rpact: Confirmatory Adaptive Clinical Trial Design and Analysis
Designing group-sequential trials with two groups and a survival endpoint with rpact
Marcel Wolbers, Gernot Wassmer, and Friedrich Pahlke
Last change: 13 November, 2019
Summary
This R Markdown document provides examples for designing trials with survival endpoints with rpact.
1 Introduction
The examples in this document are not intended to replace the official rpact documentation and help pages but rather to supplement them. They also only cover a selection of all rpact features.
General convention: In rpact, arguments containing the index “2” always refer to the control group, “1” refer to the intervention group, and treatment effects compare treatment versus control.
First, load the rpact package
## [1] '2.0.5'2 Overview of relevant rpact functions
Sample size calculation for a group-sequential trial with a survival endpoints is performed in two steps:
Define the (abstract) group-sequential design using the function
getDesignGroupSequential(). For details regarding this step, see the R markdown file “Defining group-sequential boundaries with rpact”. This step can be omitted if the trial has no interim analyses.Calculate sample size for the survival endpoint by feeding the abstract design into function
getSampleSizeSurvival(). This step is described in detail below.
Other relevant rpact functions for survival are:
getPowerSurvival(): This function is the analogue togetSampleSizeSurvival()for the calculation of power rather than sample size.getEventProbabilities(): Calculates the probability of an event depending on follow-up time. This is useful for aligning interim analyses for different time-to-event endpoints.getSimulationSurvival(): This function simulates group-sequential trials. For example, it allows to assess the power of trials with delayed treatment effects or to assess the data-dependent variability of the timing of interim analyses even if the protocol assumptions are perfectly fulfilled. It also allows to simulate hypothetical datasets for trials stopped early.
This document describes all functions mentioned above except for trial
simulation (getSimulationSurvival()) which is described in the document
Simulation-based design of group-sequential trials with a survival endpoint
with rpact.
However, before describing the functions themselves, the document describes how survival functions, drop-out, and accrual can be specified in rpact which is required for all of these functions.
3 Specifying survival distributions in rpact
rpact allows to specify survival distributions with exponential, piecewise exponential, and Weibull distributions. Exponential and piecewise exponential distributions are described below.
Weibull distributions are specified in the same way as exponential distributions
except that an additional scale parameter kappa needs to be provided which is
1 for the exponential distribution. Note that the parameters shape and
scale in the standard R functions for the Weibull distribution in the
stats-library (such as dweibull) are equivalent to kappa and 1/lambda,
respectively, in rpact.
3.1 Exponential survival distributions
3.1.1 Event probability at a specific time point known
- The time point is given by argument
eventTime. - The probability of an event (i.e., 1 minus survival function) in the control
group is given by argument
pi2. - The probability of an event in the intervention arm can either be provided
explicitly through argument
pi1or, alternatively, implicitly by specifying the target hazard ratiohazardRatio.
Example: If the intervention is expected to improve survival at 24 months
from 70% to 80%, this would be expressed through arguments
eventTime=24, pi2=0.3, pi1=0.2.
3.1.2 Exponential parameter \(\lambda\) known
- The constant hazard function in the control arm can be provided as argument
lambda2. - The hazard in the intervention arm can either be provided explicitly through
the argument
lambda1or, alternatively, implicitly by specifying the target hazard ratiohazardRatio.
3.1.3 Median survival known
Medians cannot be specified directly. However, one can exploit that the hazard rate \(\lambda\) of the exponential distribution is equal to \(\lambda=\log(2)/\text{median}\).
Example: If the intervention is expected to improve the median survival from
60 to 75 months, this would be expressed through the arguments
lambda2=log(2)/60, lambda1=log(2)/75. Alternatively, it could be specified via
lambda2=log(2)/60, hazardRatio=0.8 (as the hazard ratio is 60/75=0.8).
3.2 Weibull
Weibull distributions are specified in the same way as exponential distributions
except that an additional scale parameter kappa needs to be provided which is
1 for the exponential distribution.
3.3 Piecewise exponential survival
3.3.1 Piecewise constant hazard rate \(\lambda\) in each interval known
Example: If the survival function in the control arm is assumed to be 0.03
(events/time-unit of follow-up) for the first 6 months, 0.06 for months 6-12,
and 0.02 from month 12 onwards, this can be specified using the argument
piecewiseSurvivalTime as follows:
piecewiseSurvivalTime=list("0 - <6" =0.03,
"6 - <12"=0.06,
">=12" =0.02)Alternatively, the same distribution can be specified by giving the start times
of each interval as argument piecewiseSurvivalTime and the actual hazard rate
in that interval as argument lambda2. I.e., the relevant arguments for this
example would be:
piecewiseSurvivalTime=c(0,6,12), lambda2=c(0.03,0.06,0.02)For the intervention arm, one could either explicitly specify the hazard rate in
the same time intervals through the argument lambda1 or, more conveniently,
specify the survival function in the intervention arm indirectly via the target
hazard ratio (argument hazardRatio).
