Promizing Zone Design with rpact

Nestlé rpact Training 2024, Lausanne

May 24, 2024

A Motivating Example from Hsiao, Liu, and Mehta (Biometrical Journal, 2019)

Efficacy endpoint PFS
Assumed hazard ratio = 0.67, \(\alpha = 0.025\) and \(\beta = 0.1\) requires 263 events:

 getSampleSizeSurvival(alpha = 0.025, beta = 0.1, hazardRatio = 0.67)$maxNumberOfEvents

[1] 262.0594

280 PFS events yields power 91.8 %:

 getPowerSurvival(alpha = 0.025, hazardRatio = 0.67, directionUpper = FALSE, 
        maxNumberOfEvents = 280, maxNumberOfSubjects = 600,      
        median2 = 8.5)$overallReject

[1] 0.9178375

If 350 patients are enrolled over 28 months with a median PFS time of 8.5 months in the control group, the final analysis is expected to be after an additional follow-up of about 12 months:

getPowerSurvival(alpha = 0.025, hazardRatio = 0.67, 
            directionUpper = FALSE, 
            maxNumberOfEvents = 280, 
            maxNumberOfSubjects = 350,
            median2 = 8.5,
            accrualTime = 28)$analysisTime

         [,1]
[1,] 40.52321

500 PFS events are needed to have 90% power at HR = 0.75 with more patients and a different expected follow-up:

 getSampleSizeSurvival(alpha = 0.025, beta = 0.1, hazardRatio = 0.75)$maxNumberOfEvents

[1] 507.8443

“Milestone-based” investment:
- Two-stage approach with interim after 140 events
- Enough power for detecting HR = 0.67
- If conditional power CP for detecting HR = 0.75 falls in a “promizing zone”, an additional investment would be made that allows the trial to remain open until 420 PFS events were obtained
- Conditional power based on assumed minimum clinical relevant effect HR = 0.75

Promizing Zone Design

Number of events for the second stage between 140 and 280
If conditional power for 280 additional events at HR = 0.75 is smaller than \(cp_{min}\), set number of additional events = 140 (non-promising case)
If conditional power for 140 additional events at HR = 0.75 exceeds \(cp_{max}\), set number of additional events = 140, otherwise calculate event number according to \[CP_{HR = 0.75} = cp_{max}\] (promising case)
This defined a promizing zone for HR within the sample size may be modified.

Promizing Zone Design Using `rpact`

First, define the design

myDesign <- getDesignInverseNormal(kMax = 2, typeOfDesign = "noEarlyEfficacy")

Define the event number calculation function myEventSizeCalculationFunction()

# Define promizing zone event size function
myEventSizeCalculationFunction <- function(..., stage,
                     plannedEvents,
                     conditionalPower,
                     minNumberOfEventsPerStage,
                     maxNumberOfEventsPerStage,
                     conditionalCriticalValue,
                     estimatedTheta) {
  
  calculateStageEvents <- function(cp) {
      4 * max(0, conditionalCriticalValue + qnorm(cp))^2 / 
      log(max(1 + 1e-12, estimatedTheta))^2
  }
  
  # Calculate events required to reach maximum desired conditional power
  # cp_max (provided as argument conditionalPower)
  stageEventsCPmax <- ceiling(calculateStageEvents(cp = conditionalPower))
  
  # Calculate events required to reach minimum desired conditional power
  # cp_min (**manually set for this example to 0.8**)
  stageEventsCPmin <- ceiling(calculateStageEvents(cp = 0.8))
  
  # Define stageEvents
  stageEvents <- min(max(minNumberOfEventsPerStage[stage], stageEventsCPmax),
    maxNumberOfEventsPerStage[stage])
  
  # Set stageEvents to minimal sample size in case minimum conditional power
  # cannot be reached with available sample size
  if (stageEventsCPmin > maxNumberOfEventsPerStage[stage]) {
    stageEvents <- minNumberOfEventsPerStage[stage]
  
  }
  # return overall events for second stage 
  return(plannedEvents[1] + stageEvents)
}

Run the Simulation

by specifying calcEventsFunction = myEventSizeCalculationFunction and a range of assumed true hazard ratios

hazardRatioSeq <- seq(0.65, 0.85, by = 0.025)

simSurvPromZone <- getSimulationSurvival(design = myDesign,
                      hazardRatio = hazardRatioSeq,
                      directionUpper = FALSE, 
                      plannedEvents = c(140, 280), 
                      median2 = 8.5,
                      minNumberOfEventsPerStage = c(NA, 140),
                      maxNumberOfEventsPerStage = c(NA, 280),
                      thetaH1 = 0.75,
                      conditionalPower = 0.9,
                      accrualTime = 36, 
                      calcEventsFunction = myEventSizeCalculationFunction,
                      maxNumberOfIterations = maxNumberOfIterations,
                      longTimeSimulationAllowed = TRUE,
                      maxNumberOfSubjects = 500)

“Usual” Conditional Power Approach

Specify calcEventsFunction = NULL

simSurvCondPower <- getSimulationSurvival(design = myDesign,
                       hazardRatio = hazardRatioSeq, 
                       directionUpper = FALSE, 
                       plannedEvents = c(140, 280), 
                       median2 = 8.5,
                       minNumberOfEventsPerStage = c(NA, 140),
                       maxNumberOfEventsPerStage = c(NA, 280),
                       thetaH1 = 0.75,
                       conditionalPower = 0.9,
                       accrualTime = 36, 
                       calcEventsFunction = NULL,
                       maxNumberOfIterations = maxNumberOfIterations,
                       longTimeSimulationAllowed = TRUE,
                       maxNumberOfSubjects = 500)