Simulating multi-arm multi-stage designs
mams.sim.RdThe function simulates multi-arm multi-stage designs and estimates power and expected sample size.
Usage
mams.sim(obj=NULL,nsim=NULL, nMat=NULL,
u=NULL, l=NULL, pv=NULL, deltav=NULL, sd=NULL, ptest=NULL,
parallel=NULL, H0=NULL, K = NULL)Arguments
- obj
an object of class
MAMS. The parameters/design of the considered in the output of themams()function are considered as reference for the simulation. If other parameters are given, their values override the parameters of theMAMSobject- nsim
Number of simulations (default=`50000“).
- nMat
Jx(K+1) dimensional matrix of observed/expected sample sizes. Rows correspond to stages and columns to arms. First column is control (default:
NULL).- u
Vector of previously used upper boundaries (default=
NULL).- l
Vector of previously used upper boundaries (default=
NULL).- pv
Vector of size K of true treatment effects on the probability scale. See Details (default=
NULL).- deltav
Vector of size K of true treatment effects on the traditional scale. See Details (default=
NULL).- sd
Standard deviation. See Details (default=
NULL).- ptest
Vector of treatment numbers for determining power. For example, c(1, 2) will count rejections of one or both hypotheses for testing treatments 1 and 2 against control (default=
1).- parallel
if
TRUE(default), allows parallelization of the computation via a user-defined strategy specified by means of the functionplan. If not set differently, the default strategy issequential, which corresponds to a computation without parallelization.- H0
if
TRUE(default), the simulation also considers the case with all effect sizes set to 0.- K
Allocation for treatment arms (used only with method = "dtl")
Value
An object containing the following components:
- l
Lower boundary.
- u
Upper boundary.
- n
Sample size on control in stage 1.
- N
Maximum total sample size.
- K
Number of experimental treatments.
- J
Number of stages in the trial.
- rMat
Matrix of allocation ratios. First row corresponds to control and second row to experimental treatments.
- nsim
Number of simulation runs.
- typeI
The proportion any hypothesis is rejected.
- power
The proportion the first hypothesis is rejected and the corresponding test statistic is largest.
- ptest
The vector
ptest.- prop.rej
The proportion of times at least one of the hypothesis specified by
ptestis rejected.- exss
The expected sample size.
Details
This function simulates multi-arm multi-stage studies for a given matrix of
sample sizes and boundaries given by the vectors u and l.
The effect difference between each experimental treatment and control is
given by pv and is parameterized as
\(P(X_k > X_0 ) = p\).
That is the probability of a randomly selected person on treatment k
observing a better outcome than a random person on control.
For pv=rep(0.5,4) the experimental treatments and control perform equally
well (i.e. the global null hypothesis is true).
The advantage of this parameterization is that no knowledge about the
variance is required. To convert traditional effect sizes,
\(\delta\) to this format use
\(p=\Phi(\frac{\delta}{\sqrt{2}\sigma})\).
Alternatively, the effect size can also be specified directly on the
traditional scale of deltav with an additional specification of
the standard deviation sd.
he function returns the probability of rejecting any hypothesis (typeI),
the power to reject the first hypothesis when the first treatment has the
largest estimated effect, the proportion of rejections of the hypothesis
specified by ptest (prop.rej) as well as the expected
sample size.
References
Jaki T., Pallmann P., and Magirr D. (2019), The R Package MAMS for Designing Multi-Arm Multi-Stage Clinical Trials, Journal of Statistical Software, 88(4), 1-25. Link: doi:10.18637/jss.v088.i04
Magirr D., Jaki T., and Whitehead J. (2012), A generalized Dunnett test for multi-arm multi-stage clinical studies with treatment selection, Biometrika, 99(2), 494-501. Link: doi:10.1093/biomet/ass002
See also
mams.
