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Log-Rank Test Simulation

Source: R/lrsim.R
lrsim.Rd

Performs simulation for two-arm group sequential trials based on weighted log-rank test.

Usage

lrsim(
  kMax = 1,
  informationRates = NA,
  criticalValues = NA,
  futilityBounds = NA,
  hazardRatioH0 = 1,
  allocation1 = 1,
  allocation2 = 1,
  accrualTime = 0,
  accrualIntensity = NA,
  piecewiseSurvivalTime = 0,
  stratumFraction = 1,
  lambda1 = NA,
  lambda2 = NA,
  gamma1 = 0,
  gamma2 = 0,
  n = NA,
  followupTime = NA,
  fixedFollowup = FALSE,
  rho1 = 0,
  rho2 = 0,
  plannedEvents = NA,
  plannedTime = NA,
  maxNumberOfIterations = 1000,
  maxNumberOfRawDatasetsPerStage = 0,
  seed = 0,
  nthreads = 0
)

Arguments

kMax

The maximum number of stages.

informationRates

The information rates in terms of number of events for the conventional log-rank test and in terms of the actual information for weighted log-rank tests. Fixed prior to the trial. If left unspecified, it defaults to plannedEvents / plannedEvents[kMax] when plannedEvents is provided and to plannedTime / plannedTime[kMax] otherwise.

criticalValues

Upper boundaries on the z-test statistic scale for stopping for efficacy.

futilityBounds

Lower boundaries on the z-test statistic scale for stopping for futility at stages 1, ..., kMax-1. Defaults to rep(-6, kMax-1) if left unspecified. The futility bounds are non-binding for the calculation of critical values.

hazardRatioH0

Hazard ratio under the null hypothesis for the active treatment versus control. Defaults to 1 for superiority test.

allocation1

Number of subjects in the active treatment group in a randomization block. Defaults to 1 for equal randomization.

allocation2

Number of subjects in the control group in a randomization block. Defaults to 1 for equal randomization.

accrualTime

A vector that specifies the starting time of piecewise Poisson enrollment time intervals. Must start with 0, e.g., c(0, 3) breaks the time axis into 2 accrual intervals: \([0, 3)\) and \([3, \infty)\).

accrualIntensity

A vector of accrual intensities. One for each accrual time interval.

piecewiseSurvivalTime

A vector that specifies the starting time of piecewise exponential survival time intervals. Must start with 0, e.g., c(0, 6) breaks the time axis into 2 event intervals: \([0, 6)\) and \([6, \infty)\). Defaults to 0 for exponential distribution.

stratumFraction

A vector of stratum fractions that sum to 1. Defaults to 1 for no stratification.

lambda1

A vector of hazard rates for the event in each analysis time interval by stratum for the active treatment group.

lambda2

A vector of hazard rates for the event in each analysis time interval by stratum for the control group.

gamma1

The hazard rate for exponential dropout, a vector of hazard rates for piecewise exponential dropout applicable for all strata, or a vector of hazard rates for dropout in each analysis time interval by stratum for the active treatment group.

gamma2

The hazard rate for exponential dropout, a vector of hazard rates for piecewise exponential dropout applicable for all strata, or a vector of hazard rates for dropout in each analysis time interval by stratum for the control group.

n

Sample size.

followupTime

Follow-up time for the last enrolled subject.

fixedFollowup

Whether a fixed follow-up design is used. Defaults to FALSE for variable follow-up.

rho1

The first parameter of the Fleming-Harrington family of weighted log-rank test. Defaults to 0 for conventional log-rank test.

rho2

The second parameter of the Fleming-Harrington family of weighted log-rank test. Defaults to 0 for conventional log-rank test.

plannedEvents

The planned cumulative total number of events at each stage.

plannedTime

The calendar times for the analyses. To use calendar time to plan the analyses, plannedEvents should be missing.

maxNumberOfIterations

The number of simulation iterations. Defaults to 1000.

maxNumberOfRawDatasetsPerStage

The number of raw datasets per stage to extract.

seed

The seed to reproduce the simulation results.

nthreads

The number of threads to use in simulations (0 means the default RcppParallel behavior).

Value

An S3 class lrsim object with 3 components:

  • overview: A list containing the following information:

    • rejectPerStage: The efficacy stopping probability by stage.

    • futilityPerStage: The futility stopping probability by stage.

    • cumulativeRejection: Cumulative efficacy stopping probability by stage.

    • cumulativeFutility: The cumulative futility stopping probability by stage.

    • numberOfEvents: The average number of events by stage.

    • numberOfDropouts: The average number of dropouts by stage.

    • numberOfSubjects: The average number of subjects by stage.

    • analysisTime: The average analysis time by stage.

    • overallReject: The overall rejection probability.

    • expectedNumberOfEvents: The expected number of events for the overall study.

    • expectedNumberOfDropouts: The expected number of dropouts for the overall study.

