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Power and Sample Size for a Generic Group Sequential Design

Source: R/RcppExports.R
getDesign.Rd

Obtains the maximum information and stopping boundaries for a generic group sequential design assuming a constant treatment effect, or obtains the power given the maximum information and stopping boundaries.

Usage

getDesign(
  beta = NA_real_,
  IMax = NA_real_,
  theta = NA_real_,
  kMax = 1L,
  informationRates = NA_real_,
  efficacyStopping = NA_integer_,
  futilityStopping = NA_integer_,
  criticalValues = NA_real_,
  alpha = 0.025,
  typeAlphaSpending = "sfOF",
  parameterAlphaSpending = NA_real_,
  userAlphaSpending = NA_real_,
  futilityBounds = NA_real_,
  typeBetaSpending = "none",
  parameterBetaSpending = NA_real_,
  userBetaSpending = NA_real_,
  spendingTime = NA_real_,
  varianceRatio = 1
)

Arguments

beta

The type II error.

IMax

The maximum information. Either beta or IMax should be provided while the other one should be missing.

theta

The parameter value. Null hypothesis is at theta = 0, and the alternative hypothesis is one-sided for theta > 0.

kMax

The maximum number of stages.

informationRates

The information rates. Fixed prior to the trial. Defaults to (1:kMax) / kMax if left unspecified.

efficacyStopping

Indicators of whether efficacy stopping is allowed at each stage. Defaults to TRUE if left unspecified.

futilityStopping

Indicators of whether futility stopping is allowed at each stage. Defaults to TRUE if left unspecified.

criticalValues

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

alpha

The significance level. Defaults to 0.025.

typeAlphaSpending

The type of alpha spending. One of the following: "OF" for O'Brien-Fleming boundaries, "P" for Pocock boundaries, "WT" for Wang & Tsiatis boundaries, "sfOF" for O'Brien-Fleming type spending function, "sfP" for Pocock type spending function, "sfKD" for Kim & DeMets spending function, "sfHSD" for Hwang, Shi & DeCani spending function, "user" for user defined spending, and "none" for no early efficacy stopping. Defaults to "sfOF".

parameterAlphaSpending

The parameter value for the alpha spending. Corresponds to \(\Delta\) for "WT", \(\rho\) for "sfKD", and \(\gamma\) for "sfHSD".

userAlphaSpending

The user defined alpha spending. Cumulative alpha spent up to each stage.

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.

typeBetaSpending

The type of beta spending. One of the following: "sfOF" for O'Brien-Fleming type spending function, "sfP" for Pocock type spending function, "sfKD" for Kim & DeMets spending function, "sfHSD" for Hwang, Shi & DeCani spending function, "user" for user defined spending, and "none" for no early futility stopping. Defaults to "none".

parameterBetaSpending

The parameter value for the beta spending. Corresponds to \(\rho\) for "sfKD", and \(\gamma\) for "sfHSD".

userBetaSpending

The user defined beta spending. Cumulative beta spent up to each stage.

spendingTime

A vector of length kMax for the error spending time at each analysis. Defaults to missing, in which case, it is the same as informationRates.

varianceRatio

The ratio of the variance under H0 to the variance under H1.

Value

An S3 class design object with three components:

  • overallResults: A data frame containing the following variables:

    • overallReject: The overall rejection probability.

    • alpha: The overall significance level.

    • attainedAlpha: The attained significance level, which is different from the overall significance level in the presence of futility stopping.

    • kMax: The number of stages.

    • theta: The parameter value.

    • information: The maximum information.

    • expectedInformationH1: The expected information under H1.

    • expectedInformationH0: The expected information under H0.

    • drift: The drift parameter, equal to theta*sqrt(information).

    • inflationFactor: The inflation factor (relative to the fixed design).

  • byStageResults: A data frame containing the following variables:

    • informationRates: The information rates.

    • efficacyBounds: The efficacy boundaries on the Z-scale.

    • futilityBounds: The futility boundaries on the Z-scale.

    • rejectPerStage: The probability for efficacy stopping.

    • futilityPerStage: The probability for futility stopping.

    • cumulativeRejection: The cumulative probability for efficacy stopping.

    • cumulativeFutility: The cumulative probability for futility stopping.

    • cumulativeAlphaSpent: The cumulative alpha spent.

    • efficacyTheta: The efficacy boundaries on the parameter scale.

    • futilityTheta: The futility boundaries on the parameter scale.

    • efficacyP: The efficacy boundaries on the p-value scale.

    • futilityP: The futility boundaries on the p-value scale.

