Generalized Bayesian Optimal Phase II Design (G-BOP2)
Xinying Fang, Wanni Lei & Shouhao Zhou
2025-04-22
1. Installation
The R package GBOP2 is available for free on the
Comprehensive R Archive Network (CRAN). It offers functions for
implementing G-BOP2 design with various Particle Swarm Optimization
(PSO) algorithms (Kennedy & Eberhart 1995), including PSO-Default
(Kennedy & Eberhart, 1995), PSO-Quantum (Sun, Xu & Feng 2004),
PSO-DEXP (Stehl´ık et al. 2024), and PSO-GO. The G-BOP2 design includes
two objectives, one is the sample size minimization,
and the other is the power maximization. Parallel
computing is only used when the user explicitly sets nCore > 1. The
function defaults to sequential execution. If you plan to run parallel
computing, you need to manually controlling the parallel backend using
init_cluster(nCore) and stop_cluster(). The
parallel backend will be automatically stopped when a function
completes, so you will need to set nCore again each time before running
the function. You can load the package using the following command:
#install.packages("GBOP2")
library(GBOP2)2. Summary of functions
| Efficacy - Single Boundary | Efficacy - Dual Boundary | Toxicity and Efficacy | |
|---|---|---|---|
| Sample Size Minimization | GBOP2_minSS_singleE | GBOP2_minSS_dualE | GBOP2_minSS_TE |
| Power Maximization | GBOP2_maxP_singleE | GBOP2_maxP_dualE | GBOP2_maxP_TE |
3. Study Objective: sample size minimization
3.1 Efficacy: single boundary for futility
Objective function \(L\):
The parameters \(u\) and \(w\) are in the range \([0,1]\) and represent the pre-assigned weights for the study sample size \(N\) and how it is distributed under the null hypothesis \(H_0\) and the alternative hypothesis \(H_1\), respectively. The score \(g\) refers to some specified function on the sample size variable N and takes the unit-bounded value in \([0,1]\). The parameters \(\alpha\) and \(\beta\) refer to the Type I and Type II error. The constants \(\alpha_0\) and \(\beta_0\) refer to the nominal Type I and Type II error rates, while \(n_{\text{max}}\) is the maximum allowable sample size. While \(w\) is often set to 0, 1, or 1/2 in optimal designs to indicate diverse trial-specific considerations on \(H_0\) and \(H_1\), we recommend \(w= 2/3\) to reflect the observation that only 1/3 of phase II studies successfully transit to phase III, suggesting empirically 2/3 chance in favor of \(H_0\).
For example:
Simon’s two stage optimal design (Simon 1989): \[ g(N \mid H_.) = E(N \mid H_.)/n, u = 0, w = 1 \]
Simon’s two stage minimax design (Simon 1989): \[ g(N \mid H_.) = E(N \mid H_.)/n, u = 1, w =1 \]
Fleming 1982 \[ g(N \mid H_.) = E(N \mid H_.)/n, u = 0, w = 1 \]
The following examples illustrate the different designs identified by
the PSO-GO algorithm. Note that the PSO-GO algorithm guarantees
convergence to the global optimum almost surely when using a large
nParallel value. If the global optimum is not reached, try
increasing the nParallel value or running the algorithm
again with different seeds.
The GBOP2_minSS_single function is designed to identify
the optimal G-BOP2 design for single futility boundary. The optimal
G-BOP2 design aims to minimize the sample size target specified in the
objective function among the designs that satisfy the pre-specified
nominal Type I error and minimum statistical power. Parameters such as
nlooks, weight, maxPatients,
Nmin_cohort1, and Nmin_increase provide users
with more flexibility to adjust the number of interim looks, the weight
of the sample size under the null hypothesis, and other design
characteristics. When users choose pso_method equals “all”,
GBOP2_minSS_single function conducts PSO_Ensemble in
parallel by indicating the number of parallel through
nParallel option. Otherwise, users can specify the
pso_method option as default,
quantum, or dexp to conduct a single PSO
algorithm. The following example show the optimal 3-stage G-BOP2 optimal
design identified by the PSO-GO with \(u=1\), \(w=1\).
## PSO-GO
init_cluster(5) ## Initialize 5 parallel cluster.
