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rpact: Confirmatory Adaptive Clinical Trial Design and Analysis

Summary

This R Markdown document describes how to simulate design characterics for survival design under complex settings (incl.  non-proportional hazards) in rpact.

1 Introduction

rpact provides the function getSimulationSurvival() for simulation of group sequential trials with a time-to-event endpoint. For a given scenario, getSimulationSurvival() simulates many hypothetical group sequential trials and calculates the test results. Based on this Monte Carlo simulation, estimates of key quantities such as overall study power, stopping probabilities at each interim analysis, timing of analyses etc. can be obtained.

getSimulationSurvival() complements the analytical calculations from the function getPowerSurvival() (and getSampleSizeSurvival()) in multiple ways:

  • Simulations can be used to assess the accuracy of the analytical formulas.
  • Simulations allow to answer questions such as the following:
    • How variable is the timing of interim analysis (even if all assumptions are correct)?
    • How could a dataset of a trial that is stopped early for efficacy at an interim analysis look like?
  • Simulation is also possible for scenarios that are analytically intractable such as scenarios with delayed treatment effects.

The syntax of function getSimulationSurvival() is very similar to the function getPowerSurvival() and getSampleSizeSurvival(). Hence, this document only provides some examples and expects that the reader is familiar with the R Markdown document Designing group sequential trials with two groups and a survival endpoint with rpact which describes standard designs of a trial with a survival endpoint.

getSimulationSurvival() also supports the usage of adaptive sample size recalculation but this is not covered here. For more details, please also consult the help ?getSimulationSurvival.

First, load the rpact package

library(rpact)
packageVersion("rpact") # version should be version 2.0.5 or later
## [1] '3.3.1'

For this R Markdown document, some additional R packages are loaded which will be used to examine simulated datasets.

# Load ggplot2 for plotting
library(ggplot2) # general plotting
library(survminer) # plotting of KM curves with ggplot2
## 
## Attache Paket: 'survminer'
## Das folgende Objekt ist maskiert 'package:survival':
## 
##     myeloma
library(survival) # survival analysis routines

2 Standard analytical calculation

For comparison with the simulation-based analysis, a standard example is first calculated under the following assumptions:

  • Group-sequential design with one interim analysis after 66% of information using an O’Brien & Fleming type \(\alpha\)-spending function, one-sided Type I error 2.5%, power 80%:
design <- getDesignGroupSequential(
    informationRates = c(0.66, 1),
    typeOfDesign = "asOF", sided = 1, alpha = 0.025, beta = 0.2
)
  • Exponential PFS with a median PFS of 60 months in control (median2 = 60) and a target hazard ratio of 0.74 (hazardRatio = 0.74).
  • 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 (allocation1 = 1 and allocation2 = 1); this is how subjects are randomized in treatment groups 1 and 2 in a subsequent way). 1:1 allocation is the default and is thus not explicitly set in the function call below.
  • A fixed total sample size of 1200 (maxNumberOfSubjects = 1200).

As described in the R Markdown document which describes standard designs of a trial with a survival endpoint, sample size calculations for this design can be performed as per the code below:

sampleSizeResult <- getSampleSizeSurvival(design,
    median2 = 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
)

# Summary of results
kable(summary(sampleSizeResult))

Sample size calculation for a survival endpoint

Sequential analysis with a maximum of 2 looks (group sequential design), overall significance level 2.5% (one-sided). The sample size was calculated for a two-sample logrank test, H0: hazard ratio = 1, H1: hazard ratio = 0.74, control median(2) = 60, maximum number of subjects = 1200, accrual time = c(1, 2, 3, 4, 5, 6, 31.571), accrual intensity = c(6, 12, 18, 24, 30, 36, 42), dropout rate(1) = 0.025, dropout rate(2) = 0.025, dropout time = 12, power 80%.