Note: The sample size calculation functions discussed in this document assume
that the proportional hazards assumption holds, i.e., that lambda1 and
lambda2 are proportional if both are provided. Otherwise, the function call to
getSampleSizeSurvival() gives an error. For situations with non-proportional
hazards, please see the separate document which discusses the simulation tool
using the function getSimulationSurvival().
3.3.2 Survival function at different time points known
Piecewise exponential distribution are useful to approximate survival functions. In principle, it is possible to approximate any distribution function well with a piecewise exponential distribution if suitably narrow intervals are chosen.
Assume that the survival function is \(S(t_1)\) at time \(t_1\) and \(S(t_2)\) at time \(t_2\) with \(t_2>t_1\) and that the hazard rate in the interval \([t_1,t_2]\) is constant. Then, the hazard rate in the interval can be derived as \((\log(S(t_1))-\log(S(t_2)))/(t_2-t_1)\).
Example: Assume that it is known that the survival function is 1, 0.9, 0.7, and 0.5 at months 0, 12, 24, 48 in the control arm. Then, an interpolating piecewise exponential distribution can be derived as follows:
t <- c(0,12,24,48) # time points at which survival function is known (must include 0)
S <- c(1,0.9,0.7,0.5) # Survival function at timepoints t
# derive hazard in each intervals per the formula above
lambda <- -diff(log(S))/diff(t)
# Define parameters for piecewise exponential distribution in the control arm
# interpolating the targeted survival values
# (code for lambda1 below assumes that the hazard afer month 48 is
# identical to the hazard in interval [24,48])
piecewiseSurvivalTime <- t
lambda2 <- c(lambda,tail(lambda,n=1))
lambda2 # print hazard rates ## [1] 0.008780043 0.020942869 0.014019677 0.014019677Appendix: Random numbers, cumulative distribution, and quantiles for the piecewise exponential distribution
rpact also provides general functionality for the piecewise exponential distribution,
see ?getPiecewiseExponentialDistribution for details. Below is some example
code which shows that the derivation of the piecewise exponential distribution
in the previous example is correct.
# plot the piecewise exponential survival distribution from the example above
tp <- seq(0,72,by=0.01)
plot(tp,
1-getPiecewiseExponentialDistribution(time=tp,
piecewiseSurvivalTime = piecewiseSurvivalTime,
piecewiseLambda = lambda2),
type="l",xlab="Time (months)",ylab="Survival function S(t)",lwd=2,ylim=c(0,1),axes=F,
main="Piecewise exponential distribution for the example")
axis(1,at=seq(0,72,by=12)); axis(2,at=seq(0,1,by=0.1))
abline(v=seq(0,72,by=12),h=seq(0,1,by=0.1),col=gray(0.9))
# Calculate median survival for the example (which should give 48 months here)
getPiecewiseExponentialQuantile(0.5,piecewiseSurvivalTime = piecewiseSurvivalTime,
piecewiseLambda = lambda2)## [1] 484 Specifying dropout in rpact
rpact models dropout with an independent exponentially distributed variable. Dropout is specified by giving the probability of dropout at a specific time point. For example, an annual (12-monthly) dropout probability of 5% in both treatment arms can be specified through the following arguments:
dropoutRate1 = 0.05, dropoutRate2 = 0.05, dropoutTime = 125 Specifying accrual in rpact
rpact allows to specify arbitrarily complex recruitment scenarios. Some examples are provided below.
5.1 Absolute accrual intensity and accrual duration known
Example 1: A constant accrual intensity of 24 subjects/months over 30 months (i.e., a maximum sample size of 24*30=720 subjects) can be specified through the arguments below:
accrualTime = c(0,30), accrualIntensity=24Example 2: A flexible accrual intensity of 20 subjects/months in the first 6 months, 25 subjects/months for months 6-12, and 30 subjects per month for months 12-24 can be specified through either of the two equivalent options below:
Option 1: List-based definition:
accrualTime = list("0 - <6" = 20,
"6 - <12" = 25,
"12- <24" = 30)Option 2: Vector-based definition (note that the length of the accrualTime
vector is 1 larger than the length of the accrualIntensity as the end time of
the accrual period is also included):
accrualTime = c(0,6,12,24), accrualIntensity=c(20,25,30)5.2 Absolute accrual intensity known, accrual duration unknown
Example 1: A constant accrual intensity of 24 subjects/months with unspecified end of recruitment is specified through the arguments below:
accrualTime = 0, accrualIntensity=24
# Note: accrualTime is the start time of accrual here which must be explicitly set to 0Example 2: A flexible accrual intensity of 20 subjects/months in the first 6 months, 25 subjects/months from month 6-12, and 30 subjects thereafter can be specified through either of the two equivalent options below:
Option 1: List-based definition:
accrualTime = list("0 - <6" = 20,
"6 - <12" = 25,
">=12" = 30)Option 2: Vector-based definition (note that the length of the accrualTime
vector is equal to the length of the accrualIntensity vector as only the start
time of the last accrual intensity is provided):
accrualTime = c(0,6,12), accrualIntensity=c(20,25,30)6 Sample size calculation for superiority trials without interim analyses
Given the specification of the survival distributions, drop-out and accrual as
described above, sample size calculations can be readily performed in rpact
using the function getSampleSizeSurvival(). Importantly, in survival trials,
the number of events (and not the sample size) determines the power of the
trial. Hence, the choice of the sample size to reach this target number of
events is based on study-specific trade-offs between the costs per recruited
patient, study duration, and desired minimal follow-up duration.