## Note: Users must explicitly set the number of cores using init_cluster()
## each time before running GBOP2_minSS_singleE(). Parallel computing will be
## automatically terminated once the function completes. This setup complies
## with CRAN's strict checks on parallel computing.
optimal_GO_single <- GBOP2_minSS_singleE(
design = "optimal", ## "optimal" or "minimax", "unified"
unified.u = 1, ## specify when design = "unified", u in [0, 1]
weight = 1, ## weight of sample size under null, w in [0, 1]
nlooks = 2, ## number of interim looks. For 3-stage design, nlooks = 2.
p0 = 0.2, ## response rate in null hypothesis
p1 = 0.4, ## response rate in alternative hypothesis
err1 = 0.05, ## Type Ierror
minPower = 0.8, ## power
maxPatients = 50, ## maximum number of patients
Nmin_cohort1 = 5, ## minimum number of first cohort
Nmin_increase = 5, ## minimum number of increase in each cohort
pso_method = "all", ## choose from "all", "default", "quantum" or "dexp"
nParallel = 3, ## number of PSO-ensemble, only effective when pso_method = "all"
seed = 456,
nSwarm = 64, ## nSwarm in PSO
maxIter = 200) ## maxIter in PSO
Interpretation:
The GBOP2_minSS_singleE function successfully identified an
optimal three-stage design with a power of over 80% and
a Type I error rate of less than 5% . The optimized
interim times are 8, 22, and 38 respectively, and the
optimized stopping boundaries for each stage are 1, 5, and
11. The trial will be terminated and the drug will be
considered not promising if the observed number of responses at any
stage is less than or equal to the corresponding stopping boundary.
Otherwise, the drug will be declared promising. The
parameters section provides key values used in
determining the optimal boundaries.
summary(optimal_GO_single)
#> ------------------------------------------------
#> PSOGO for optimal/minimax design with single boundary
#> design: optimal
#> weight: 1
#> method: default
#> cpu time: 1.59 seconds
#>
#> parameters:
#> lambda1: 0.829704
#> lambda2: 0.946774
#> gamma: 0.269488
#>
#> cohort sizes: 8, 22, 38
#> boundary values: stop if <= 1, stop if <= 5, stop if <= 11
#>
#> type I error: 0.0482192
#> power: 0.801947
#> expected sample size: 18.4166
#> utility value: 18.4166
#>
#> ------------------------------------------------3.2 Efficacy: dual boundary for futility monitoring and superiority detecting
The GBOP2_minSS_dual function is designed to identify
the optimal and minimax designs for dual futility boundaries, which
meets the pre-specified Type I and Type II error rates by using PSO_GO
algorithm. When pso_method equals “all”,
GBOP2_minSS_dual function conducts PSO_Ensemble in
parallel, otherwise, users can specify the pso_method
option as default, quantum, or
dexp to conduct a single PSO algorithm. The following
example show the optimal design for dual futility boundary identified by
the PSO-GO.
## PSO-GO
init_cluster(5) ## Initialize 5 parallel cluster
## Note: Users must explicitly set the number of cores using init_cluster()
## each time before running GBOP2_minSS_dualE(). Parallel computing will be
## automatically terminated once the function completes. This setup complies
## with CRAN's strict checks on parallel computing.
optimal_GO_dual <- GBOP2_minSS_dualE(
design = "optimal", ## choose from "optimal", "minimax" and "unified"
unified.u = 1, ## specify when design = "unified", u in [0, 1]
weight = 0.5, ## weight of sample size under null
nlooks = 1, ## number of interim looks (R)
p0 = 0.2, ## response rate in null hypothesis
p1 = 0.4, ## response rate in alternative hypothesis
err1 = 0.05, ## Type I error
minPower = 0.8, ## power
maxPatients = 50, ## maximum number of patients
Nmin_cohort1 = 10, ## minimum number of first cohort
Nmin_increase = 5, ## minimum number of increase in each cohort
pso_method = "all", ## choose from "all", ""default", "quantum" or "dexp"
nParallel = 3, ## number of PSO_ensemble
seed = 123,
nSwarm = 64, ## nSwarm in PSO
maxIter = 400) ## maxIter in PSO
Interpretation:
The GBOP2_minSS_dualE function successfully identified an
optimal two-stage design with a power exceeding 80% and
a Type I error rate below 5%. The optimized interim
times are 14 and 37, respectively. The stopping
boundaries for futility are 3 and 11, and the
boundaries for efficacy are 7 and 12.The trial will be
terminated for futility, and the drug will be considered not promising
if the observed number of responses is \(≤\) 3 in the first stage
or \(≤\) 11 in the
second stage. Conversely, the trial will be terminated for efficacy, and
the drug will be considered promising if the observed number of
responses is \(≥\) 7
in the first stage or \(≥\)
12 in the second stage.The parameters
section provides key values used in determining the optimal
boundaries.
