Power Calculation With Stratification Variables

This R Markdown document illustrates the power calculation in the presence of stratification variables. This example is taken from EAST 6.4 section 56.7 on lung cancer patients comparing two treatment groups in a target patient population with some prior therapy. There are three stratification variables:

We consider a three stage Lan-DeMets O’Brien-Fleming group sequential design. The stratum fractions are

p1 = c(0.28, 0.13, 0.25, 0.34)
p2 = c(0.28, 0.72)
p3 = c(0.43, 0.37, 0.2)
stratumFraction = p1 %x% p2 %x% p3
stratumFraction = stratumFraction/sum(stratumFraction)

Using the small cancer cell, age <=50, and performance status score <=50 as the reference stratum, the hazard ratios are

theta1 = c(1, 2.127, 0.528, 0.413)
theta2 = c(1, 0.438)
theta3 = c(1, 0.614, 0.159)

If the hazard rate of the reference stratum is 0.009211, then the hazard rate for the control group is

lambda2 = 0.009211*exp(log(theta1) %x% log(theta2) %x% log(theta3))

The hazard ratio of the active treatment group versus the control group is 0.4466.

In addition, we assume an enrollment period of 24 months with a constant enrollment rate of 12 patients per month to enroll 288 patients, and the target number of events of 66.

caltime(nevents = 66, accrualDuration = 24, accrualIntensity = 12,
        stratumFraction = stratumFraction, 
        lambda1 = 0.4466*lambda2, lambda2 = lambda2, 
        followupTime = 100)
#> [1] 54.92196

Therefore, the follow-up time for the last enrolled patient is 30.92 months. Now we can evaluate the power using the lrpower function.

lrpower(kMax = 3, 
        informationRates = c(0.333, 0.667, 1), 
        alpha = 0.025, typeAlphaSpending = "sfOF", 
        accrualIntensity = 12,
        stratumFraction = stratumFraction,
        lambda1 = 0.4466*lambda2, 
        lambda2 = lambda2, 
        accrualDuration = 24, 
        followupTime = 30.92)
#>                                                                        
#> Group-sequential design with 3 stages for log-rank test                
#> Overall power: 0.882, overall significance level (1-sided): 0.025      
#> Maximum # events: 66, expected # events: 53.8                          
#> Maximum # dropouts: 0, expected # dropouts: 0                          
#> Maximum # subjects: 288, expected # subjects: 288                      
#> Maximum information: 16.42, expected information: 13.41                
#> Total study duration: 54.9, expected study duration: 46.2              
#> Accrual duration: 24, follow-up duration: 30.9, fixed follow-up: FALSE 
#> Allocation ratio: 1                                                    
#> Alpha spending: Lan-DeMets O'Brien-Fleming, beta spending: None        
#>                                                                        
#>                        Stage 1 Stage 2 Stage 3
#> Information rate       0.333   0.667   1.000  
#> Efficacy boundary (Z)  3.712   2.511   1.993  
#> Cumulative rejection   0.0284  0.5247  0.8824 
#> Cumulative alpha spent 0.0001  0.0061  0.0250 
#> Number of events       22.0    44.0    66.0   
#> Number of dropouts     0.0     0.0     0.0    
#> Number of subjects     288.0   288.0   288.0  
#> Analysis time          24.9    39.0    54.9   
#> Efficacy boundary (HR) 0.183   0.446   0.594  
#> Efficacy boundary (p)  0.0001  0.0060  0.0231 
#> Information            5.49    10.98   16.42  
#> HR                     0.447   0.447   0.447

Therefore, the overall power is about 88% for the stratified analysis. This is confirmed by the simulation below.

lrsim(kMax = 3, 
      informationRates = c(0.333, 0.667, 1), 
      criticalValues = c(3.712, 2.511, 1.993), 
      accrualIntensity = 12,
      stratumFraction = stratumFraction,
      lambda1 = 0.4466*lambda2, 
      lambda2 = lambda2, 
      accrualDuration = 24, 
      followupTime = 30.92,
      plannedEvents = c(22, 44, 66),
      maxNumberOfIterations = 1000, 
      seed = 314159)
#>                                                         
#> Group-sequential design with 3 stages for log-rank test 
#> Overall power: 0.882                                    
#> Expected # events: 54.6                                 
#> Expected # dropouts: 0                                  
#> Expected # subjects: 287.8                              
#> Expected study duration: 46.6                           
#> Accrual duration: 24, fixed follow-up: FALSE            
#>                                                         
#>                      Stage 1 Stage 2 Stage 3
#> Cumulative rejection 0.013   0.504   0.882  
#> Cumulative futility  0.000   0.000   0.118  
#> Number of events     22.0    44.0    66.0   
#> Number of dropouts   0.0     0.0     0.0    
#> Number of subjects   279.3   288.0   288.0  
#> Analysis time        24.8    39.0    55.0

Power Calculation With Stratification Variables

Kaifeng Lu

12/15/2021