    • information: The cumulative information.

    • efficacyStopping: Whether to allow efficacy stopping.

    • futilityStopping: Whether to allow futility stopping.

    • rejectPerStageH0: The probability for efficacy stopping under H0.

    • futilityPerStageH0: The probability for futility stopping under H0.

    • cumulativeRejectionH0: The cumulative probability for efficacy stopping under H0.

    • cumulativeFutilityH0: The cumulative probability for futility stopping under H0.

  • settings: A list containing the following input parameters:

    • typeAlphaSpending: The type of alpha spending.

    • parameterAlphaSpending: The parameter value for alpha spending.

    • userAlphaSpending: The user defined alpha spending.

    • typeBetaSpending: The type of beta spending.

    • parameterBetaSpending: The parameter value for beta spending.

    • userBetaSpending: The user defined beta spending.

    • spendingTime: The error spending time at each analysis.

    • varianceRatio: The ratio of the variance under H0 to the variance under H1.

References

Christopher Jennison, Bruce W. Turnbull. Group Sequential Methods with Applications to Clinical Trials. Chapman & Hall/CRC: Boca Raton, 2000, ISBN:0849303168

Author

Kaifeng Lu, kaifenglu@gmail.com

Examples


# Example 1: obtain the maximum information given power
(design1 <- getDesign(
  beta = 0.2, theta = -log(0.7),
  kMax = 2, informationRates = c(0.5,1),
  alpha = 0.025, typeAlphaSpending = "sfOF",
  typeBetaSpending = "sfP"))
#>                                                                              
#> Group-sequential design with 2 stages                                        
#> theta: 0.357, maximum information: 71.97                                     
#> Overall power: 0.8, overall alpha (1-sided): 0.025, attained alpha: 0.0205   
#> Drift parameter: 3.026, inflation factor: 1.167                              
#> Expected information under H1: 60.12, expected information under H0: 41.78   
#> Alpha spending: Lan-DeMets O'Brien-Fleming, beta spending: Lan-DeMets Pocock 
#>                                                                              
#>                               Stage 1 Stage 2
#> Information rate              0.500   1.000  
#> Efficacy boundary (Z)         2.963   1.969  
#> Futility boundary (Z)         0.985   1.969  
#> Cumulative rejection          0.2053  0.8000 
#> Cumulative futility           0.1240  0.2000 
#> Cumulative alpha spent        0.0015  0.0250 
#> Efficacy boundary (theta)     0.494   0.232  
#> Futility boundary (theta)     0.164   0.232  
#> Efficacy boundary (p)         0.0015  0.0245 
#> Futility boundary (p)         0.1624  0.0245 
#> Information                   35.99   71.97  
#> Cumulative rejection under H0 0.0015  0.0205 
#> Cumulative futility under H0  0.8376  0.9795 

# Example 2: obtain power given the maximum information
(design2 <- getDesign(
  IMax = 72.5, theta = -log(0.7),
  kMax = 3, informationRates = c(0.5, 0.75, 1),
  alpha = 0.025, typeAlphaSpending = "sfOF",
  typeBetaSpending = "sfP"))
#>                                                                              
#> Group-sequential design with 3 stages                                        
#> theta: 0.357, maximum information: 72.5                                      
#> Overall power: 0.771, overall alpha (1-sided): 0.025, attained alpha: 0.0187 
#> Drift parameter: 3.037, inflation factor: 1.263                              
#> Expected information under H1: 51.82, expected information under H0: 39.54   
#> Alpha spending: Lan-DeMets O'Brien-Fleming, beta spending: Lan-DeMets Pocock 
#>                                                                              
#>                               Stage 1 Stage 2 Stage 3
#> Information rate              0.500   0.750   1.000  
#> Efficacy boundary (Z)         2.963   2.359   2.014  
#> Futility boundary (Z)         1.076   1.511   2.014  
#> Cumulative rejection          0.2075  0.6018  0.7710 
#> Cumulative futility           0.1420  0.1896  0.2290 
#> Cumulative alpha spent        0.0015  0.0096  0.0250 
#> Efficacy boundary (theta)     0.492   0.320   0.237  
#> Futility boundary (theta)     0.179   0.205   0.237  
#> Efficacy boundary (p)         0.0015  0.0092  0.0220 
#> Futility boundary (p)         0.1410  0.0654  0.0220 
#> Information                   36.25   54.38   72.50  
#> Cumulative rejection under H0 0.0015  0.0093  0.0187 
#> Cumulative futility under H0  0.8590  0.9488  0.9813