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Keywords

with, time, design, based, designs, survival, Weibull, event-free, Huang, Optimal, sample, two-stage, function, Thomas, OptimDes, control, distributions, FixDes, that, accrual, Package, optimal, binomial, exact, target, test., projected, OptInterim-package, distribution, treatment

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Package ‘OptInterim’
February 19, 2015
Type Package
Title Optimal Two and Three Stage Designs for Single-Arm and Two-Arm
Randomized Controlled Trials with a Long-Term Binary Endpoint
Version 3.0.1
Date 2012-9-30
Author Bo Huang and Neal Thomas

Maintainer Bo Huang
Description Optimal two and three stage designs monitoring
time-to-event endpoints at a specified timepoint
Depends mvtnorm, clinfun, stats, graphics, R (>= 2.14.1)
License GPL (>= 2)
Repository CRAN
Date/Publication 2012-12-12 20:39:06
NeedsCompilation no
R topics documented:
OptInterim-package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
FixDes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
np.OptimDes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
OptimDes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
OptimDesControl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
plot.OptimDes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
print.OptimDes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
SimDes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
TestStage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
weibPmatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
weibull.plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Index 25
1
2 OptInterim-package
OptInterim-package Optimal Two and Three Stage Designs with Time-to-event Endpoints
Evaluated at a Pre-specified Time.
Description
OptimDes is the primary function in the package OptInterim. Detailed documentation of the design
and a package vignette with examples can be found under "directory" on the package HTML help
page. The user supplies the projected accrual distribution and distributions for time to event under
the null and alternative hypotheses. OptimDes returns the total sample size, the interim time and
sample size, and boundaries for decision rules.
plot.OptimDes can be applied to the output of OptimDes to evaluate other two-stage or threestage designs that may achieve near optimal results for both of the implemented optimality criteria.
print.OptimDes produces a summary of the design with a comparison to a standard single-stage
design (which can also be obtained from FixDes) Design parameters for nearly optimal designs can
be evaluated with np.OptimDes. Statistical tests for a two-stage or three-stage design are implemented in TestStage.
The function SimDes produces simulations under the assumed design by default to check the accuracy of asymptotic approximations. By supplying alterative accrual and survival distributions, the
performance of the design and estimation can be checked for robustness.
The function weibPmatch can be used to find parameters of a Weibull distribution that match a
event-free rate at a specified time. Function weibull.plot displays a null-alternative pair of hypothesized Weibull survival functions.
Details
Package: OptInterim
Type: Package
Version: 3.0
Date: 2012-12-03
License: GPL version 2 or later
Author(s)
Bo Huang <> and Neal Thomas <>
See Also
Package Survival
FixDes 3
FixDes Construct a Single-stage Design for a time-to-event Endpoint Evaluated at a Pre-specifed Time Versus Either a Known Standard Control
or a Randomized Comparative Control
Description
Find the sample size, duration of accrual, and test boundary for a single-stage design with an eventfree endpoint versus either a known standard control or a randomized comparative control. Testing
is one-sided based on the Kaplan-Meier estimator evaluated at a user-specified time point.
Usage
FixDes(B.init, m.init, alpha, beta, param, x, num.arm, r=0.5)
Arguments
B.init A vector of user-specified time points (B1, ..., Bb) that determine a set of time
intervals with uniform accrual.
m.init The projected number of patients that can be accrued within the time intervals
determined by B.init.
alpha Type I error.
beta Type II error.
param Events should be defined as poor outcomes (e.g. death, progression). Computations and reporting are based on the proportion without an event at a prespecified time, x. For constructing an optimal design, complete event-free distributions at all times must be specified for the control condition (Null), and for
the alternative "effective" treatment. Weibull distributions are currently implemented. param is a vector of length 4: (shape null, scale null, shape alternative,
scale alternative). The R parameterization of the Weibull distribution is used.
x Pre-specified time for the event-free endpoint (e.g., 1 year).
num.arm Number of treatment arms. num.arm=1 for single-arm trial assuming a known
standard control. num.arm=2 for two-arm randomized trial with a comparative
control arm.
r Proportion of patients randomized to the treatment arm. By default, r=0.5.
Details
Estimation is based on the Kaplan-Meier or Nelson-Aalen estimators evaluated at a target time (e.g.,
1 year). The event-free rates at the target time are computed from Weibull distributions assumed for
the treatment and control distributions, as is done in function OptimDes. The design depends only
on the event-free rates at the target time (except for small changes due to rounding with different
survival functions). The duration of accrual depends on the projected maximum accrual rates.
4 FixDes
Value
A list with components:
n0 Fixed design sample size.
DA Duration of accrual.
SL Total study length (time, DA+x).
n0E n0 based on exact binomial test.
DAE DA based on exact binomial test.
SLE SL based on exact binomial test.
C Rejection cutpoint for the test statistic.
Note
The single-stage sample size is used as the starting value for evaluating the optimal n for a two-stage
design in OptimDes.
Author(s)
Bo Huang <> and Neal Thomas <>
References
Huang B., Talukder E. and Thomas N. Optimal two-stage Phase II designs with long-term endpoints. Statistics in Biopharmaceutical Research, 2(1), 51–61.
Case M. D. and Morgan T. M. (2003) Design of Phase II cancer trials evaluating survival probabilities. BMC Medical Research Methodology, 3, 7.
Lin D. Y., Shen L., Ying Z. and Breslow N. E. (1996) Group seqential designs for monitoring
survival probabilities. Biometrics, 52, 1033–1042.
Simon R. (1989) Optimal two-stage designs for phase II clinical trials. Controlled Clinical Trials,
10, 1–10.