summary(optimal_GO_dual)
#> ------------------------------------------------------
#> PSOGO for minimizing sample size with dual-boundary
#> Design: optimal
#> Weight: 0.5
#> Method: default
#>
#> Cohort values: 14, 37
#> Boundary values:
#> Boundary futile: stop if <= 3, stop if <= 11
#> Boundary effi: stop if >= 7, stop if >= 12
#>
#> Type I Error: 0.0474947
#> Power: 0.802168
#> Expected Sample Size: 23.871
#> Utility Value: 23.871
#>
#> Parameters:
#> Lambda1: 0.853079
#> Lambda_grad1: 0.682598
#> Lambda_grad2: 0.942373
#> Gamma_1: 0.30064
#> Gamma_2: 0.181021
#> Gamma_3: 0.922944
#> Delta0: 0.0786016
#> Delta1: 0.148326
#>
#> ------------------------------------------------3.3 Toxicity and Efficacy: joint safety and futility monitoring
The GBOP2_minSS_TE function is designed to identify the
optimal and minimax designs for toxicity and efficacy boundaries, which
meets the pre-specified Type I and Type II error rates by using PSO_GO
algorithm. When pso_method equals “all”,
GBOP2_minSS_dual function conducts PSO_Ensemble in
parallel, otherwise, users can specify the pso_method
option as default, quantum, or
dexp to conduct a single PSO algorithm. The following
example show the optimal design for dual futility boundaries identified
by the PSO-GO.
init_cluster(5) ## Initialize 5 parallel cluster
## Note: Users must explicitly set the number of cores using init_cluster()
## each time before running GBOP2_minSS_TE(). Parallel computing will be
## automatically terminated once the function completes. This setup complies
## with CRAN's strict checks on parallel computing.
Optimal_GO_TE <- GBOP2_minSS_TE(
design = "optimal", ## choose from "optimal", "minimax" and "unified"
unified.u = 1, ## specify when design = "unified", u in [0, 1]
nlooks = 1, ## number of interim looks (R)
skip_efficacy = NULL, ## A vector with a length equal to the number of stages. The default is NULL (no skip for efficacy). If skipping is enabled, set the corresponding stage to 1; otherwise, set it to 0. For example, c(1,0) indicates that futility monitoring is skipped in the first stage but applied in the second stage.
skip_toxicity = NULL, ## A vector with a length equal to the number of stages. The default is NULL (no skip for toxicity). If skipping is enabled, set the corresponding stage to 1; otherwise, set it to 0. For example, c(1,0) indicates that safety monitoring is skipped in the first stage but applied in the second stage.
maxPatients = 26, ## maximum number of patients
Nmin_cohort1 = 13, ## minimum number of first cohort
Nmin_increase = 13, ## minimum number of increase in each cohort
p01 = 0.3, ## efficacy under null
p02 = 0.4, ## toxicity under null
p03 = 0.2, ## efficacy and toxicity under null, quantifying the correlation between efficacy and toxicity
p11 = 0.6, ## efficacy under alternative
p12 = 0.2, ## toxicity under alternative
p13 = 0.15,## efficacy and toxicity under alternative, quantifying the correlation between efficacy and toxicity
err_eff = 0.1, ## Type I error for futile
err_tox = 0.1, ## Type I error for toxic
err_all = 0.05, ## Type I futile and toxic,
power_eff = 0.8, ## power for futile
power_tox = 0.8, ## power for toxic
power_all = 0.8, ## power for futile and toxic
pso_method = "all", ## choose from "all", ""default", "quantum" or "dexp"
nParallel = 3, ## number of PSO_ensemble
nSwarm = 32, ## nSwarm in PSO
maxIter = 100 ## maxIter in PSO
)
Interpreting \(H_{01}\) represents that the treatment is
safe but futile, \(H_{10}\) represents
that the treatment is efficacious but toxic, and \(H_{00}\) represents the global null that
the treatment is futile and toxic. \(H_{11}\) denote claiming that the treatment
is efficacious and safe. The GBOP2_minSS_TE function
identified an optimal two-stage design with interim times of 13
and 26 patients. The trial stops early for futility if efficacy
responses are \(≤\) 3
at stage 1 or \(≤\) 10
at stage 2, and stops early for toxicity if \(≥\) 5 toxicities occur at
stage 1 or \(≥\) 8 at
stage 2. This design achieves over 80% power and
efficiently balancing safety and effectiveness.