Stage 1 2
Information rate 66% 100%
Efficacy boundary (z-value scale) 2.524 1.992
Overall power 0.4074 0.8000
Expected number of subjects 1200.0
Number of subjects 1200.0 1200.0
Cumulative number of events 231.3 350.5
Expected study duration 47.9
Cumulative alpha spent 0.0058 0.0250
One-sided local significance level 0.0058 0.0232
Efficacy boundary (t) 0.718 0.808
Exit probability for efficacy (under H0) 0.0058
Exit probability for efficacy (under H1) 0.4074

Legend:

  • (t): treatment effect scale

By design, the power of the trial is 80%. The interim analysis is after 232 events which is expected to occur after 39.5 months, and the final analysis is after 351 events which is expected to occur after 53.7 months.

Use getPowerSurvival() to calculate the power achieved for the ceiled number of events. Note that the direction of the alternative needs to be specified. Here, the alternative is towards hazard ratios < 1 which is specified as directionUpper = FALSE.

powerResult <- getPowerSurvival(design,
    maxNumberOfEvents = ceiling(sampleSizeResult$maxNumberOfEvents),
    directionUpper = FALSE,
    median2 = 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
)

# Summary of results
kable(summary(powerResult))

Power calculation for a survival endpoint

Sequential analysis with a maximum of 2 looks (group sequential design), overall significance level 2.5% (one-sided). The results were calculated for a two-sample logrank test, H0: hazard ratio = 1, power directed towards smaller values, H1: hazard ratio = 0.74, control median(2) = 60, maximum number of subjects = 1200, maximum number of events = 351, accrual time = c(1, 2, 3, 4, 5, 6, 31.571), accrual intensity = c(6, 12, 18, 24, 30, 36, 42), dropout rate(1) = 0.025, dropout rate(2) = 0.025, dropout time = 12.

Stage 1 2
Information rate 66% 100%
Efficacy boundary (z-value scale) 2.524 1.992
Overall power 0.4080 0.8005
Expected number of subjects 1200.0
Number of subjects 1200.0 1200.0
Expected number of events 302.3
Cumulative number of events 231.7 351.0
Expected study duration 47.9
Cumulative alpha spent 0.0058 0.0250
One-sided local significance level 0.0058 0.0232
Efficacy boundary (t) 0.718 0.808
Exit probability for efficacy (under H0) 0.0058
Exit probability for efficacy (under H1) 0.4080

Legend:

  • (t): treatment effect scale

These numbers will now be compared to simulations.

3 Simulation under proportional hazards

The call getSimulationSurvival() uses the same arguments as getSampleSizeSurvival() with the following changes:

  • The maximum number of patients (maxNumberOfSubjects = 1200) is always provided to allow the simulation.
  • The number of events at each analysis is specified as per the analytical calculation above (plannedEvents = c(232,351)).
  • The direction of the alternative is specified as directionUpper = FALSE.
  • The number of simulated trials is specified (maxNumberOfIterations = 10000 in the example below).
  • By default, raw datasets from simulation runs are not extracted. However, in this example, it is specifies that one raw dataset that led to stopping after each stage, respectively, will be stored: maxNumberOfRawDatasetsPerStage = 1.
  • For reproducibility, it is useful to set the random seed which is set to seed = 123 in the example.
  • For simulation of trials with unequal randomization, integer arguments allocation1 and allocation2 must be provided to the function getSimulationSurvival (instead of argument allocationRatioPlanned in the function getSampleSizeSurvival). allocation1 and allocation2 specify the number of consecutively enrolled subjects in the intervention and control groups, respectively, before another subject from the opposite group is recruited. For example, allocation1 = 2, allocation2 = 1 refers to 2 (intervention):1 (control) randomization and allocation1 = 1, allocation2 = 2 to 1 (intervention):2 (control) randomization.
simulationResult <- getSimulationSurvival(design,
    median2 = 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),
    plannedEvents = c(232, 351),
    directionUpper = FALSE,
    maxNumberOfSubjects = 1200,
    maxNumberOfIterations = 10000,
    maxNumberOfRawDatasetsPerStage = 1,
    seed = 234
)