See Also
OptimDes, TestStage, SimDes
Examples
B.init <- c(1, 2, 3, 4, 5)
m.init <- c(15, 20, 25, 20, 15)
alpha <- 0.05
beta <- 0.1
param <- c(1, 1.09, 2, 1.40)
x <- 1
# H0: S0=0.40 H1: S1=0.60
FixDes(B.init, m.init, alpha, beta, param, x,num.arm=1)
np.OptimDes 5
m.init <- 5*c(15, 20, 25, 20, 15)
FixDes(B.init, m.init, alpha, beta, param, x,num.arm=2)
np.OptimDes Optimal Two-Stage or Three-Stage Designs with User-specified Combined Sample Size or Study Length
Description
Construct a two-stage or three-stage design with a time-to-event endpoint evaluated at a pre-specified
time (e.g., 6-month progression-free survival) comparing treatment versus either a historical control
rate with possible stopping for futility (single-arm), or an active control arm with possible stopping
for both futility and superiority (two-arm), after the end of Stage 1 utilizing time to event data. The
design minimizes either the expected duration of accrual (EDA), the expected sample size (ES),
or the expected total study length (ETSL). The maximum combined sample size for both stages is
pre-specifed by the user.
Usage
np.OptimDes(
B.init, m.init, alpha, beta, param, x, n = NULL, pn = NULL,
pt = NULL, target = c("EDA", "ETSL","ES"), sf=c("futility","OF","Pocock"),
num.arm,r=0.5,num.stage=2,pause=0,
control = OptimDesControl(), ...)
Arguments
B.init A vector of user-specified time points (B1, ..., Bb) that determine a set of time
intervals with uniform accrual.
m.init The projected number of patients that can be accrued within the time intervals
determined by B.init.
alpha Type I error.
beta Type II error.
param Events should be defined as poor outcomes (e.g. death, progression). Computations and reporting are based on the proportion without an event at a prespecified time, x. For constructing an optimal design, complete event-free distributions at all times must be specified for the control condition (Null), and for
the alternative "effective" treatment. Weibull distributions are currently implemented. param is a vector of length 4: (shape null, scale null, shape alternative,
scale alternative). The R parameterization of the Weibull distribution is used.
x Pre-specified time for the event-free endpoint (e.g., 1 year).
n User-specified combined sample size for both stages.
6 np.OptimDes
pn Combined sample size for both stages specified by the ratio of the targetted twostage sample size to the correponding sample size for a single-stage design.
pt Combined sample size for both stages specified by the ratio of the targetted twostage study length to the correponding study length for a single-stage design.
target The expected duration of accrual (EDA) is minimized with target="EDA", the
expected total study length is minimized with target="ETSL", and the expected
sample size with target="ES".
sf Spending function for alpha at the end of Stage 1. There are three types of
spending functions: no efficacy stopping with sf="futility", O’Brien-Fleming
boundaries with sf="OF", and Pocock boundaries with sf="Pocock".
num.arm Number of arms: a single-arm design with num.arm=1, or a randomized twoarm design with num.arm=2.
r Proportion of patients randomized to the treatment arm when num.arm=2. By
default, r=0.5.
num.stage Number of stages: a two-stage design with num.stage=2, or a three-stage design
with num.stage=3.
pause The pause in accrual following the scheduled times for interim analyses. Data
collected during the pause on the previously accrued patients are included in the
interim analysis conducted at the end of the pause. Accrual resumes after the
pause without interuption as if no pause had occurred. Default is pause=0.
control An optional list of control settings. See OptimDesControl for the parameters
that can be set and their default values.
... No additional optional parameters are currently implemented
Details
Plots (plot.OptimDes) based on the ouput of OptimDes can be used to find compromise designs
based on different combined sample sizes with near optimal values for both ETSL ES, and EDA.
np.OptimDes can be used to compute ETSL, ES, EDA, and the other design parameters for any
specified total sample size.
The targeted combined sample size must be specified by one of three equivalent approaches: n,
pn, and pt. The design calculations assume Weibull distributions for the event-free endpoint in the
treatment group, and for the (assumed known, "Null") control distribution.
The function weibPmatch can be used to select Weibull parameters that yield a target event-free
rate at a specified time.
Value
A list of class OptimDes with the same output as function OptimDes.
Author(s)
Bo Huang <> and Neal Thomas <>
OptimDes 7
References
Huang B., Talukder E. and Thomas N. (2010). Optimal two-stage Phase II designs with long-term
endpoints. Statistics in Biopharmaceutical Research, 2, 51–61.
Case M. D. and Morgan T. M. (2003) Design of Phase II cancer trials evaluating survival probabilities. BMC Medical Research Methodology, 3, 7.
Lin D. Y., Shen L., Ying Z. and Breslow N. E. (1996) Group seqential designs for monitoring
survival probabilities. Biometrics, 52, 1033–1042.
Simon R. (1989) Optimal two-stage designs for phase II clinical trials. Controlled Clinical Trials,
10, 1–10.