summary(Optimal_GO_TE)
#> ------------------------------------------------
#> PSOGO for minimizing sample size with TE boundary
#> Design: optimal
#> Method: default
#>
#> Cohort Sizes: 13, 26
#> Boundary Values:
#> Efficacy: stop if <= 3, stop if <= 10
#> Toxicity: stop if >= 5, stop if >= 8
#>
#> Type I Errors:
#> Type I (H01): 0.0806731
#> Type I (H10): 0.0978768
#> Type I (H00): 0.00300503
#>
#> Expected Sample Size: 20.9633
#> Power (H11): 0.80488
#> Utility Value: 20.9633
#>
#> Parameters:
#> Lambda_e1: 0.648456
#> Lambda_e2: 0.878973
#> Lambda_t1: 0.741246
#> Lambda_t2: 0.847259
#> Gamma: 1
#>
#> ------------------------------------------------4. Study Objective: power maximization
Objective function \(L\):
The parameters \(u\) and \(w\) are in the range \([0,1]\) and represent the pre-assigned weights for the study sample size \(N\) and how it is distributed under the null hypothesis \(H_0\) and the alternative hypothesis \(H_1\), respectively. The score \(g\) refers to some specified function on the sample size variable N and takes the unit-bounded value in \([0,1]\). The parameters \(\alpha\) and \(\beta\) refer to the Type I and Type II error. The constants \(\alpha_0\) and \(\beta_0\) refer to the nominal Type I and Type II error rates, while \(n_{\text{max}}\) is the maximum allowable sample size.
4.1 Efficacy: single boundary for futility
The GBOP2_maxP_singleE function is designed to identify
the design which maximizing the power for a single futility boundary,
which meets the pre-specified Type I and Type II error rates by using
PSO_GO algorithm. The parameters are the same as in the
GBOP2_minSS_single function except for the
totalPatients, which represents the total number of
patients. Similarly, when users choose pso_method equals
“all”, GBOP2_maxP_single function conducts PSO_Ensemble in
parallel by indicating the number of parallel through
nParallel option. Otherwise, users can specify the
pso_method option as default,
quantum, or dexp to conduct a single PSO
algorithm. The following example show the design maximizes the power
identified by the PSO-GO.
## PSO-GO
init_cluster(5) ## Initialize 5 parallel cluster
## Note: Users must explicitly set the number of cores using init_cluster()
## each time before running GBOP2_maxP_singleE(). Parallel computing will be
## automatically terminated once the function completes. This setup complies
## with CRAN's strict checks on parallel computing.
power_GO_single <- GBOP2_maxP_singleE(
nlooks = 3, ## number of interim analyses
p0 = 0.2, ## response rate under null hypothesis
p1 = 0.4, ## response rate under alternative hypothesis
err1 = 0.05, ## Type I error rate
minPower = 0.8, ## desired minimum statistical power
totalPatients = 50, ## total number of patients
Nmin_cohort1 = 15, ## minimum cohort size for the first stage
Nmin_increase = 8, ## minimum increment in cohort size for subsequent stages
pso_method = "all", ## choose from "all", "default", "quantum" or "dexp"
nParallel = 3, ## number of parallel runs for global optimization
seed = 1024,
nSwarm = 64, ## nSwarm in PSO
maxIter = 200 ## maxIter in PSO
)
Interpretation The GBOP2_maxP_singleE
function successfully identified an optimal four-stage design for
maximizing power under a single boundary setting. The design specifies
interim analyses after recruiting 25, 34, 42, and 50
patients. The optimized stopping boundaries are set at 5, 8, 11,
and 14 responses at each stage. Specifically, the trial will
terminate early and declare the drug not promising if the number of
observed responses at each stage is less than or equal to these
respective boundaries. Otherwise, the trial proceeds to the next stage
or concludes positively at the final stage. This design achieves a high
statistical power over 80% while maintaining a Type I
error rate below 5%, balancing the ability to detect
treatment efficacy with resource efficiency.