In practice, a range of sample sizes need to be explored manually or via suitable graphs to find an optimal trade off. Given a plausible absolute accrual intensity (usually provided by operations), a simple approach in rpact is to use one or both of the two options below:
- Perform the calculation by trying multiple maximum sample sizes (argument
maxNumberOfSubjects). - It often makes sense to require a minimal follow-up time (argument
followUpTime) for all patients in the trial at the time of the analysis. In this case, rpact automatically determines the maximum sample size.
6.1 Example - exponential survival, flexible accrual intensity, no interim analyses
- Exponential PFS with a median PFS of 60 months in control
(
lambda2=log(2)/60) and a target hazard ratio of 0.74 (hazardRatio=0.74). - Log-rank test at the two-sided 5%-significance level (
sided=2, alpha=0.05), power 80% (beta=0.2). - Annual drop-out of 2.5% in both arms (
dropoutRate1 = 0.025, dropoutRate2 = 0.025, dropoutTime = 12). - Recruitment is 42 patients/month from month 6 onwards after linear ramp up.
(
accrualTime = c(0,1,2,3,4,5,6), accrualIntensity=c(6,12,18,24,30,36,42)) - Randomization ratio 1:1 (
allocationRatioPlanned=1). This is the default and is thus not explicitly set in the function call below. - Two sample size choices will be initially explored:
- A fixed total sample size of 1200 (
maxNumberOfSubjects = 1200). - Alternatively, the total sample size will be implicitly determined by
specifying that every subject must have a minimal follow-up duration of
at 12 months at the time of the analysis (
followUpTime=12).
- A fixed total sample size of 1200 (
Based on this, the required number of events and timing of interim analyses for the fixed total sample size of 1200 can be determined as follows:
sampleSize1 <- getSampleSizeSurvival(sided=2,alpha=0.05,beta=0.2,
lambda2=log(2)/60,hazardRatio=0.74,
dropoutRate1 = 0.025, dropoutRate2 = 0.025, dropoutTime = 12,
accrualTime = c(0,1,2,3,4,5,6),
accrualIntensity=c(6,12,18,24,30,36,42),
maxNumberOfSubjects = 1200)
sampleSize1## Design plan parameters and output for survival data:
##
## Design parameters:
## Significance level : 0.0500
## Type II error rate : 0.2
## Two-sided power : FALSE
## Test : two-sided
##
## User defined parameters:
## lambda (2) : 0.0116
## Hazard ratio : 0.740
## Maximum number of subjects : 1200.0
## Accrual intensity : 6.0, 12.0, 18.0, 24.0, 30.0, 36.0, 42.0
## Drop-out rate (1) : 0.025
## Drop-out rate (2) : 0.025
##
## Default parameters:
## Type of computation : Schoenfeld
## Theta H0 : 1
## Planned allocation ratio : 1
## Event time : 12
## Piecewise survival times : 0.00
## Drop-out time : 12.00
##
## Sample size and output:
## Direction upper : FALSE
## pi (1) : 0.0975
## pi (2) : 0.129
## median (1) : 81.1
## median (2) : 60.0
## lambda (1) : 0.00855
## Number of events : 346.3
## Accrual time : 1.00, 2.00, 3.00, 4.00, 5.00, 6.00, 31.57
## Total accrual time : 31.57
## Follow up time : 21.54
## Calculate follow up time : TRUE
## Number of subjects fixed : 1200.0
## Number of subjects fixed (1) : 600.0
## Number of subjects fixed (2) : 600.0
## Analysis times : 53.11
## Study duration : 53.11
## Lower critical values (effect scale) : 0.810
## Upper critical values (effect scale) : 1.234
## Local two-sided significance levels : 0.0500
##
## Legend:
## (i): values of treatment arm iThus, the required number of events is
347 and the MDD corresponds to an
observed HR of
.The
1200 subjects will be recruited over
31.57 months and the
total study duration is
53.11 months. The user
could now vary maxNumberSubject to further optimize the trade-off between
sample size and study duration.
