Study Objective: sample size minimization
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
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: 1.586 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
#>
#> ------------------------------------------------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
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
#>
#> ------------------------------------------------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
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: 0.755894
#> Lambda_e2: 0.840601
#> Lambda_t1: 0.696103
#> Lambda_t2: 0.888233
#> Gamma: 0.944276
#>
#> ------------------------------------------------