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Simulation for a Binary and a Time-to-Event Endpoint in Group Sequential Trials

Source: R/lrsim.R
binary_tte_sim.Rd

Performs simulation for two-endpoint two-arm group sequential trials.

  • Endpoint 1: Binary endpoint, analyzed using the Mantel-Haenszel test for risk difference.

  • Endpoint 2: Time-to-event endpoint, analyzed using the log-rank test for treatment effect.

The analysis times for the binary endpoint are based on calendar times, while the time-to-event analyses are triggered by reaching the pre-specified number of events. The binary endpoint is assessed at the first post-treatment follow-up visit (PTFU1).

Usage

binary_tte_sim(
  kMax1 = 1,
  kMax2 = 1,
  riskDiffH0 = 0,
  hazardRatioH0 = 1,
  allocation1 = 1,
  allocation2 = 1,
  accrualTime = 0,
  accrualIntensity = NA,
  piecewiseSurvivalTime = 0,
  stratumFraction = 1,
  globalOddsRatio = NA,
  pi1 = NA,
  pi2 = NA,
  lambda1 = NA,
  lambda2 = NA,
  gamma1 = NA,
  gamma2 = NA,
  delta1 = NA,
  delta2 = NA,
  upper1 = NA,
  upper2 = NA,
  n = NA,
  plannedTime = NA,
  plannedEvents = NA,
  maxNumberOfIterations = 1000,
  maxNumberOfRawDatasetsPerStage = 0,
  seed = 0,
  nthreads = 0
)

Arguments

kMax1

Number of stages for the binary endpoint.

kMax2

Number of stages for the time-to-event endpoint.

riskDiffH0

Risk difference under the null hypothesis for the binary endpoint.

hazardRatioH0

Hazard ratio under the null hypothesis for the time-to-event endpoint.

allocation1

Number of subjects in the 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.

globalOddsRatio

Global odds ratio of the Plackett copula linking the two endpoints.

pi1

Response probabilities by stratum for the treatment group for the binary endpoint.

pi2

Response probabilities by stratum for the control group for the binary endpoint.

lambda1

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

lambda2

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

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 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.

delta1

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

delta2

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

upper1

Maximim protocol-specified treatment duration for the treatment group.

upper2

Maximum protocol-specified treatment duration for the control group.

n

Sample size.

plannedTime

Calendar times for the analyses of the binary endpoint.

plannedEvents

Target cumulative number of events for the time-to-event analyses.

maxNumberOfIterations

Number of simulation iterations to perform.

maxNumberOfRawDatasetsPerStage

Number of subject-level datasets to retain per stage. Set to 0 to skip raw data saving.

seed

The seed to reproduce the simulation results.

nthreads

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

Value

A list with 4 components:

  • sumdataBIN: A data frame of summary data by iteration and stage for the binary endpoint:

    • iterationNumber: The iteration number.

    • 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.

    • source1: The total number of subjects with response status determined by the underlying latent response variable.

    • source2: The total number of subjects with response status (non-responder) determined by experiencing the event for the time-to-event endpoint.

    • source3: The total number of subjects with response status (non-responder) determined by dropping out prior to the PTFU1 visit.

    • n1: The number of subjects included in the analysis of the binary endpoint for the treatment group.

    • n2: The number of subjects included in the analysis of the binary endpoint for the control group.

    • n: The total number of subjects included in the analysis of the binary endpoint at the stage.

    • y1: The number of responders for the binary endpoint in the treatment group.

    • y2: The number of responders for the binary endpoint in the control group.

    • y: The total number of responders for the binary endpoint at the stage.

    • riskDiff: The estimated risk difference for the binary endpoint.

    • seRiskDiff: The standard error for risk difference based on the Sato approximation.

    • mhStatistic: The Mantel-Haenszel test Z-statistic for the binary endpoint.

  • sumdataTTE: A data frame of summary data by iteration and stage for the time-to-event endpoint:

    • iterationNumber: The iteration number.

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

    • 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 for the time-to-event endpoint.

    • vscore: The variance of the log-rank test statistic for the time-to-event endpoint.

    • logRankStatistic: The log-rank test Z-statistic for the time-to-event endpoint.

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

    • iterationNumber: The iteration number.

    • stageNumber: The stage under consideration.

    • 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 time-to-event endpoint for the subject.

    • dropoutTime: The underlying dropout time for the time-to-event endpoint for the subject.

    • trtDiscTime: The underlying treatment discontinuation time for the binary endpoint for the subject.

    • trtDurUpperLimit: The maximum protocol-specified treatment duration for the subject based on the treatment group assignment.

    • ptfu1Time:The underlying assessment time for the binary endpoint for the subject.

    • timeUnderObservation: The time under observation since randomization for the binary endpoint for the subject.

    • latentResponse: The underlying latent response variable for the binary endpoint for the subject, which determines the response status for the binary endpoint at PTFU1 visit.

    • responder: Whether the subject is a responder for the binary endpoint.

    • source: The source of the determination of responder status for the binary endpoint: = 1 based on the underlying latent response variable, = 2 based on the occurrence of the time-to-event endpoint before the assessment time of the binary endpoint (imputed as a non-responder), = 3 based on the dropout before the assessment time of the binary endpoint (imputed as a non-responder), = 4 excluded from analysis due to administrative censoring.

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

    • iterationNumber: The iteration number.

    • stageNumber: The stage under consideration.

    • 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 time-to-event endpoint for the subject.

    • dropoutTime: The underlying dropout time for the time-to-event endpoint for the subject.

    • timeUnderObservation: The time under observation since randomization for the time-to-event endpoint for the subject.

    • event: Whether the subject experienced the event for the time-to-event endpoint.

    • dropoutEvent: Whether the subject dropped out for the time-to-event endpoint.

Details

We consider dual primary endpoints with endpoint 1 being a binary endpoint and endpoint 2 being a time-to-event endpoint. The analyses of endpoint 1 will be based on calendar times, while the analyses of endpoint 2 will be based on the number of events. Therefore, the analyses of the two endpoints are not at the same time points. The correlation between the two endpoints is characterized by the global odds ratio of the Plackett copula. In addition, the time-to-event endpoint will render the binary endpoint as a non-responder, and so does the dropout. In addition, the treatment discontinuation will impact the number of available subjects for analysis. The administrative censoring will exclude subjects from the analysis of the binary endpoint.

Author

Kaifeng Lu, kaifenglu@gmail.com

Examples


tcut = c(0, 12, 36, 48)
surv = c(1, 0.95, 0.82, 0.74)
lambda2 = (log(surv[1:3]) - log(surv[2:4]))/(tcut[2:4] - tcut[1:3])

sim1 = binary_tte_sim(
  kMax1 = 1,
  kMax2 = 2,
  accrualTime = seq(0, 8),
  accrualIntensity = 40/9 * seq(1, 9),
  piecewiseSurvivalTime = c(0, 12, 36),
  globalOddsRatio = 1,
  pi1 = 0.80,
  pi2 = 0.65,
  lambda1 = 0.65*lambda2,
  lambda2 = lambda2,
  gamma1 = -log(1-0.04)/12,
  gamma2 = -log(1-0.04)/12,
  delta1 = -log(1-0.02)/12,
  delta2 = -log(1-0.02)/12,
  upper1 = 15*28/30.4,
  upper2 = 12*28/30.4,
  n = 640,
  plannedTime = 20 + 15*28/30.4,
  plannedEvents = c(130, 173),
  maxNumberOfIterations = 1000,
  maxNumberOfRawDatasetsPerStage = 1,
  seed = 314159,
  nthreads = 1)