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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

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

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

OptimDesControl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

plot.OptimDes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

print.OptimDes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

weibull.plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Index 25

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 <

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 <

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,

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

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 <

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,

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

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 <

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,

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

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 <

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,

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 <

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,

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 <

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,

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 <

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,

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

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,

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

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 <

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,

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

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 <

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 <

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,

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

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