Examples
# \donttest{
# Note that some of these examples may take a few minutes to run
# 2-stage design with O'Brien & Fleming efficacy and zero futility boundary
# with equal sample size per arm and stage. Design can be found using
# mams(K=4, J=2, alpha=0.05, power=0.9, r=1:2, r0=1:2, ushape="obf",
# lshape="fixed",
# lfix=0, p=0.65, p0=0.55)
# under global null hypothesis (using the pv scale)
mams.sim(nsim=10000, nMat=matrix(c(44, 88), nrow=2, ncol=5),
u=c(3.068, 2.169),
l=c(0.000, 2.169), pv=rep(0.5, 4), ptest=1)
#> Warning: Standard deviation set to 1
#>
#> Design parameters for a 2 stage trial with 4 treatments:
#>
#> Stage 1 Stage 2
#> Cumulative sample size per stage (control): 44 88
#> Cumulative sample size per stage (active): 44 88
#>
#> Maximum total sample size: 440
#>
#> Stage 1 Stage 2
#> Upper bound: 3.068 2.169
#> Lower bound: 0.000 2.169
#>
#>
#> Simulated error rates based on 10000 simulations:
#>
#> Prop. rejecting at least 1 hypothesis: 0.047
#> Prop. rejecting first hypothesis (Z_1>Z_2,...,Z_K) 0.011
#> Prop. rejecting hypothesis 1: 0.013
#> Expected sample size: 341.889
#>
# under global null hypothesis (using the deltav scale)
mams.sim(nsim=10000, nMat=matrix(c(44, 88), nrow=2, ncol=5),
u=c(3.068, 2.169),
l=c(0.000, 2.169), pv=NULL, deltav=rep(0, 4), sd=1, ptest=1)
#>
#> Design parameters for a 2 stage trial with 4 treatments:
#>
#> Stage 1 Stage 2
#> Cumulative sample size per stage (control): 44 88
#> Cumulative sample size per stage (active): 44 88
#>
#> Maximum total sample size: 440
#>
#> Stage 1 Stage 2
#> Upper bound: 3.068 2.169
#> Lower bound: 0.000 2.169
#>
#>
#> Simulated error rates based on 10000 simulations:
#>
#> Prop. rejecting at least 1 hypothesis: 0.050
#> Prop. rejecting first hypothesis (Z_1>Z_2,...,Z_K) 0.013
#> Prop. rejecting hypothesis 1: 0.016
#> Expected sample size: 342.184
#>
# under LFC
mams.sim(nsim=10000, nMat=matrix(c(44, 88), nrow=2, ncol=5),
u=c(3.068, 2.169),
l=c(0.000, 2.169), pv=c(0.65, 0.55, 0.55, 0.55), ptest=1:2)
#> Warning: Standard deviation set to 1
#>
#> Design parameters for a 2 stage trial with 4 treatments:
#>
#> Stage 1 Stage 2
#> Cumulative sample size per stage (control): 44 88
#> Cumulative sample size per stage (active): 44 88
#>
#> Maximum total sample size: 440
#>
#> Stage 1 Stage 2
#> Upper bound: 3.068 2.169
#> Lower bound: 0.000 2.169
#>
#>
#> Simulated error rates based on 10000 simulations:
#>
#> Prop. rejecting at least 1 hypothesis: 0.928
#> Prop. rejecting first hypothesis (Z_1>Z_2,...,Z_K) 0.906
#> Prop. rejecting hypotheses 1 or 2: 0.922
#> Expected sample size: 346.386
#>
# when all treatments doing similarly well
mams.sim(nsim=10000, nMat=matrix(c(44, 88), nrow=2, ncol=5),
u=c(3.068, 2.169),
l=c(0.000, 2.169), pv=c(0.63, 0.62, 0.60, 0.61), ptest=4)
#> Warning: Standard deviation set to 1
#>
#> Design parameters for a 2 stage trial with 4 treatments:
#>
#> Stage 1 Stage 2
#> Cumulative sample size per stage (control): 44 88
#> Cumulative sample size per stage (active): 44 88
#>
#> Maximum total sample size: 440
#>
#> Stage 1 Stage 2