summary(power_GO_single)
#> ------------------------------------------------
#> PSOGO for maximizing power with single boundary
#> design: Optimal
#> method: default
#> cpu time: 3.171 seconds
#>
#> parameters:
#> lambda1: 0.9876
#> lambda2: 0.9067
#> gamma: 0.7931
#>
#> cohort sizes: 25, 34, 42, 50
#> boundary values: stop if <= 5, stop if <= 8, stop if <= 11, stop if <= 14
#>
#> type I error: 0.0499
#> power: 0.9174
#> expected sample size (en(p0)): 30.9182
#> utility value: -0.9174
#>
#> ------------------------------------------------4.2 Efficacy: dual boundary for futility monitoring and superiority detecting
The GBOP2_maxP_dual function is designed to identify the
design that maximizing power for dual futility boundaries, which meets
the pre-specified Type I and Type II error rates by using PSO_GO
algorithm. When pso_method equals “all”,
GBOP2_maxP_dual function conducts PSO_Ensemble in parallel,
otherwise, users can specify the pso_method option as
default, quantum, or dexp to
conduct a single PSO algorithm. The following example show the design
that maximizing power for dual futility boundary identified by the
PSO-GO.
## PSO-GO
init_cluster(5) ## Initialize 5 parallel cluster
## Note: Users must explicitly set the number of cores using init_cluster()
## each time before running GBOP2_maxP_dualE(). Parallel computing will be
## automatically terminated once the function completes. This setup complies
## with CRAN's strict checks on parallel computing.
power_GO_dual <- GBOP2_maxP_dualE(
design = "optimal", ## fixed as "optimal", which can not be modified by user
nlooks = 1, ## number of interim looks
p0 = 0.2, ## response rate under the null
p1 = 0.4, ## response rate under the alternative
err1 = 0.05, ## Type I error
minPower = 0.8, ## minimum desired power
totalPatients = 50, ## total number of patients
Nmin_cohort1 = 10, ## minimum cohort size at the first stage
Nmin_increase = 5, ## minimum increment of cohort size at subsequent stages
pso_method = "all", ## choose from "all", "default", "quantum" or "dexp"
nParallel = 3, ## number of parallel PSO_Ensemble
seed = 1024,
nSwarm = 64, ## nSwarm in PSO
maxIter = 400 ## maxIter in PSO
)
Interpretation
The GBOP2_maxP_dualE function identified an optimal
two-stage design with interim times of 44 and 50
patients, maximizing power under dual boundaries. The trial will
terminate early for futility if the number of responses is \(≤\) 7 at stage 1 or \(≤\) 15 at stage 2, or
terminate early for efficacy if the number of responses is \(≥\) 14 at stage 1 or \(≥\) 16 at stage 2. This
design achieves high statistical power (over 80%) while
maintaining a controlled Type I error (less than 5%),
effectively balancing trial efficiency, safety, and the ability to
detect a true treatment effect.
summary(power_GO_dual)
#> ------------------------------------------------------
#> PSOGO for power maximization with dual-boundary
#> Design: optimal
#> Method: dexp
#>
#> Cohort values: 44, 50
#> Boundary values:
#> Boundary futile: stop if <= 7, stop if <= 15
#> Boundary effi: stop if >= 14, stop if >= 16
#>
#> Type I Error: 0.0491397
#> Power: 0.924656
#> Expected Sample Size: 47.8006
#> Utility Value: -0.924656
#>
#> Parameters:
#> Lambda1: 0.773015
#> Lambda_grad1: 0.842724
#> Lambda_grad2: 0.964223
#> Gamma_1: 0.399839
#> Gamma_2: 0.152698
#> Gamma_3: 0.963263
#> Delta0: 0
#> Delta1: 0.150164
#>
#> ------------------------------------------------4.3 Toxicity and Efficacy: joint safety and futility monitoring
The GBOP2_maxP_TE function is designed to identify the
design that maximizing power for TE boundary, which meets the
pre-specified Type I and Type II error rates by using PSO_GO algorithm.