See Also
OptimDes, plot.OptimDes, weibPmatch
Examples
## Not run:
B.init <- c(1, 2, 3, 4, 5)
m.init <- c(15, 20, 25, 20, 15)
alpha <- 0.05
beta <- 0.1
param <- c(1, 1.09, 2, 1.40)
x <- 1
# H0: S0=0.40 H1: S1=0.60
object14 <- np.OptimDes(B.init,m.init,alpha,beta,param,x,pt=1.1,target="ETSL",sf="futility",num.arm=1,num.stage=2)
## End(Not run)
OptimDes Construct Optimal Two-stage or Three-stage Designs with Time-toevent Endpoints Evaluated at a Pre-specified Time
Description
Construct an optimal two-stage or three-stage designs with a time-to-event endpoint evaluated at
a pre-specified time (e.g., 6 months) comparing treatment versus either a historical control rate
with possible stopping for futility (single-arm), or an active control arm with possible stopping for
both futility and superiority (two-arm), after the end of Stage I utilizing time to event data. The
design minimizes either the expected duration of accrual (EDA), expected sample size (ES), or the
expected total study length (ETSL).
8 OptimDes
Usage
OptimDes(
B.init, m.init, alpha, beta, param, x, target = c("EDA", "ETSL","ES"),
sf=c("futility","OF","Pocock"), num.arm,r=0.5, num.stage=2,
pause=0,
control = OptimDesControl(),...)
Arguments
B.init A vector of user-specified time points (B1, ..., Bb) that determine a set of time
intervals with uniform accrual.
m.init The projected number of patients that can be accrued within the time intervals
determined by B.init. A large number of potential patients results in long execution times for OptimDes, so unrealistically large values should not be entered.
alpha Type I error.
beta Type II error.
param Events should be defined as poor outcomes (e.g. death, progression). Computations and reporting are based on the proportion without an event at a prespecified time, x. For constructing an optimal design, complete event-free distributions at all times must be specified for the control condition (Null), and for
the alternative "effective" treatment. Weibull distributions are currently implemented. param is a vector of length 4: (shape null, scale null, shape alternative,
scale alternative). The R parameterization of the Weibull distribution is used.
x Pre-specified time for the event-free endpoint (e.g., 1 year).
target The expected duration of accrual (EDA) is minimized with target="EDA", the
expected total study length is minimized with target="ETSL", or the expected
sample size with target="ES".
sf Spending function for alpha at the end of Stage 1. There are three types of
spending functions: no efficacy stopping with sf="futility", O’Brien-Fleming
boundaries with sf="OF", and Pocock boundaries with sf="Pocock".
num.arm Number of arms: a single-arm design with num.arm=1, or a randomized twoarm design with num.arm=2.
r Proportion of patients randomized to the treatment arm when num.arm=2. By
default, r=0.5.
num.stage Number of stages: a two-stage design with num.stage=2, or a three-stage design
with num.stage=3.
pause The pause in accrual following the scheduled times for interim analyses. Data
collected during the pause on the previously accrued patients are included in the
interim analysis conducted at the end of the pause. Accrual resumes after the
pause without interuption as if no pause had occurred. Default is pause=0.
control An optional list of control settings. See OptimDesControl. for the parameters
that can be set and their default values.
... No additional optional parameters are currently implemented.
OptimDes 9
Details
OptimDes finds an two-stage or three-stage design with a time to event endpoint evaluated at a
pre-specified time with potential stopping after the first stage.
For single arm designs, it implements the Case and Morgan (2003) and Huang, Talukder and
Thomas (2010) generalizaton of the Simon (1989) two-stage design for comparing a treatment to a
known standard rate with possible stopping for futility at the interim.
For randomized two-arm comparative designs, it allows an early stopping for both futility and superiority. The spending function for superiority can be chosen with argument sf.
The design minimizes either the expected duration of accrual (EDA), expected sample size (ES), or
the expected total study length (ETSL).
The design calculations assume Weibull distributions for the event-free endpoint in the treatment
group, and for the (assumed known, "Null") control distribution. The function weibPmatch can be
used to select Weibull parameters that yield a target event-free rate at a specified time. Estimation is
based on the Kaplan-Meier or Nelson-Aalen estimators evaluated at a target time (e.g., 1 year). The
full treatment and control distributions and the accrual distribution affect power (and alpha level in
some settings), see Case and Morgan (2003)).
Accrual rates are specified by the user. These rates can differ across time intervals specified by the
user (this generalizes the results in Case and Morgan).
A package vignette as user manual can be found in the /doc subdirectory of the OptInterim package. It can be accessed from the HTML help page for the package.
Value
A list with components:
target The optimizaton target ("EDA" or "ETSL" or "ES").
sf The alpha spending function ("futility" or "OF" or "Pocock").
test A vector giving the type I error alpha, type II error beta, Weibull parameters
param and survival time of interest x.
design A vector giving the number of study arms num.arm, treatment randomization
rate r, the number of study stages num.stage, the pause in accrual before an
interim analysis pause.
accrual A list containing the input vectors B.init and m.init.
result A 5-element vector containing the expected duration of accrual (EDA), the expected total study length (ETSL), the expected sample size of the optimal design
(ES), and the probability of stopping under the Null (pstopNULL) and Alternative hypotheses (pstopAlt).
n A two (or three)-element vector containing the sample size for the interim analysis and the sample size if all stages are completed.
stageTime A 3 (or 4)-element vector giving the times for the interim and final analyses, and
maximum duration of accrual. Interim times are given for the beginning of any
pause before the analysis occurs.
boundary A vector giving the rejection cutpoints (see Test2stage) for the test statistic
and decision rules.
10 OptimDes
se A vector of length 4 (or 6) with the asymptotic standard errors at the iterim
and final analysis under the null hypothesis, followed by the corresponding SEs
under the alternative hypothesis. These SEs must be divided by the square root
of sample size.
u A two (or three)-element vector giving means of interim test statistics under H1.