Alternatively, specifying a minimum follow-up duration of 12 months leads to the following result:
sampleSize2 <- getSampleSizeSurvival(sided=2,alpha=0.05,beta=0.2,
lambda2=log(2)/60,hazardRatio=0.74,
dropoutRate1 = 0.025, dropoutRate2 = 0.025,dropoutTime = 12,
accrualTime = c(0,1,2,3,4,5,6),
accrualIntensity=c(6,12,18,24,30,36,42),
followUpTime=12)
sampleSize2## Design plan parameters and output for survival data:
##
## Design parameters:
## Significance level : 0.0500
## Type II error rate : 0.2
## Two-sided power : FALSE
## Test : two-sided
##
## User defined parameters:
## lambda (2) : 0.0116
## Hazard ratio : 0.740
## Accrual intensity : 6.0, 12.0, 18.0, 24.0, 30.0, 36.0, 42.0
## Follow up time : 12.00
## Drop-out rate (1) : 0.025
## Drop-out rate (2) : 0.025
##
## Default parameters:
## Type of computation : Schoenfeld
## Theta H0 : 1
## Planned allocation ratio : 1
## Event time : 12
## Piecewise survival times : 0.00
## Drop-out time : 12.00
##
## Sample size and output:
## Direction upper : FALSE
## pi (1) : 0.0975
## pi (2) : 0.129
## median (1) : 81.1
## median (2) : 60.0
## lambda (1) : 0.00855
## Maximum number of subjects : 1433.7
## Number of events : 346.3
## Accrual time : 1.00, 2.00, 3.00, 4.00, 5.00, 6.00, 37.13
## Total accrual time : 37.13
## Calculate follow up time : TRUE
## Number of subjects fixed : 1433.7
## Number of subjects fixed (1) : 716.8
## Number of subjects fixed (2) : 716.8
## Analysis times : 49.13
## Study duration : 49.13
## Lower critical values (effect scale) : 0.810
## Upper critical values (effect scale) : 1.234
## Local two-sided significance levels : 0.0500
##
## Legend:
## (i): values of treatment arm iThis specification leads to a higher sample size of 1434 subjects which will be recruited over 37.13 months and the total study duration is 49.13 months.
With the generic summary() function the two calculations are summarized as
follows:
## Sample size calculation for a survival endpoint
##
## Fixed sample analysis.
## The sample size was calculated for a two-sample logrank test (two-sided),
## hazard ratio 0.74, control lambda (2) = 0.012, allocation ratio = 1, and power 80%.
##
## Efficacy boundary (z-value scale) 1.960
## Number of subjects 1200
## Number of events 346.3
## Analysis time 53.1
## Two-sided local significance level 0.0500
## Lower efficacy boundary (t) 0.810
## Upper efficacy boundary (t) 1.234
##
## Legend:
## (t): approximate treatment effect scale## Sample size calculation for a survival endpoint
##
## Fixed sample analysis.
## The sample size was calculated for a two-sample logrank test (two-sided),
## hazard ratio 0.74, control lambda (2) = 0.012, allocation ratio = 1, and power 80%.
##
## Efficacy boundary (z-value scale) 1.960
## Number of subjects 1434
## Number of events 346.3
## Analysis time 49.1
## Two-sided local significance level 0.0500
## Lower efficacy boundary (t) 0.810
## Upper efficacy boundary (t) 1.234
##
## Legend:
## (t): approximate treatment effect scaleTo further explore the possible trade-offs, one could visualize recruitment and study durations for a range of sample sizes as illustrated in the code below:
# set up data frame which contains sample sizes and corresponding durations
sampleSizeDuration <- data.frame(
maxNumberSubjects=seq(600,1800,by=50),
accrualTime=NA,
studyDuration=NA)
# calculate recruitment and study duration for each sample size
for (i in 1:nrow(sampleSizeDuration)){
sampleSizeResult <- getSampleSizeSurvival(
sided = 2,alpha = 0.05,beta = 0.2,
lambda2 = log(2)/60,hazardRatio = 0.74,
dropoutRate1 = 0.025, dropoutRate2 = 0.025,dropoutTime = 12,
accrualTime = c(0,1,2,3,4,5,6),
accrualIntensity = c(6,12,18,24,30,36,42),
maxNumberOfSubjects = sampleSizeDuration$maxNumberSubjects[i])
sampleSizeDuration$accrualTime[i] <- sampleSizeResult$totalAccrualTime
sampleSizeDuration$studyDuration[i] <- sampleSizeResult$maxStudyDuration
}
# plot result
plot(sampleSizeDuration$maxNumberSubjects,
sampleSizeDuration$studyDuration,type="l",
xlab="Total sample size",
ylab="Duration (months)",
main="Recruitment and study duration vs sample size",
ylim=c(0,max(sampleSizeDuration$studyDuration)),
col="blue",lwd=1.5)
lines(sampleSizeDuration$maxNumberSubjects,
sampleSizeDuration$accrualTime,col="red",lwd=1.5)
legend(x=1000,y=100,
legend=c("Study duration under H1",
"Recruitment duration"),
col=c("blue","red"),lty=1,lwd=1.5)
7 Sample size calculation for non-inferiority trials without interim analyses
Sample size calculation proceeds in the same fashion as for superiority trials
except that the role of the null and the alternative hypothesis are reversed.