When pso_method equals “all”, GBOP2_maxP_TE
function conducts PSO_Ensemble in parallel, otherwise, users can specify
the pso_method option as default,
quantum, or dexp to conduct a single PSO
algorithm. The following example show the design that maximizing power
for TE boundary identified by the PSO-GO.
## PSO-GO
init_cluster(5) ## Initialize 5 parallel cluster
## Note: Users must explicitly set the number of cores using init_cluster()
## each time before running GBOP2_maxP_TE(). Parallel computing will be
## automatically terminated once the function completes. This setup complies
## with CRAN's strict checks on parallel computing.
power_GO_TE <- GBOP2_maxP_TE(pso_method = "all", ## choose from "all", "default", "quantum" or "dexp"
nlooks =1 , ## number of interim analyses
skip_efficacy = NULL, ## Default: NULL no skipping for efficacy.
## Input a vector as the same length as total stages, 1 is skip.
## Ex: skip for efficacy for the first stage of a two stage design input c(1,0).
skip_toxicity = NULL, ## Default: NULL no skipping for toxicity.
## Input a vector as the same length as total stages, 1 is skip.
## Ex: skip for toxicity. for the first stage of a two stage design input c(1,0).
totalPatients = 26, ## total number of patients
Nmin_cohort1 = 10, ## minimum number of patients in the first stage
Nmin_increase = 5, ## minimum increment in cohort size per stage
p01 = 0.3, ## efficacy rate under null hypothesis
p02 = 0.4, ## toxicity rate under null hypothesis
p03 = 0.2, ## joint efficacy-toxicity correlation under null
p11 = 0.6, ## efficacy rate under alternative hypothesis
p12 = 0.2, ## toxicity rate under alternative hypothesis
p13 = 0.15, ## joint efficacy-toxicity correlation under alternative
err_eff = 0.1, ## Type I error for efficacy (futile but safe)
err_tox = 0.1, ## Type I error for toxicity (effective but toxic)
err_all = 0.05, ## Type I error for futile and toxic outcomes
power_eff = 0.8, ## power target for efficacy
power_tox = 0.8, ## power target for toxicity
power_all = 0.8, ## power target for joint efficacy-toxicity
nSwarm = 32, ## nSwarm in PSO
nParallel = 3, ## number of parallel PSO_Ensemble
seed = 123,
maxIter = 100 ## maxIter in PSO
)
Interpretation
The GBOP2_maxP_TE function identified an optimal
two-stage design with interim times of 21 and 26
patients to maximize power under joint toxicity-efficacy (TE)
boundaries. The trial stops early for futility (ineffectiveness) if
efficacy responses are \(≤\)
8 at stage 1 or \(≤\)
9 at stage 2, and stops early for toxicity (safety
concerns) if toxicities are \(≥\)
7 at stage 1 or \(≥\)
8 at stage 2. This design achieves a statistical power
over 80%, effectively controlling Type I error rates
and balancing patient safety, efficacy detection, and trial
efficiency.
summary(power_GO_TE)
#> ------------------------------------------------
#> PSOGO for maximizing power with TE boundaries
#> Design:
#> Method: dexp
#>
#> Cohort Sizes: 21, 26
#> Boundary Values:
#> Efficacy: stop if <= 8, stop if <= 9
#> Toxicity: stop if >= 7, stop if >= 8
#>
#> Type I Errors:
#> Type I (H01): 0.0938436
#> Type I (H10): 0.0991404
#> Type I (H00): 0.00389506
#>
#> Power (H11): 0.814415
#> Utility Value: 0.814415
#>
#> Parameters:
#> Lambda_e1: 0.8675
#> Lambda_e2: 0.765116
#> Lambda_t1: 0.794352
#> Lambda_t2: 0.909879
#> Gamma: 0.263865
#> ------------------------------------------------stop_cluster() ## Stops the currently running parallel cluster and reverts to sequential execution.
get_cluster() ## Parallel execution terminated — make sure returning NULL.
#> NULL5. Reference
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