See detailed description. It is also used to compute conditional power.
exposure The expected total exposure of patients at the time of the planned interim analysis (including any accrual pause). Patient exposure is truncated by both the
interim analysis time (including any pause) and the target surival time (i.e., no
exposure after x). Exposure is a vector of length 1 or 2. The first value is the
expected exposure at the first interim analysis. For two-stage, single-group designs, the second value is exposure with the Case-Morgan finite sample adjustment. For 3-stage designs, the second value is the exposure at the second interim
analysis. For two-stage, two-group designs, exposure is a scalar indicating the
expected exposure at the first interim analysis.
all.info A data frame containing the results for all of the evaluated sample sizes.
single.stage A six-element vector giving the sample size fix.n, duration of accrual DA, study
length SL, and corresponding values based on the exact binomial test for a onearm single-stage design and the Fisher exact test for a two-arm single-stage design with the design distributional assumptions.
Note
The algorithm will search for the optimal n between the sample size for a single-stage design and
the user specified maximum sample size sum(m.init).
When the length of B.init or m.init is 1, the accrual rate is constant as in Lin et al. (1996), Case
and Morgan (2003).
Author(s)
Bo Huang <> and Neal Thomas <>
References
Huang B., Talukder E. and Thomas N. (2010). Optimal two-stage Phase II designs with long-term
endpoints. Statistics in Biopharmaceutical Research, 2, 51–61.
Case M. D. and Morgan T. M. (2003). Design of Phase II cancer trials evaluating survival probabilities. BMC Medical Research Methodology, 3, 7.
Lin D. Y., Shen L., Ying Z. and Breslow N. E. (1996). Group seqential designs for monitoring
survival probabilities. Biometrics, 52, 1033–1042.
Simon R. (1989). Optimal two-stage designs for phase II clinical trials. Controlled Clinical Trials,
10, 1–10.
See Also
np.OptimDes, print.OptimDes, plot.OptimDes, weibPmatch
OptimDesControl 11
Examples
## Not run:
B.init <- c(1, 2, 3, 4, 5)
m.init <- c(15, 20, 25, 20, 15)
alpha <- 0.05
beta <- 0.1
param <- c(1, 1.09, 2, 1.40)
x <- 1
# H0: S0=0.40 H1: S1=0.60
object12 <- OptimDes(B.init,m.init,alpha,beta,param,x,target="EDA",
sf="futility",num.arm=1,num.stage=2,control=OptimDesControl(n.int=c(1,5),trace=TRUE))
print(object12)
m.init <- 4*c(15, 20, 25, 20, 15)
object2 <- OptimDes(B.init,m.init,alpha,beta,param,x,target="EDA",sf="futility",num.arm=2)
print(object2)
object23O <- OptimDes(B.init,m.init,alpha,beta,param,x,target="ETSL",sf="OF",
num.arm=2,num.stage=3,control=OptimDesControl(trace=TRUE,aboveMin=c(1.05,1.10)))
print(object3)
## End(Not run)
OptimDesControl Set parameters controlling numerical methods for OptimDes and
np.OptimDes
Description
Set parameters controlling numerical methods for OptimDes and np.OptimDes
Usage
OptimDesControl(trace=TRUE,tol=0.01,n.int=c(1,5),aboveMin=c(1.05,1.10))
Arguments
trace A logical value indicating if a trace of the iteration progress should be printed.
Default is FALSE. If TRUE the sample size n, the corresponding optimal correlation rho and minimized EDA(ETSL) are printed at the conclusion of each iteration.
tol tol is the desired accuracy for optimize. Default is 0.01.
n.int A two-element vector containing the grid search interval for maximum sample
size of one-arm and two-arm designs. Default is c(1, 5).
12 plot.OptimDes
aboveMin The minimization method searches by increasing order from the single-stage
sample size until the criteria exceeds the current minimum by a multiplicative
factor of aboveMin. The search will also terminate if sum(m.init) is reached.
Default is 1.05 for one-arm, 1.10 for two-arm designs.
Author(s)
Bo Huang <> and Neal Thomas <>
References
Huang B., Talukder E. and Thomas N. Optimal two-stage Phase II designs with long-term endpoints. Statistics in Biopharmaceutical Research, 2(1), 51–61.
Case M. D. and Morgan T. M. (2003) Design of Phase II cancer trials evaluating survival probabilities. BMC Medical Research Methodology, 3, 7.
Lin D. Y., Shen L., Ying Z. and Breslow N. E. (1996) Group seqential designs for monitoring
survival probabilities. Biometrics, 52, 1033–1042.
Simon R. (1989) Optimal two-stage designs for phase II clinical trials. Controlled Clinical Trials,
10, 1–10.
See Also
OptimDes, np.OptimDes
plot.OptimDes Plot efficiency of optimal two-stage or three-stage designs as a function of the total sample size or study length
Description
Output from function OptimDes is used to display the ETSL, ES and EDA for a two-stage or threestage design relative to a single-stage design as a function of the combined-stage sample size or
study length.
Usage
## S3 method for class 'OptimDes'
plot(x, xscale = "t", l.type = 1:5, l.col =
c("blue", "green", "purple", "red", "dark red"), CMadj=F,...)
Arguments
x Output from function OptimDes.
xscale Scale of the x-axis. "t" for combined-stage study length. "n" for combined
sample size. Default is t.
l.type Line types for the plot. Default is 1-5.
plot.OptimDes 13
l.col Line colors for the plot. Default is "blue" for ETSL, "green" for EDA, "purple"
for ES, "red" for t1 and "dark red" for t2 if it’s a three-stage design.