# Summary of simulation results
kable(summary(simulationResult))

Simulation of a survival endpoint

Sequential analysis with a maximum of 2 looks (group sequential design), overall significance level 2.5% (one-sided). The results were simulated for a two-sample logrank test, H0: hazard ratio = 1, power directed towards smaller values, H1: hazard ratio = 0.74, control median(2) = 60, planned cumulative events = c(232, 351), maximum number of subjects = 1200, accrual time = c(1, 2, 3, 4, 5, 6, 31.571), accrual intensity = c(6, 12, 18, 24, 30, 36, 42), dropout rate(1) = 0.025, dropout rate(2) = 0.025, dropout time = 12, simulation runs = 10000, seed = 234.

Stage 1 2
Fixed weight 0.66 1
Efficacy boundary (z-value scale) 2.524 1.992
Overall power 0.3974 0.7959
Expected number of subjects 1200.0
Number of subjects 1200.0 1200.0
Expected number of events 303.7
Cumulative number of events 232.0 119.0
Expected study duration 48.1
Exit probability for efficacy 0.3974 0.3985

According to the output, the simulated overall power is 79.6% and the probability to cross the efficacy boundary at the interim analysis is 39.7%. These are both within 1% of the analytical power.

The mean simulated analysis times are after 39.61 months for the interim analysis and after 53.66 for the final analysis. Both timings differ by <0.1 months from the analytical calculation (Difference analysis times = 0.06, 0.01).

You can show median [range] and mean+/-sd of the trial results across the simulated datasets with the print() command together showStatistics = TRUE:

# Print of simulation results showing trial results statistics
print(simulationResult, showStatistics = TRUE, markdown = TRUE)

Simulation of survival data (group sequential design)

Design parameters

  • Information rates: 0.660, 1.000
  • Critical values: 2.524, 1.992
  • Futility bounds (non-binding): -Inf
  • Cumulative alpha spending: 0.005798, 0.025000
  • Local one-sided significance levels: 0.005798, 0.023210
  • Significance level: 0.0250
  • Test: one-sided

User defined parameters

  • Maximum number of iterations: 10000
  • Seed: 234
  • Direction upper: FALSE
  • Planned cumulative events: 232, 351
  • median(2): 60.0
  • Maximum number of subjects: 1200
  • Accrual time: 1.00, 2.00, 3.00, 4.00, 5.00, 6.00, 31.57
  • 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
  • Hazard ratio: 0.740

Default parameters

  • Planned allocation ratio: 1
  • Conditional power: NA
  • Drop-out time: 12.00
  • Theta H0: 1
  • Allocation 1: 1
  • Allocation 2: 1
  • kappa: 1

Results

  • Assumed treatment rate: 0.0975
  • Assumed control rate: 0.129
  • median(1): 81.1
  • lambda(1): 0.00855
  • lambda(2): 0.0116
  • Analysis times [1]: 39.61
  • Analysis times [2]: 53.66
  • Expected study duration: 48.09
  • Events not achieved [1]: 0.000
  • Events not achieved [2]: 0.000
  • Number of subjects [1]: 1200
  • Number of subjects [2]: 1200
  • Number of events per stage [1]: 232
  • Number of events per stage [2]: 119
  • Cumulative number of events [1]: 232
  • Cumulative number of events [2]: 351
  • Iterations [1]: 10000
  • Iterations [2]: 6026
  • Overall reject: 0.7959
  • Reject per stage [1]: 0.3974
  • Reject per stage [2]: 0.3985
  • Futility stop per stage: 0.0000
  • Early stop: 0.3974
  • Expected number of subjects: 1200
  • Expected number of events: 303.7
  • Conditional power (achieved) [1]: NA
  • Conditional power (achieved) [2]: 0.5446

