Medicine, 43(27), 5047-5059. doi:10.1002/sim.10220

Zhang, L. and Rosenberger, W. F. (2006). Response-adaptive randomization for clinical trials with continuous outcomes. Biometrics, 62(2), 562-569. 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 = 120

### for entry time and response time simulation
eTime = 7
gsize.param = 5
rspT.param = rep(10, 2)
rspT.dist = "exponential"

## Arguments for the design
n0 = 10
target.alloc = "Neyman"

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

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

Randomized Pólya urn procedure

Description

Simulating the randomized Pólya urn procedure (number of arms \ge 2) with two-sided hypothesis testing in a clinical trial context.

Usage

PolyaUrn(k, p, ssn, Y0 = NULL, nsim = 2000, alpha = 0.05, test.fun = NULL,
         typeI = FALSE, seed = NULL)

Arguments

Details

The randomized Pólya urn (RPU) procedure can be described as follows. An urn initially contains at least one ball of each of the K treatment types. A ball is drawn from the urn with replacement. If a type i ball is drawn, i=1, \ldots, K, then treatment i is assigned to the next patient. If the response is a success, a ball of type i is added to the urn. Otherwise the urn remains unchanged.

Value

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

The randomized Pólya urn tends to concentrate on one arm, so some arms can end with very few patients. The asymptotic tests are then far from their nominal level (the rejection rate under the null hypothesis can be well above alpha, especially for K \ge 3), and trials in which an arm has no patients give no test. Use typeI = TRUE to check the type I error.

References

Durham, S. D., Flournoy, N. and Li, W. (1998). A sequential design for maximizing the probability of a favourable response. Canadian Journal of Statistics, 26(3), 479-495. doi:10.2307/3315771

Examples

## a simple use
Polya.res <- PolyaUrn(k = 3, p = c(0.6, 0.7, 0.6), ssn = 200, Y0 = NULL,
                      nsim = 100, alpha = 0.05)

## view the output
Polya.res

  ## view all simulation settings
  Polya.res[["method"]]
  Polya.res[["parameter"]]

  ## view the simulation results
  Polya.res[["propotion"]]
  Polya.res[["failure rate"]]
  Polya.res[["power"]]
  Polya.res[["data: assignment"]]
  

Randomized Play-the-winner Rule

Description

Simulating the randomized play-the-winner rule (two arms) with two-sided hypothesis testing in a clinical trial context.

Usage

RPWRule(k, p, ssn, Y0 = NULL, nsim = 2000, alpha = 0.05, test.fun = NULL,
        typeI = FALSE, seed = NULL)

Arguments

Details

The randomized play-the-winner rule allocates future subjects in a clinical trial to treatment groups based on the performance of previously treated subjects. A ball is drawn from the urn to assign the next subject. A success on a treatment adds a ball of the same type to the urn, and a failure adds a ball of the other type. This increases the chance that future patients are assigned to the better-performing treatment.

Value

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

References

Wei, L. J. and Durham, S. (1978). The randomized play-the-winner rule in medical trials. Journal of the American Statistical Association, 73(364), 840-843. doi:10.1080/01621459.1978.10480109

Examples

## a simple use
RPW.res <- RPWRule(k = 2, p = c(0.7, 0.8), ssn = 200, Y0 = NULL, nsim = 100, alpha = 0.05)
## view the output
RPW.res


  ## view all simulation settings
  RPW.res[["method"]]
  RPW.res[["parameter"]]

  ## view the simulation results
  RPW.res[["propotion"]]
  RPW.res[["failure rate"]]
  RPW.res[["power"]]
  RPW.res[["data: assignment"]]
  

Wei's Urn: Randomized Play-the-winner Rule with Multiple Arms

Description

Simulating Wei's urn, the randomized play-the-winner rule for multiple arms (number of arms \ge 2), with two-sided hypothesis testing in a clinical trial context.