CMadj If true, the sample sizes and times are adjusted by the ratio of the exact binomial
to asymptotic normal sample size for the single stage design, as in Case and
Morgan (2003). Proportional adjustment of times and sample sizes are made
even if the accrual rates are not constant. This adjustment is valid for two-stage
1-group designs. Default is false.
... Additional graphical parameters passed to function plot.
Details
The plot displays the tradeoff between ETSL, EDA and ES as a function of the combined sample
size or study length. Robustness of the optimal design to deviations from the target sample size can
be explored. The plots often suggest compromised designs achieving near-optimal results for both
EDA and ETSL may be a better choice. Test boundary values (C1L, C1U, etc), and numerical values
of other design parameters, can be obtained for a design selected from the plots using function
np.OptimDes.
The plot also includes the times of the interim analysis (t1, t1) as a ratio to the time for a corresonding single-stage analysis.
Author(s)
Bo Huang <> and Neal Thomas <>
References
Huang B., Talukder E. and Thomas N. Optimal two-stage Phase II designs with long-term endpoints. Statistics in Biopharmaceutical Research, 2(1), 51–61.
Case M. D. and Morgan T. M. (2003) Design of Phase II cancer trials evaluating survival probabilities. BMC Medical Research Methodology, 3, 7.
Lin D. Y., Shen L., Ying Z. and Breslow N. E. (1996) Group seqential designs for monitoring
survival probabilities. Biometrics, 52, 1033–1042.
Simon R. (1989) Optimal two-stage designs for phase II clinical trials. Controlled Clinical Trials,
10, 1–10.
See Also
print.OptimDes, OptimDes, np.OptimDes
14 print.OptimDes
print.OptimDes Print Objects Output By OptimDes
Description
Print an object output by OptimDes.
Usage
## S3 method for class 'OptimDes'
print(x, dig = 3, all=FALSE, condPow=F,
CMadj=F, ...)
Arguments
x Object output by OptimDes.
dig Number of digits printed.
all If TRUE, results are printed for all total sample sizes evaluated by OptimDes.
condPow The conditional probability of rejecting the null hypothesis computed assuming
the alternative distributions and an interim Z statistic equal to the interim test
boundary C1 is reported when condPow=T.
CMadj If true, the sample sizes and times are adjusted by the ratio of the exact binomial
to asymptotic normal sample size for the single stage design, as in Case and
Morgan (2003). Proportional adjustments of times and sample sizes are made
even if the accrual rates are not constant. Default is false.
... Optional print arguments, see print.default.
Author(s)
Bo Huang <> and Neal Thomas <>
References
Huang B., Talukder E. and Thomas N. Optimal two-stage Phase II designs with long-term endpoints. Statistics in Biopharmaceutical Research, 2(1), 51–61.
Case M. D. and Morgan T. M. (2003) Design of Phase II cancer trials evaluating survival probabilities. BMC Medical Research Methodology, 3, 7.
Lin D. Y., Shen L., Ying Z. and Breslow N. E. (1996) Group seqential designs for monitoring
survival probabilities. Biometrics, 52, 1033–1042.
Simon R. (1989) Optimal two-stage designs for phase II clinical trials. Controlled Clinical Trials,
10, 1–10.
See Also
plot.OptimDes, OptimDes
SimDes 15
SimDes Simulation Studies for Two-Stage or Three-Stage Designs from function OptimDes
Description
Simulation experiments to compare the alpha level, power and other features of two-stage or threestage designs from function OptimDes with the targetted values.
Usage
SimDes(object,B.init,m.init,weib0,weib1,interimRule='e1',
sim.n=1000,e1conv=1/365,CMadj=F,attainI=1,attainT=1,FixDes="F",
Rseed)
Arguments
object Output object of function OptimDes.
B.init A vector of user-specified time points (B1, ..., Bb) that determine a set of time
intervals with uniform accrual. This vector needs to be specified only if its
values differ from the call to OptimDes.
m.init The projected number of patients that can be accrued within the time intervals
determined by B.init. This vector needs to be specified only if its values differ
from the call to OptimDes.
weib0 A vector with the shape and scale for the Weibull distribution under the null
hypothesis. These need to be specified only if they differ from the input to
OptimDes.
weib1 A vector with the shape and scale for the Weibull distribution under the alternative hypothesis. These need to be specified only if they differ from the input to
OptimDes.
interimRule The interim analysis is performed when the planned n1 patients are accrued
regardless of the time required when interimRule=’n1’. The interim analysis
is performed at the planned time t1 regardless of the number of patients accrued when interimRule=’t1’. The interim analysis is performed when the truncated (by x) total exposure matches the total expected exposure when interimRule=’e1’. The default is ’e1’.
sim.n The number of simulation replications.
e1conv Convergence criteria for matching the truncated exposure when interimRule='e1'.
The default is 1/365, which is appropriate provided B.init is specified in years
CMadj If true, the n, n1, and t1 are adjusted by the ratio of the exact binomial to asymptotic normal sample size for the single stage design, as in Case and Morgan
(2003). Proportional adjustment of times and sample sizes are made even if the
accrual rates are not constant. The adjustment to the mda is made through the
adjustment to n rather than by multiplication to ensure consistency with accrual
boundaries. The truncated exposure time is matched to the adjusted time of the
interim analysis. Default is false.
16 SimDes
attainI Samples sizes and times of the interim analyses often differ from the exact targetted values for operational reasons. The attainI permits simulations with a
different interim time or sample size (depending on interimRule) by a specified
fraction.
attainT Simulations with a total sample size (assuming the trial does not stop based on
the interim analysis) that differs from the planned total by a specified fraction.