I.e., in this case, the non-inferiority margin \(\Delta\) corresponds to the
treatment effect under the null hypothesis (thetaH0). Testing for
non-inferiority trials is always one-sided.
In the example code below, we assume that the non-inferiority margin corresponds
to a 20% increase of the hazard function and that “in reality”, i.e., under the
alternative hypothesis, the survival functions in the two groups are equal. In
this case, the “null hypothesis” is thetaH0=1.2 versus the “alternative”
hazardRatio=1. For illustration, all other assumptions were chosen as per the
sample size calculations for superiority trials above.
sampleSizeNonInf <- getSampleSizeSurvival(sided = 1,alpha = 0.025,beta = 0.2,
lambda2 = log(2)/60,
thetaH0=1.2, hazardRatio = 1,
dropoutRate1 = 0.025, dropoutRate2 = 0.025,dropoutTime = 12,
accrualTime = c(0,1,2,3,4,5,6),
accrualIntensity = c(6,12,18,24,30,36,42),
followUpTime = 12)
sampleSizeNonInf## Design plan parameters and output for survival data:
##
## Design parameters:
## Significance level : 0.0250
## Type II error rate : 0.2
## Test : one-sided
##
## User defined parameters:
## Theta H0 : 1.2
## lambda (2) : 0.0116
## Hazard ratio : 1.000
## Accrual intensity : 6.0, 12.0, 18.0, 24.0, 30.0, 36.0, 42.0
## Follow up time : 12.00
## Drop-out rate (1) : 0.025
## Drop-out rate (2) : 0.025
##
## Default parameters:
## Type of computation : Schoenfeld
## Planned allocation ratio : 1
## Event time : 12
## Piecewise survival times : 0.00
## Drop-out time : 12.00
##
## Sample size and output:
## Direction upper : FALSE
## pi (1) : 0.129
## pi (2) : 0.129
## median (1) : 60.0
## median (2) : 60.0
## lambda (1) : 0.0116
## Maximum number of subjects : 2609.2
## Number of events : 944.5
## Accrual time : 1.00, 2.00, 3.00, 4.00, 5.00, 6.00, 65.12
## Total accrual time : 65.12
## Calculate follow up time : TRUE
## Number of subjects fixed : 2609.2
## Number of subjects fixed (1) : 1304.6
## Number of subjects fixed (2) : 1304.6
## Analysis times : 77.12
## Study duration : 77.12
## Critical values (effect scale) : 1.056
## Local one-sided significance levels : 0.0250
##
## Legend:
## (i): values of treatment arm iIn this example, the required number of events is 945. The specification above leads to a sample size of 2610 subjects recruited over 65.12 months and an expected total study duration of 77.12 months.
8 Sample size calculation for trials with interim analyses
Calculations can be performed as for trials without interim analyses except that
the group-sequential design needs to be specified using
getDesignGroupSequential before calling getSampleSizeSurvival. For details
regarding the specification of the group-sequential design, see the R Markdown
file “Defining group-sequential boundaries with rpact”.
In general, rpact supports both one-sided and two-sided group-sequential designs. However, if futility boundaries are specified, only one-sided tests are permitted. For simplicity, it is often preferred to use one-sided tests for group-sequential designs (typically with \(\alpha=0.025\)).
One additional consideration for phase III trials with efficacy interim analyses is that they should only be performed at time-points when the data is sufficiently mature to allow for a potential filing. For example, it is often sensible to perform efficacy interim analyses only after all subjects have been recruited. This aspect can be further explored by looking at simulated data for (hypothetical) trials that would have stopped at the interim analysis. This can easily be implemented using the simulation tool in rpact and we refer to the R Markdown document Simulation-based design of group-sequential trials with a survival endpoint with rpact for further details.
8.1 Example - piece-wise exponential survival, constant accrual intensity, group-sequential design
- 1:1 randomized.
- 80% power at the one-sided 2.5% significance level.