#> Upper bound: 3.068 2.169
#> Lower bound: 0.000 2.169
#>
#>
#> Simulated error rates based on 10000 simulations:
#>
#> Prop. rejecting at least 1 hypothesis: 0.942
#> Prop. rejecting first hypothesis (Z_1>Z_2,...,Z_K) 0.401
#> Prop. rejecting hypothesis 4: 0.497
#> Expected sample size: 363.119
#>
##
## example considering different parallelisation strategies
##
# parallel = FALSE (future framework not used)
set.seed(1)
system.time(
print(mams.sim(nsim=25000, nMat=matrix(c(44, 88), nrow=2, ncol=5),
u=c(3.068, 2.169),
l=c(0.000, 2.169), pv=c(0.65, 0.55, 0.55, 0.55),
ptest=1:2, parallel=FALSE))
)
#> Warning: Standard deviation set to 1
#>
#> Design parameters for a 2 stage trial with 4 treatments:
#>
#> Stage 1 Stage 2
#> Cumulative sample size per stage (control): 44 88
#> Cumulative sample size per stage (active): 44 88
#>
#> Maximum total sample size: 440
#>
#> Stage 1 Stage 2
#> Upper bound: 3.068 2.169
#> Lower bound: 0.000 2.169
#>
#>
#> Simulated error rates based on 25000 simulations:
#>
#> Prop. rejecting at least 1 hypothesis: 0.928
#> Prop. rejecting first hypothesis (Z_1>Z_2,...,Z_K) 0.905
#> Prop. rejecting hypotheses 1 or 2: 0.921
#> Expected sample size: 346.755
#>
#> user system elapsed
#> 5.83 0.22 6.35
# parallel = TRUE (default) with default strategy (sequential computation)
plan(sequential)
set.seed(1)
system.time(
print(mams.sim(nsim=25000, nMat=matrix(c(44, 88), nrow=2, ncol=5),
u=c(3.068, 2.169),
l=c(0.000, 2.169), pv=c(0.65, 0.55, 0.55, 0.55), ptest=1:2))
)
#> Warning: Standard deviation set to 1
#>
#> Design parameters for a 2 stage trial with 4 treatments:
#>
#> Stage 1 Stage 2
#> Cumulative sample size per stage (control): 44 88
#> Cumulative sample size per stage (active): 44 88
#>
#> Maximum total sample size: 440
#>
#> Stage 1 Stage 2
#> Upper bound: 3.068 2.169
#> Lower bound: 0.000 2.169
#>
#>
#> Simulated error rates based on 25000 simulations:
#>
#> Prop. rejecting at least 1 hypothesis: 0.928
#> Prop. rejecting first hypothesis (Z_1>Z_2,...,Z_K) 0.905
#> Prop. rejecting hypotheses 1 or 2: 0.921
#> Expected sample size: 346.755
#>
#> user system elapsed
#> 8.84 0.45 9.70
# parallel = TRUE (default) with multisession strategy (parallel computation)
plan(multisession)
set.seed(1)
system.time(
print(mams.sim(nsim=25000, nMat=matrix(c(44, 88), nrow=2, ncol=5),
u=c(3.068, 2.169),
l=c(0.000, 2.169), pv=c(0.65, 0.55, 0.55, 0.55), ptest=1:2))
)
#> Warning: Standard deviation set to 1
#>
#> Design parameters for a 2 stage trial with 4 treatments:
#>
#> Stage 1 Stage 2
#> Cumulative sample size per stage (control): 44 88
#> Cumulative sample size per stage (active): 44 88
#>
#> Maximum total sample size: 440
#>
#> Stage 1 Stage 2
#> Upper bound: 3.068 2.169
#> Lower bound: 0.000 2.169
#>
#>
#> Simulated error rates based on 25000 simulations:
#>
#> Prop. rejecting at least 1 hypothesis: 0.928
#> Prop. rejecting first hypothesis (Z_1>Z_2,...,Z_K) 0.905
#> Prop. rejecting hypotheses 1 or 2: 0.921
#> Expected sample size: 346.755
#>
#> user system elapsed
#> 6.72 0.54 9.78
plan("default")
# }