FixDes If FixDes="E" or "N", a fixed design is simulated with the sample size determined by the Exact or Normal approximation. All other options for modifying
the simuations are ignored. The alpha level and power based on an exact test
and the normal approximation are returned. All other output variables are 0.
The default is "F"
Rseed Optional integer for input to function set.seed. If unspecified, the random seed
status at the time of the function call is used.
Details
sim.n(default is 1000) simulation experiments are conducted to assess how close the empirical type
I and II error rates come to the target values.
Simulation studies can also be used to assess the performance of the optimal design under misspecification of the design parameters. For example, if the Weibull shape and scale parameters
of the time to event distributions are changed, or if the accrual rates are changed. (see Case and
Morgan, 2003, for discussion of this topic).
The function weibPmatch can be used to select Weibull parameters that yield a target event-free
rate at a specified time.
Value
A vector with:
alphaExact Estimated alpha level using an exact test for the final test. It is NA if the design
allows interim stopping for superiority.
alphaNorm Estimated alpha level using approximately normal tests.
powerExact Estimated power using an exact test for the final test. It is NA if the design allows
interim stopping for superiority.
powerNorm Estimated power using approximately normal tests.
eda Estimated mean duration of accrual under the null hypothesis.
etsl Estimated mean total study length under the null hypothesis.
es Estimated mean total sample size under the null hypothesis.
edaAlt Estimated mean duration of accrual under the alternative hypothesis.
etslAlt Estimated mean total study length under the alternative hypothesis.
esAlt Estimated mean total sample size under the alternative hypothesis.
pstopNull The proportion of trials stopped for futility at the interim analysis under the null
hypothesis.
pstopAlt The proportion of trials stopped for futility at the interim analysis under the
alternative hypothesis.
SimDes 17
pstopENull The proportion of trials stopped for efficacy at the interim analysis under the
null hypothesis.
pstopEAlt The proportion of trials stopped for efficacy at the interim analysis under the
alternative hypothesis.
aveE Average total (truncated at x) exposure at time of interim analysis.
pinfoNull The proportion of the total information obtained at the interim analysis under
the null hypothesis.
pinfoNull2 The proportion of the total information obtained at the second interim analysis
under the null hypothesis when num.stage=3.
pinfoAlt The proportion of the total information obtained at the interim analysis under
the alternative hypothesis.
n1 Average sample size at interim.
n2 Average sample size at second interim.
t1 Average time at interim.
t2 Average time at second interim.
difIntSupL Lowest interim survival rate difference stopped for efficacy.
difIintSupH Highest interim survival rate difference not stopped for efficacy.
difIntFutL Lowest interim survival rate difference continued to final analysis based on the
normal approximation.
difIntFutH Highest interim survival rate difference resulting in futility terimination based
on the normal approximation.
difFinSupL Lowest final survival rate difference rejecting null based on the normal approximation.
difFinFutH Highest final survival rate difference without rejecting null based on the normal
approximation.
Author(s)
Bo Huang <> and Neal Thomas <>
References
Huang B., Talukder E. and Thomas N. Optimal two-stage Phase II designs with long-term endpoints. Statistics in Biopharmaceutical Research, 2(1), 51–61.
Case M. D. and Morgan T. M. (2003) Design of Phase II cancer trials evaluating survival probabilities. BMC Medical Research Methodology, 3, 7.
Lin D. Y., Shen L., Ying Z. and Breslow N. E. (1996) Group seqential designs for monitoring
survival probabilities. Biometrics, 52, 1033–1042.
Simon R. (1989) Optimal two-stage designs for phase II clinical trials. Controlled Clinical Trials,
10, 1–10.
See Also
OptimDes, TestStage, weibPmatch
18 TestStage
Examples
## Not run:
B.init <- c(1, 2, 3, 4, 5)
m.init <- c(15, 20, 25, 20, 15)
alpha <- 0.05
beta <- 0.1
param <- c(1, 1.09, 2, 1.40)
x <- 1
# H0: S0=0.40 H1: S1=0.60
object1 <- OptimDes(B.init,m.init,alpha,beta,param,x,target="EDA",sf="futility",num.arm=1)
SimDes(object1,sim.n=100)
### Stopping based on pre=planned time of analysis
SimDes(object1,interimRule='t1',sim.n=100)
### accrual rates differ from planned
SimDes(object1,m.init=c(5,5,25,25,25),sim.n=100)
## End(Not run)
TestStage Statistical test for two-stage or three-stage designs from function OptimDes
Description
This function performs the hypothesis tests for the two-stage or three-stage designs with event-free
endpoint from OptimDes.
Usage
TestStage(tan,tstage,x,num.arm,num.stage,
Y1,T1,Y0=NULL,T0=NULL,p0=NULL,
C1L=NULL,C1U=NULL,C2L=NULL,C2U=NULL,C3U=NULL,
printTest=TRUE,
cen1=rep(1,length(T1)),cen0=rep(1,length(T0)))
Arguments
tan Study time (from first accrual) of the analysis.
tstage tstage=1 for the first interim analysis. tstage=2 for the second analysis interim analysis when num.stage=3, or the final analysis when num.stage=2.
tstage=3 for the final analysis when num.stage=3.
x Pre-specified time for the event-free endpoint (e.g., 1 year).
TestStage 19
num.arm Number of treatment arms. num.arm=1 for single-arm trial and num.arm=2 for
a two-arm randomized trial.
num.stage Number of trial stages: num.stage=2 or num.stage=3.
Y1 A vector containing the study start times (measured from the beginning of the
study) of patients in the treatment arm. If times occuring after the analysis time
tan are included, they are appropriately censored.
T1 A vector containing the event times corresponding to Y1.