Usage

WeiUrn(k, p, ssn, Y0 = NULL, nsim = 2000, alpha = 0.05, test.fun = NULL,
       typeI = FALSE, seed = NULL)

Arguments

Details

Wei's urn procedure extends the randomized play-the-winner rule (Wei and Durham, 1978) from k = 2 to k > 2 treatments, which makes it usable in multi-arm clinical trials. A success on treatment k adds one ball of type k to the urn, and a failure on treatment k adds 1/(K-1) balls of each of the other K-1 types. With k = 2 it reduces to the randomized play-the-winner rule.

Value

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

References

Wei, L. J. (1979). The generalized Polya's urn design for sequential medical trials. The Annals of Statistics, 7(2), 291-296. doi:10.1214/aos/1176344614

Wei, L. J. and Durham, S. (1978). The randomized play-the-winner rule in medical trials. Journal of the American Statistical Association, 73(364), 840-843.

Examples

## a simple use
wei.res <- WeiUrn(k = 3, p = c(0.7, 0.8, 0.7), ssn = 200, Y0 = NULL, nsim = 100, alpha = 0.05)

## view the output
wei.res


  ## view all simulation settings
  wei.res[["method"]]
  wei.res[["parameter"]]

  ## view the simulation results
  wei.res[["propotion"]]
  wei.res[["failure rate"]]
  wei.res[["power"]]
  wei.res[["data: assignment"]]
  

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

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.

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. 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. 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. 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. 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. 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. 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

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

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.

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. 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. 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. 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. 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. 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. 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

Allocation Probabilities for an Ongoing Trial

Description

Computes the allocation probabilities of the next patient, or of the next group of patients, in an ongoing response-adaptive randomized trial from the assignments and responses observed so far, and optionally draws the assignments. The same estimates, targets and allocation functions as in the simulation functions (for example DBCD_Bin and Group.DBCD_Bin) are used.

Usage

nextAlloc(alloc, outcome, k, response = "binary", target.alloc = NULL,
          allocation = "DBCD", r = 2, erade.alpha = 0.5, lower.bound = 0,
          theta0 = NULL, size = 1)

Arguments

Details

The parameters of each arm are estimated from the observed responses only (NA values are ignored): smoothed success rates for binary responses, and the mean and variance (which needs at least two responses) for continuous responses. The current allocation proportions use all enrolled patients, including those whose responses are not available yet. If the target cannot be estimated, equal allocation is used as the target, and with no enrolled patients every arm has probability 1/k. All patients of a group are assigned independently with the same probabilities, as in the group designs of Zhai, Li, Zhang and Hu (2024).

Value

A list with the following elements.

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. 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. 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. doi:10.1214/08-AOS655

Zhai, G., Li, Y., Zhang, L. and Hu, F. (2024). Group response-adaptive randomization with delayed and missing responses. Statistics in Medicine, 43(27), 5047-5059. doi:10.1002/sim.10220

See Also

DBCD_Bin, DBCD_Cont, Group.DBCD_Bin for simulating the designs.

Examples

## 30 patients enrolled in a three-arm trial, the last three responses are pending
set.seed(1)
alloc <- sample(1:3, 30, replace = TRUE)
outcome <- rbinom(30, 1, c(0.6, 0.7, 0.8)[alloc])
outcome[28:30] <- NA

## probabilities for the next patient with the DBCD and the RSIHR target
nextAlloc(alloc, outcome, k = 3, target.alloc = "RSIHR")

## assignments for the next group of 5 patients with ERADE
nextAlloc(alloc, outcome, k = 3, target.alloc = "RSIHR", allocation = "ERADE",
          size = 5)$assignment

## continuous responses
y <- rnorm(30, mean = c(13, 15, 14)[alloc], sd = 3)
nextAlloc(alloc, y, k = 3, response = "continuous", target.alloc = "Neyman")$prob

Print and Summarize Simulation Results

Description

Methods for the objects of class "grouprar" returned by the simulation functions of the package, such as DBCD_Bin or WeiUrn.

