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

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`

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%:

```
<- getDesignGroupSequential(
design 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:

```
<- getSampleSizeSurvival(design,
sampleSizeResult 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`

.

```
<- getPowerSurvival(design,
powerResult 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.

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.

```
<- getSimulationSurvival(design,
simulationResult 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

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
<- getData(simulationResult)
aggregateSimulationData # 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
<- ggplot(aggregateSimulationData, aes(factor( p
```