- Efficacy interim analyses at 50% and 75% of total information using an O’Brien&Fleming type \(\alpha\)-spending function, no futility interim analyses.
- Target HR for primary endpoint (PFS) is 0.75.
- PFS in the control arm follows a piece-wise exponential distribution, with
the hazard rate h(t) estimated using historical controls as follows:
- h(t) = 0.025 for t between 0 and 6 months,
- h(t) = 0.04 for t between 6 and 9 months,
- h(t) = 0.015 for t between 9 and 15 months,
- h(t) = 0.01 for t between 15 and 21 months,
- h(t) = 0.007 for t beyond 21 months.
- An annual dropout probability of 5%.
- Recruitment of 42 patients/month up to a maximum of 1000 subjects.
8.1.1 Sample size calculation
The group-sequential design (including Type I and II error) can be specified as follows:
design <- getDesignGroupSequential(sided = 1, alpha = 0.025, beta = 0.2,
informationRates = c(0.5,0.75,1),
typeOfDesign="asOF")The piecewise exponential survival is defined as described above:
piecewiseSurvivalTime <- list(
"0 - <6" = 0.025,
"6 - <9" = 0.04,
"9 - <15" = 0.015,
"15 - <21" = 0.01,
">=21" = 0.007)Now, the sample size calculation can be performed:
sampleSizeResult <- getSampleSizeSurvival(
design = design,
piecewiseSurvivalTime = piecewiseSurvivalTime,hazardRatio = 0.75,
dropoutRate1 = 0.05, dropoutRate2 = 0.05, dropoutTime = 12,
accrualTime = 0, accrualIntensity=42, maxNumberOfSubjects = 1000)The sample size result can be summarized by displaying the default rpact output:
## Design plan parameters and output for survival data:
##
## Design parameters:
## Significance level : 0.0250
## Type II error rate : 0.2
## Test : one-sided
##
## User defined parameters:
## lambda (2) : 0.025, 0.040, 0.015, 0.010, 0.007
## Hazard ratio : 0.750
## Maximum number of subjects : 1000.0
## Accrual intensity : 42.0
## Piecewise survival times : 0.00, 6.00, 9.00, 15.00, 21.00
## Drop-out rate (1) : 0.050
## Drop-out rate (2) : 0.050
##
## Default parameters:
## Type of computation : Schoenfeld
## Theta H0 : 1
## Planned allocation ratio : 1
## kappa : 1
## Drop-out time : 12.00
##
## Sample size and output:
## Direction upper : FALSE
## lambda (1) : 0.01875, 0.03000, 0.01125, 0.00750, 0.00525
## Maximum number of subjects (1) : 500.0
## Maximum number of subjects (2) : 500.0
## Maximum number of events : 386.8
## Accrual time : 23.81
## Follow up time : 36.19
## Calculate follow up time : TRUE
## Information rates [1] : 0.500
## Information rates [2] : 0.750
## Information rates [3] : 1.000
## Analysis times [1] : 23.17
## Analysis times [2] : 33.28
## Analysis times [3] : 60.00
## Expected study duration under H1 : 43.87
## Maximal study duration : 60.00
## Number of events by stage [1] : 193.4
## Number of events by stage [2] : 290.1
## Number of events by stage [3] : 386.8
## Expected number of events under H0 : 385.7
## Expected number of events under H0/H1 : 371.7
## Expected number of events under H1 : 318.3
## Number of subjects [1] : 973.2
## Number of subjects [2] : 1000.0
## Number of subjects [3] : 1000.0
## Expected number of subjects under H1 : 995.5
## Reject per stage [1] : 0.168
## Reject per stage [2] : 0.372
## Reject per stage [3] : 0.260
## Early stop : 0.54
## Critical values (effect scale) [1] : 0.653
## Critical values (effect scale) [2] : 0.758
## Critical values (effect scale) [3] : 0.815
## Local one-sided significance levels [1] : 0.001525
## Local one-sided significance levels [2] : 0.009162
## Local one-sided significance levels [3] : 0.022000
##
## Legend:
## (i): values of treatment arm i
## [k]: values at stage kIn addition, the generic summary() function summarizes the results per
stage in a different format and also adds results which are stored in the
underlying group-sequential design such as exit probabilities:
## Sample size calculation for a survival endpoint
##
## Sequential analysis with a maximum of 3 looks (group sequential design).
## The sample size was calculated for a two-sample logrank test (one-sided),
## hazard ratio 0.75, piecewise survival distribution, allocation ratio = 1, and power 80%.
##
## Stage 1 2 3
## Information rate 50% 75% 100%
## Efficacy boundary (z-value scale) 2.963 2.359 2.014
## Number of subjects 974 1000 1000
## Cumulative number of events 193.4 290.1 386.8
## Analysis time 23.2 33.3 60.0
## Cumulative alpha spent 0.0015 0.0096 0.0250
## Cumulative power 0.1680 0.5400 0.8000
## One-sided local significance level 0.0015 0.0092 0.0220
## Efficacy boundary (t) 0.653 0.758 0.815
## Exit probability for efficacy (under H0) 0.0015 0.0081
## Exit probability for efficacy (under H1) 0.1680 0.3720
##
## Legend:
## (t): approximate treatment effect scale8.1.2 Sample size plots
rpact provides a large number of plots, see the R Markdown document How to create admirable plots with rpact for examples of all plots. Below, we just show two selected plots.