Y0 A vector containing the study start times (measured from the beginning of the
study) of patients in the control arm. It does not need to be set for 1-arm trials.
If times occuring after the analysis time tan are included, they are appropriately
censored.
T0 A vector containing the event times corresponding to Y0.
p0 The event rate under the null hypothesis.
C1L The study is terminated for futility after the first stage if the Z-statistic is <=C1.
C1U The study is terminated for efficacy after the first stage if the Z-statistic is
>=C1U.
C2L For a three-stage design, stop for futility after the second stage if Z<=C2.
C2U For a three-stage design, stop for efficacy after the second stage if the Z>=C2U.
For a two-stage design, reject the null hypothesis at the final stage if the Z>=C2U.
C3U For a three-stage design, reject the null hypothesis at the final stage if the Z>=C3U.
printTest If TRUE (default), the result of the test and the interim decision is printed.
cen1 The times in T1 are regarded as events unless they are set to censored by setting
the corresponding value in cen1 to zero.
cen0 The times in T0 are regarded as events unless they are set to censored by setting
the corresponding value in cen0 to zero.
Details
The hypothesis tests are performed in two stages as described in Huang, Talukder and Thomas
(2010) and Case and Morgan (2003) for single-arm designs, and extended to the randomized twoarm two-stage and three-stage designs.
For two-stage designs:
Stage 1. Accrue patients between time 0 and time t1. Each patient will be followed until failure,
or for x years or until time t1, whichever is less. Calculate the normalized interim test statistic Z1.
If Z1<=C1, stop the study for futility; For randomized two-arm trials, if Z1>=C1U, stop the study for
efficacy; otherwise, continue to the next stage.
Stage 2. Accrue patients between t1 and MDA. Follow all patients until failure or for x years, then
calculate the normalized final test statistic Z2, and reject H0 if Z2>=C2.
For three-stage designs:
Stage 1. Accrue patients between time 0 and time t1. Each patient will be followed until failure,
or for x years or until time t1, whichever is less. Calculate the normalized interim test statistic Z1.
If Z1<=C1, stop the study for futility; For randomized two-arm trials, if Z1>=C1U, stop the study for
efficacy; otherwise, continue to the next stage.
20 TestStage
Stage 2. Accrue patients between t1 and t2. Follow all patients until failure or for x years, then
calculate the normalized final test statistic Z2. If Z2<=C2, stop the study for futility; For randomized
two-arm trials, if Z2>=C2U, stop the study for efficacy; otherwise, continue to the next stage.
Stage 3. Accrue patients between t2 and MDA. Follow all patients until failure or for x years, then
calculate the normalized final test statistic Z3, and reject H0 if Z3>=C3.
The test statistic is based on the Nelson-Aalen estimator of the cumulative hazard function.
Value
A vector containing results for the interim analysis or the final analysis:
z The test statistic
se Standard error of sum of the cummulative hazards (not log cummulative hazards) at time x.
cumL A two-element vector of cummulative hazard estimators at time x.
Author(s)
Bo Huang <> and Neal Thomas <>
References
Huang B., Talukder E. and Thomas N. Optimal two-stage Phase II designs with long-term endpoints. Statistics in Biopharmaceutical Research, 2(1), 51–61.
Case M. D. and Morgan T. M. (2003) Design of Phase II cancer trials evaluating survival probabilities. BMC Medical Research Methodology, 3, 7.
Lin D. Y., Shen L., Ying Z. and Breslow N. E. (1996) Group seqential designs for monitoring
survival probabilities. Biometrics, 52, 1033–1042.
Simon R. (1989) Optimal two-stage designs for phase II clinical trials. Controlled Clinical Trials,
10, 1–10.
See Also
OptimDes, SimDes
Examples
## Not run:
### single arm trial
B.init <- c(1, 2, 3, 4, 5)
m.init <- c(15, 20, 25, 20, 15)
alpha <- 0.05
beta <- 0.1
param <- c(1, 1.09, 2, 1.40)
x <- 1
# H0: S0=0.40 H1: S1=0.60
TestStage 21
shape0 <- param[1]
scale0 <- param[2]
shape1 <- param[3]
scale1 <- param[4]
object1 <- OptimDes(B.init,m.init,alpha,beta,param,x,target="EDA",sf="futility",num.arm=1,num.stage=2)
n <- object1$n[2]
t1 <- object1$stageTime[1]
C1 <- object1$boundary[1]
C1U <- object1$boundary[2]
C2 <- object1$boundary[3]
b <- length(B.init)
l <- rank(c(cumsum(m.init),n),ties.method="min")[b+1]
mda <- ifelse(l>1,B.init[l-1]+(B.init[l]-B.init[l-1])*(n-sum(m.init[1:(l-1)]))/m.init[l],B.init[l]*(n/m.init[l]))
### set up values to create a stepwise uniform distribution for accrual
B <- B.init[1:l]
B[l] <- mda
xv <- c(0,B)
M <- m.init[1:l]
M[l] <- ifelse(l>1,n-sum(m.init[1:(l-1)]),n)
yv <- c(0,M/(diff(xv)*n),0)
# density function of accrual
dens.Y <- stepfun(xv,yv,f=1,right=TRUE)
# pool of time points to be simulated from
t.Y <- seq(0,mda,by=0.01)
# simulate study times of length n
sample.Y <- sample(t.Y,n,replace=TRUE,prob=dens.Y(t.Y))
# simulate failure times of length n under the alternative hypothesis
sample.T <- rweibull(n,shape=shape1,scale=scale1)
Y1 <- sample.Y[sample.Y<=t1]
T1 <- sample.T[sample.Y<=t1]
Y2 <- sample.Y[sample.Y>t1]
T2 <- sample.T[sample.Y>t1]
# event rate under null hypothesis
p0<-pweibull(x,shape=shape0,scale=scale0)
# interim analysis
TestStage(x, C1, C1U, C2, tan=t1,num.arm=1,num.stage=2,Y11=Y1, T11=T1, p0=p0)
# final analysis if the study continues
TestStage(x, C1, C1U, C2, tan=t1,num.arm=1,num.stage=2,Y11=Y1, T11=T1, p0=p0)
# simulate failure times of length n under the null hypothesis
sample.T <- rweibull(n,shape=shape0,scale=scale0)
Y1 <- sample.Y[sample.Y<=t1]
22 weibPmatch
T1 <- sample.T[sample.Y<=t1]
Y2 <- sample.Y[sample.Y>t1]
T2 <- sample.T[sample.Y>t1]
# interim analysis
TestStage(x, C1, C1U, C2, tan=t1,num.arm=1,num.stage=2,Y11=Y1, T11=T1, p0=p0)
# final analysis if the study continues
TestStage(x, C1, C1U, C2, tan=mda+x,num.arm=1,num.stage=2,Y11=Y1, T11=T1, p0=p0,Y21=Y2,T21=T2)
## End(Not run)
weibPmatch Compute the shape or scale parameter for a Weibull distribution so it
has a specified event-free rate at a specified time.