Usage

## S3 method for class 'grouprar'
print(x, ...)

## S3 method for class 'grouprar'
summary(object, ...)

## S3 method for class 'summary.grouprar'
print(x, digits = 3, ...)

Arguments

Details

print shows the summary of the object. The full simulation results, including the per-simulation data, remain available as list elements, for example x[["data: allocation"]].

Value

summary returns an object of class "summary.grouprar", a list with the elements

The print methods return their argument invisibly.

See Also

DBCD_Bin, CRDesign.

Examples

res <- CRDesign(k = 3, p = c(0.6, 0.7, 0.8), ssn = 100, nsim = 50, seed = 1)
res
s <- summary(res)
s$arms
print(s, digits = 2)

Group Sequential Monitoring of Response-Adaptive Randomized Trials

Description

sqMonitor specifies the interim looks and the alpha spending function of a group sequential design, to be passed as the monitor argument of DBCD_Bin, DBCD_Cont, Group.DBCD_Bin and Group.DBCD_Cont. sqBoundary computes the corresponding two-sided boundaries for the standardized test statistic.

Usage

sqMonitor(t, spend = "OBF")

sqBoundary(t, alpha = 0.05, spend = "OBF")

Arguments

Details

The spending functions of Lan and DeMets (1983) are used, with each tail spending the one-sided function at level \alpha/2 (Proschan, Lan and Wittes, 2006), as in Zhu and Hu (2010). For one tail at level a = \alpha/2 they are 2\{1 - \Phi(z_{a/2}/\sqrt{t})\} for "OBF", a\log\{1 + (e - 1)t\} for "Pocock" and a t for "Linear". The boundary c_j of look j is chosen so that, under the null hypothesis, the probability that |Z| first reaches the boundary at look j equals the alpha spent between looks j-1 and j, where Z at the information times t_1 < t_2 < \cdots is a standard normal process with correlation \sqrt{t_i/t_j}. Zhu and Hu (2010) showed that the sequential test statistics of a two-arm trial randomized by the doubly adaptive biased coin design have asymptotically this structure, so these boundaries keep the type I error. The boundaries are computed by recursive numerical integration of the density of Z\sqrt{t} over the continuation region (Armitage, McPherson and Rowe, 1969), which is fast for any number of looks.

Sequential monitoring is available for two arms only, because the theory of Zhu and Hu (2010) covers the comparison of two treatments.

Value

sqMonitor returns an object of class "sqMonitor", a list with the sorted information times t and the spending function spend.

sqBoundary returns a numeric vector with the boundary for |Z| at each look, named look 1, look 2, ... A boundary is Inf if no alpha is spent at that look.

References

Lan, K. K. G. and DeMets, D. L. (1983). Discrete sequential boundaries for clinical trials. Biometrika, 70(3), 659-663. doi:10.1093/biomet/70.3.659

Armitage, P., McPherson, C. K. and Rowe, B. C. (1969). Repeated significance tests on accumulating data. Journal of the Royal Statistical Society, Series A, 132(2), 235-244. doi:10.2307/2343787

Proschan, M. A., Lan, K. K. G. and Wittes, J. T. (2006). Statistical Monitoring of Clinical Trials: A Unified Approach. Springer.

Zhu, H. and Hu, F. (2010). Sequential monitoring of response-adaptive randomized clinical trials. The Annals of Statistics, 38(4), 2218-2241. doi:10.1214/10-AOS796

See Also

DBCD_Bin, DBCD_Cont, Group.DBCD_Bin, Group.DBCD_Cont.

Examples

## the boundaries of Zhu and Hu (2010): 4.877, 2.963 and 1.969
sqBoundary(c(0.2, 0.5, 1), alpha = 0.05, spend = "OBF")
sqBoundary(c(0.2, 0.5, 1), alpha = 0.05, spend = "Pocock")

## two interim looks after 1/3 and 2/3 of the patients
m <- sqMonitor(c(1/3, 2/3), spend = "OBF")
res <- DBCD_Bin(n0 = 20, p = c(0.6, 0.8), k = 2, ssn = 150, target.alloc = "RSIHR",
                nsim = 30, monitor = m, seed = 1)
res[["stopping probability"]]