## Source data of the plot:
## x-axis: sampleSizeResult$eventsPerStage
## y-axes:
## y1: sampleSizeResult$criticalValuesEffectScaleUpper[, 1]
## y2: sampleSizeResult$futilityBoundsEffectScale[, 1]
9 Power calculation
The function getPowerSurvival() is the analogue of getSampleSizeSurvival()
for power calculations and has a very similar syntax. We illustrate its usage
for the above example below.
9.1 Example - piece-wise exponential survival, constant accrual intensity, group-sequential design (continued)
What is the power of the trial from the example above if the true hazard ratio is 0.7 rather than the targeted hazard ratio of 0.75 above?
The corresponding power calculation is given below. The function call uses the
same arguments as the call to function getSampleSizeSurvival() above except
that the maximum sample size of 1000 and the maximum number of events of 387
from above are provided as arguments and the hazard ratio is changed to 0.7. In
addition, the direction of the test (directionUpper) needs to be specified.
powerResult <- getPowerSurvival(
design = design,
piecewiseSurvivalTime = piecewiseSurvivalTime,hazardRatio = 0.7,
dropoutRate1 = 0.05, dropoutRate2 = 0.05, dropoutTime = 12,
accrualTime = 0, accrualIntensity=42,
maxNumberOfSubjects = 1000,maxNumberOfEvents=387,directionUpper = FALSE)
powerResult## Design plan parameters and output for survival data:
##
## Design parameters:
## Significance level : 0.0250
## Test : one-sided
##
## User defined parameters:
## Direction upper : FALSE
## lambda (2) : 0.025, 0.040, 0.015, 0.010, 0.007
## Hazard ratio : 0.700
## Maximum number of subjects : 1000.0
## Maximum number of events : 387.0
## Accrual intensity : 42.0
## Piecewise survival times : 0.00, 6.00, 9.00, 15.00, 21.00
## Drop-out rate (1) : 0.050
## Drop-out rate (2) : 0.050
##
## Default parameters:
## Type of computation : Schoenfeld
## Theta H0 : 1
## Planned allocation ratio : 1
## kappa : 1
## Drop-out time : 12.00
##
## Sample size and output:
## lambda (1) : 0.0175, 0.0280, 0.0105, 0.0070, 0.0049
## Accrual time : 23.81
## Follow up time : 39.56
## Analysis times [1] : 23.58
## Analysis times [2] : 34.72
## Analysis times [3] : 63.37
## Expected study duration : 38.26
## Maximal study duration : 63.37
## Number of events by stage [1] : 193.5
## Number of events by stage [2] : 290.2
## Number of events by stage [3] : 387.0
## Expected number of events : 283.6
## Number of subjects [1] : 990.4
## Number of subjects [2] : 1000.0
## Number of subjects [3] : 1000.0
## Expected number of subjects : 997.0
## Reject per stage [1] : 0.315
## Reject per stage [2] : 0.439
## Reject per stage [3] : 0.181
## Overall reject : 0.935
## Early stop : 0.754
## Critical values (effect scale) [1] : 0.653
## Critical values (effect scale) [2] : 0.758
## Critical values (effect scale) [3] : 0.815
## Local one-sided significance levels [1] : 0.001525
## Local one-sided significance levels [2] : 0.009162
## Local one-sided significance levels [3] : 0.022000
##
## Legend:
## (i): values of treatment arm i
## [k]: values at stage kThis yields an overall power of 0.935. Note also that the timing of the interim analyses is slightly deferred now (as the events now occur more slowly in the intervention arm) but the expected study duration is decreased from 43.87 to 38.26 months because chances to stop the trial early have increased.
The impact of the true hazard ratio on power and study duration can also be
calculated by entering the hazardRatio argument as a vector. Below, the hazard
ratio is varied from 0.6 to 1 (in increments of 0.02) and the results are
displayed graphically.
powerResult2 <- getPowerSurvival(
design = design,
piecewiseSurvivalTime = piecewiseSurvivalTime,hazardRatio = seq(0.6,1,by=0.02),
dropoutRate1 = 0.05, dropoutRate2 = 0.05, dropoutTime = 12,
accrualTime = 0, accrualIntensity=42,
maxNumberOfSubjects = 1000,maxNumberOfEvents=387,directionUpper = FALSE)
# Plot true hazard ratio vs overall power, probability of an early rejection,
# and expected number of events
plot(powerResult2, type = 6, legendPosition = 3) # legend in left bottom
For examples of all available plots in rpact, see the R vignette “How to create admirable plots with rpact” (written by rpact, link).