Description
Determine the shape or scale parameter of a Weibull distribution so it has event-free rate P0 at time
x. If the shape is specified, the scale parameter is computed, and if the scale is specified, the shape
parameter is computed.
Usage
weibPmatch(x, p0, shape, scale)
Arguments
x Pre-specified time for the event-free endpoint (e.g., 1 year).
p0 Event-free rate at time x.
shape If specified, the necessary scale parameter is computed
scale If specified, the necessary shape parameter is computed
Details
The time and event-free rate must be supplied. Either the shape or scale parameter must also be
specified, but not both. The R parameterization of the Weibull distribution is used.
Value
A single numerical value is returned, either the shape or scale parameter, depending on which is
specified by the user.
Author(s)
Bo Huang <> and Neal Thomas <>
weibull.plot 23
References
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume
1, chapter 21. Wiley, New York.
See Also
weibull.plot,pweibull, OptimDes
Examples
param <- c(1, 1.09, 2, 1.40)
x<-1
p0<-pweibull(x,param[1],param[2],lower=FALSE)
p1<-pweibull(x,param[3],param[4],lower=FALSE)
weibull.plot(param,x)
### equivalent to simple call
paramNew<-c(param[1], weibPmatch(x,p0,param[1]), param[3], weibPmatch(x,p1,param[3]))
weibull.plot(paramNew, x)
### null curve with different shape
paramNew<-c(3.0, weibPmatch(x,p0,3.0), param[3], weibPmatch(x,p1,param[3]))
weibull.plot(paramNew, x)
### alternative curve with different scale
paramNew<-c(param[1], param[2], weibPmatch(x,p1,scale=2), 2)
weibull.plot(paramNew, x)
weibull.plot Plot Weibull Survival Curves
Description
Plot Weibull survival curves with differences at a target time highlighted. Designed for use with the
param values input to function OptimDes.
Usage
weibull.plot(param, x, l.type = 1:3, l.col = c("blue", "red"), ...)
24 weibull.plot
Arguments
param Events should be defined as poor outcomes. Computations and reporting are
based on the proportion without an event at a pre-specified time, x. For constructing an optimal design, complete event-free distributions at all times must
be specified for the control condition (Null), and for the alternative "effective"
treatment. Weibull distributions are currently implemented. param is a vector
of length 4: (shape null, scale null, shape alternative, scale alternative). The R
parameterization of the Weibull distribution is used.
x Pre-specified time for the event-free endpoint (e.g., 1 year).
l.type Line types for the plot. Default is 1-3.
l.col Line colors for the plot. Default is "blue" for the null survival curve, "red" for
the alternative survival curve.
... Further graphical arguments, see plot.default.
Author(s)
Bo Huang <> and Neal Thomas <>
References
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume
1, chapter 21. Wiley, New York.
See Also
dweibull, OptimDes, weibPmatch
Examples
param <- c(1, 1.09, 2, 1.40)
x <- 1
weibull.plot(param,x)
Index
∗Topic design
FixDes, 3
np.OptimDes, 5
OptimDes, 7
OptimDesControl, 11
∗Topic hplot
plot.OptimDes, 12
weibPmatch, 22
weibull.plot, 23
∗Topic htest
TestStage, 18
∗Topic iteration
SimDes, 15
∗Topic optimize
np.OptimDes, 5
OptimDes, 7
OptimDesControl, 11
plot.OptimDes, 12
SimDes, 15
TestStage, 18
∗Topic package
OptInterim-package, 2
∗Topic print
print.OptimDes, 14
dweibull, 24
FixDes, 3
np.OptimDes, 5, 10–13
OptimDes, 3, 4, 6, 7, 7, 11–15, 17, 18, 20, 23,
24
OptimDesControl, 6, 8, 11
OptimInterim-package
(OptInterim-package), 2
OptInterim-package, 2
plot.OptimDes, 6, 7, 10, 12, 14
print.OptimDes, 10, 13, 14
pweibull, 23
SimDes, 4, 15, 20
TestStage, 4, 17, 18
weibPmatch, 7, 10, 17, 22, 24
weibull.plot, 23, 23
25

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