    • expectedNumberOfSubjects: The expected number of subjects for the overall study.

    • expectedStudyDuration: The expected study duration.

    • hazardRatioH0: Hazard ratio under the null hypothesis for the active treatment versus control.

    • useEvents: whether the analyses are planned based on the number of events or calendar time.

    • numberOfIterations: The number of simulation iterations.

    • n: Sample size.

    • fixedFollowup: Whether a fixed follow-up design is used.

    • rho1: The first parameter of the Fleming-Harrington family of weighted log-rank test. Defaults to 0 for conventional log-rank test.

    • rho2: The second parameter of the Fleming-Harrington family of weighted log-rank test. Defaults to 0 for conventional log-rank test.

    • kMax: The maximum number of stages.

  • sumdata: A data frame of summary data by iteration and stage:

    • iterationNumber: The iteration number.

    • eventsNotAchieved: Whether the final target number of events is not achieved for the iteration.

    • stopStage: The stage at which the trial stops.

    • stageNumber: The stage number, covering all stages even if the trial stops at an interim look.

    • analysisTime: The time for the stage since trial start.

    • accruals1: The number of subjects enrolled at the stage for the treatment group.

    • accruals2: The number of subjects enrolled at the stage for the control group.

    • totalAccruals: The total number of subjects enrolled at the stage.

    • events1: The number of events at the stage for the treatment group.

    • events2: The number of events at the stage for the control group.

    • totalEvents: The total number of events at the stage.

    • dropouts1: The number of dropouts at the stage for the treatment group.

    • dropouts2: The number of dropouts at the stage for the control group.

    • totalDropouts: The total number of dropouts at the stage.

    • uscore: The numerator of the log-rank test statistic.

    • vscore: The variance of the log-rank test statistic.

    • logRankStatistic: The log-rank test Z-statistic.

    • rejectPerStage: Whether to reject the null hypothesis at the stage.

    • futilityPerStage: Whether to stop the trial for futility at the stage.

  • rawdata (exists if maxNumberOfRawDatasetsPerStage is a positive integer): A data frame for subject-level data for selected replications, containing the following variables:

    • iterationNumber: The iteration number.

    • stopStage: The stage at which the trial stops.

    • stageNumber: The stage number, covering all stages even if the trial stops at an interim look.

    • analysisTime: The time for the stage since trial start.

    • subjectId: The subject ID.

    • arrivalTime: The enrollment time for the subject.

    • stratum: The stratum for the subject.

    • treatmentGroup: The treatment group (1 or 2) for the subject.

    • survivalTime: The underlying survival time for the subject.

    • dropoutTime: The underlying dropout time for the subject.

    • timeUnderObservation: The time under observation since randomization.

    • event: Whether the subject experienced the event.

    • dropoutEvent: Whether the subject dropped out.

Author

Kaifeng Lu, kaifenglu@gmail.com

Examples

# Example 1: analyses based on number of events

sim1 = lrsim(
  kMax = 2, informationRates = c(0.5, 1),
  criticalValues = c(2.797, 1.977),
  accrualIntensity = 11,
  lambda1 = 0.018, lambda2 = 0.030,
  n = 132,
  plannedEvents = c(60, 120),
  maxNumberOfIterations = 1000,
  maxNumberOfRawDatasetsPerStage = 1,
  seed = 314159,
  nthreads = 1)

# summary statistics
sim1
#>                                                         
#> Group-sequential design with 2 stages for log-rank test 
#> Empirical power: 0.791                                  
#> Expected # events: 106.9                                
#> Expected # dropouts: 0                                  
#> Expected # subjects: 132                                
#> Expected study duration: 94.3                           
#> n: 132, fixed follow-up: FALSE                          
#> Number of simulations: 1000                             
#>                                                         
#>                      Stage 1 Stage 2
#> Cumulative rejection 0.2190  0.7910 
#> Cumulative futility  0.0000  0.2090 
#> Number of events     60.0    120.0  
#> Number of dropouts   0.0     0.0    
#> Number of subjects   132.0   132.0  
#> Analysis time        31.9    111.6  