10 Prediction of number of events over time
The function getEventProbabilities calculates the probability that a randomly
chosen subject has an observed event up to a specied time after the start of
the study. By multiplying this probability with the maximal sample size, we
obtain the expected number of events up to that time point.
The arguments for the function getEventProbabilities are similar to those for
the function getSampleSizeSurvival discussed above. One difference is that the
maximal sample size must always be provided in function getEventProbabilities.
This can either be done explicitly by setting argument maxNumberOfSubjects or
implicitly by specifying the accrual duration as described in Section
5.1.
For illustration, we revisit the example from Section 6.1. In that example, the sample size was 1000 and the expected timing of the final analysis after observing 387 events was after 60.00 months. The code below re-calculates the expected number of events until that time point.
# Sample size per arm and overall for the example
maxNumberOfSubjects1 <- 500
maxNumberOfSubjects2 <- 500
maxNumberOfSubjects <- maxNumberOfSubjects1+maxNumberOfSubjects2
# Probability that a randomly chosen subject has an observed event
# until 60 months after study start
probEvent <- getEventProbabilities(time = 60,
piecewiseSurvivalTime = piecewiseSurvivalTime,hazardRatio = 0.75,
dropoutRate1 = 0.05, dropoutRate2 = 0.05, dropoutTime = 12,
accrualTime = 0, accrualIntensity = 42,maxNumberOfSubjects = maxNumberOfSubjects)
probEvent## Event probabilities at given time:
##
## User defined parameters:
## Accrual intensity : 42.0
## Piecewise survival times : 0.00, 6.00, 9.00, 15.00, 21.00
## lambda (2) : 0.025, 0.040, 0.015, 0.010, 0.007
## Hazard ratio : 0.750
## Drop-out rate (1) : 0.050
## Drop-out rate (2) : 0.050
## Maximum number of subjects : 1000.0
##
## Default parameters:
## kappa : 1
## Planned allocation ratio : 1
## Drop-out time : 12.00
##
## Time and output:
## Time : 60.00
## Accrual time : 23.81
## lambda (1) : 0.01875, 0.03000, 0.01125, 0.00750, 0.00525
## Overall event probabilities : 0.3868
## Event probabilities (1) : 0.3444
## Event probabilities (2) : 0.4292
##
## Legend:
## (i): values of treatment arm i# Total expected number of events until 60 months
probEvent$overallEventProbabilities*maxNumberOfSubjects## [1] 386.7958# Expected number of events per group until 60 months
probEvent$eventProbabilities1*maxNumberOfSubjects1 # intervention## [1] 172.1793## [1] 214.6165The function getEventProbabilities() also allows a vector argument for time
and the code below illustrates how to calculate (and plot) expected event
numbers over time (overall and by group). In addition, the number of recruited
subjects is added to the plot which can be obtained with function
getNumberOfSubjects().
time <- 1:60
# Expected number of events overall and per group at 1-60 months after study start
probEvent <- getEventProbabilities(time = time,
piecewiseSurvivalTime = piecewiseSurvivalTime,hazardRatio = 0.75,
dropoutRate1 = 0.05, dropoutRate2 = 0.05, dropoutTime = 12,
accrualTime = 0, accrualIntensity = 42,maxNumberOfSubjects = maxNumberOfSubjects)
expEvent <- probEvent$overallEventProbabilities*maxNumberOfSubjects # overall
expEventInterv <- probEvent$eventProbabilities1*maxNumberOfSubjects1 # intervention
expEventCtrl <- probEvent$eventProbabilities2*maxNumberOfSubjects2 # control
# Cumulative number of recruited patients over time
nSubjects <- getNumberOfSubjects(time=time, accrualTime = 0, accrualIntensity=42,
maxNumberOfSubjects = maxNumberOfSubjects)
# plot result
plot(time,nSubjects$numberOfSubjects,type="l",lwd=2,
main="Number of patients and expected number of events over time",
xlab="Time from study start (months)",
ylab="Number of patients / events")
lines(time,expEvent,col="blue",lwd=2)
lines(time,expEventCtrl,col="green")
lines(time,expEventInterv,col="orange")
legend(x=28,y=900,
legend=c("Recruitment","Expected events - All subjects",
"Expected events - Control","Expected events - Intervention"),
col=c("black","blue","green","orange"),lty=1,lwd=c(2,2,1,1))