Simulated data

  • Analysis times [1] : median [range]: 39.578 [33.988 - 45.552]; mean +/-sd: 39.606 +/-1.47
  • Analysis times [2] : median [range]: 53.616 [46.872 - 61.013]; mean +/-sd: 53.659 +/-2.007
  • Number of subjects [1] : median [range]: 1200 [1200 - 1200]; mean +/-sd: 1200 +/-0
  • Number of subjects [2] : median [range]: 1200 [1200 - 1200]; mean +/-sd: 1200 +/-0
  • Number of events per stage [1] : median [range]: 232 [232 - 232]; mean +/-sd: 232 +/-0
  • Number of events per stage [2] : median [range]: 119 [119 - 119]; mean +/-sd: 119 +/-0
  • Test statistic [1] : median [range]: 2.267 [-1.52 - 5.611]; mean +/-sd: 2.269 +/-0.99
  • Test statistic [2] : median [range]: 2.314 [-0.877 - 4.832]; mean +/-sd: 2.282 +/-0.773
  • Log-rank statistic [1] : median [range]: 2.267 [-1.52 - 5.611]; mean +/-sd: 2.269 +/-0.99
  • Log-rank statistic [2] : median [range]: 2.314 [-0.877 - 4.832]; mean +/-sd: 2.282 +/-0.773
  • Hazard ratio estimate LR [1] : median [range]: 0.743 [0.479 - 1.221]; mean +/-sd: 0.749 +/-0.098
  • Hazard ratio estimate LR [2] : median [range]: 0.781 [0.597 - 1.098]; mean +/-sd: 0.787 +/-0.066
  • Conditional power (achieved) [2] : median [range]: 0.607 [0 - 0.972]; mean +/-sd: 0.545 +/-0.336

Legend

  • (i): values of treatment arm i
  • [k]: values at stage k

3.1 Accessing trial summaries per stage for each simulation

Summary results for each trial can be obtained from the simulation object using the function getData(). Similarly, raw data from individual trials that were stopped at each stage can be obtained using function getRawData() (if maxNumberOfRawDatasetsPerStage was set > 0). The format of these datasets is described in the help ?getSimulationSurvival and illustrated now.

# get aggregate datasets from all simulation runs
aggregateSimulationData <- getData(simulationResult)
# show the first 6 records of the dataset
head(aggregateSimulationData)
iterationNumber 1 1 2 3 3 4
stageNumber 1 2 1 1 2 1
pi1 0 0 0 0 0 0
pi2 0 0 0 0 0 0
hazardRatio 0.74 0.74 0.74 0.74 0.74 0.74
analysisTime 40.07 53.85 41.22 40.62 54.54 40.74
numberOfSubjects 1200 1200 1200 1200 1200 1200
overallEvents1 100 159 94 107 165 98
overallEvents2 132 192 138 125 186 134
eventsPerStage 232 119 232 232 119 232
rejectPerStage 0 1 1 0 0 1
eventsNotAchieved 0 0 0 0 0 0
futilityPerStage 0 0 0 0 0 0
testStatistic 2.474 2.328 3.162 1.446 1.558 2.724
logRankStatistic 2.474 2.328 3.162 1.446 1.558 2.724
conditionalPowerAchieved NA 0.9644 NA NA 0.3572 NA
pValuesSeparate NA NA NA NA NA NA
trialStop FALSE TRUE TRUE FALSE TRUE TRUE
hazardRatioEstimateLR 0.7226 0.7800 0.6602 0.8271 0.8468 0.6993
# The aggregated dataset contains one record for each of the 10'000 simulated
# interim datasets (stageNumber = 1), and one additional record for the final
# analysis (stageNumber = 2) for all simulated trials which were not stopped at
# the interim analysis
table(aggregateSimulationData$stageNumber)
## 
##     1     2 
## 10000  6026
# One possible analysis which uses the aggregated dataset:
# display the distribution of the timing of the analyses (by stage)
# across simulation runs
p <- ggplot(aggregateSimulationData, aes(factor(