# summary for each simulated data set
head(sim1$sumdata)
#>   iterationNumber stopStage eventsNotAchieved stageNumber analysisTime
#> 1               1         1             FALSE           1     33.16275
#> 2               1         1             FALSE           2    110.48736
#> 3               2         2             FALSE           1     33.22616
#> 4               2         2             FALSE           2    109.62981
#> 5               3         2             FALSE           1     28.33555
#> 6               3         2             FALSE           2    115.48439
#>   accruals1 accruals2 totalAccruals events1 events2 totalEvents dropouts1
#> 1        66        66           132      21      39          60         0
#> 2        66        66           132      56      64         120         0
#> 3        66        66           132      29      31          60         0
#> 4        66        66           132      54      66         120         0
#> 5        66        66           132      25      35          60         0
#> 6        66        66           132      58      62         120         0
#>   dropouts2 totalDropouts     uscore   vscore logRankStatistic rejectPerStage
#> 1         0             0 -11.660878 14.80310       -3.0307840           TRUE
#> 2         0             0 -18.892453 27.16956       -3.6244912           TRUE
#> 3         0             0  -1.199027 14.98331       -0.3097597          FALSE
#> 4         0             0 -12.829339 28.87298       -2.3875830           TRUE
#> 5         0             0  -6.362696 14.94729       -1.6457354          FALSE
#> 6         0             0 -11.331095 28.85624       -2.1093662           TRUE
#>   futilityPerStage
#> 1            FALSE
#> 2            FALSE
#> 3            FALSE
#> 4            FALSE
#> 5            FALSE
#> 6            FALSE

# raw data for selected replication
head(sim1$rawdata)
#>   iterationNumber stopStage stageNumber analysisTime subjectId arrivalTime
#> 1               1         1           1     33.16275         1  0.06481234
#> 2               1         1           1     33.16275         2  0.13397351
#> 3               1         1           1     33.16275         3  0.18561876
#> 4               1         1           1     33.16275         4  0.37951139
#> 5               1         1           1     33.16275         5  0.39512177
#> 6               1         1           1     33.16275         6  0.49481613
#>   stratum treatmentGroup survivalTime dropoutTime timeUnderObservation event
#> 1       1              1    23.977669         Inf            23.977669  TRUE
#> 2       1              2    46.952068         Inf            33.028772 FALSE
#> 3       1              2    21.301206         Inf            21.301206  TRUE
#> 4       1              1    32.783234         Inf            32.783234  TRUE
#> 5       1              1     6.212465         Inf             6.212465  TRUE
#> 6       1              2   122.679540         Inf            32.667930 FALSE
#>   dropoutEvent
#> 1        FALSE
#> 2        FALSE
#> 3        FALSE
#> 4        FALSE
#> 5        FALSE
#> 6        FALSE


# Example 2: analyses based on calendar time have similar power

sim2 = lrsim(
  kMax = 2, informationRates = c(0.5, 1),
  criticalValues = c(2.797, 1.977),
  accrualIntensity = 11,
  lambda1 = 0.018, lambda2 = 0.030,
  n = 132,
  plannedTime = c(31.9, 113.2),
  maxNumberOfIterations = 1000,
  maxNumberOfRawDatasetsPerStage = 1,
  seed = 314159,
  nthreads = 1)

# summary statistics
sim2
#>                                                         
#> Group-sequential design with 2 stages for log-rank test 
#> Empirical power: 0.79                                   
#> Expected # events: 107.7                                
#> Expected # dropouts: 0                                  
#> Expected # subjects: 132                                
#> Expected study duration: 96.6                           
#> n: 132, fixed follow-up: FALSE                          
#> Number of simulations: 1000                             
#>                                                         
#>                      Stage 1 Stage 2
#> Cumulative rejection 0.2040  0.7900 
#> Cumulative futility  0.0000  0.2100 
#> Number of events     59.7    119.9  
#> Number of dropouts   0.0     0.0    
#> Number of subjects   132.0   132.0  
#> Analysis time        31.9    113.2  

# summary for each simulated data set
head(sim2$sumdata)
#>   iterationNumber stopStage eventsNotAchieved stageNumber analysisTime
#> 1               1         1             FALSE           1         31.9
#> 2               1         1             FALSE           2        113.2
#> 3               2         2             FALSE           1         31.9
#> 4               2         2             FALSE           2        113.2
#> 5               3         2             FALSE           1         31.9
#> 6               3         2             FALSE           2        113.2
#>   accruals1 accruals2 totalAccruals events1 events2 totalEvents dropouts1
#> 1        66        66           132      20      39          59         0
#> 2        66        66           132      57      64         121         0
#> 3        66        66           132      28      29          57         0
#> 4        66        66           132      55      66         121         0
#> 5        66        66           132      28      40          68         0
#> 6        66        66           132      57      61         118         0
#>   dropouts2 totalDropouts     uscore   vscore logRankStatistic rejectPerStage
#> 1         0             0 -11.972719 14.56264       -3.1374202           TRUE
#> 2         0             0 -18.767453 27.31019       -3.5912283           TRUE
#> 3         0             0  -0.632938 14.22525       -0.1678152          FALSE
#> 4         0             0 -12.970248 28.77936       -2.4177294           TRUE
#> 5         0             0  -8.140083 16.88593       -1.9809173          FALSE
#> 6         0             0 -10.971205 28.42533       -2.0577926           TRUE
#>   futilityPerStage
#> 1            FALSE
#> 2            FALSE
#> 3            FALSE
#> 4            FALSE
#> 5            FALSE
#> 6            FALSE