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Power and Sample Size for Group Sequential Design With Futility Stopping Under Null Hypothesis

Source: R/RcppExports.R
getDesign2.Rd

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

Usage

getDesign2(
  beta = NA_real_,
  IMax = NA_real_,
  theta = NA_real_,
  kMax = 1L,
  informationRates = NA_real_,
  efficacyStopping = NA_integer_,
  futilityStopping = NA_integer_,
  criticalValues = NULL,
  alpha = 0.025,
  typeAlphaSpending = "sfOF",
  parameterAlphaSpending = NA_real_,
  userAlphaSpending = NA_real_,
  symmetricBounds = TRUE,
  astar = 0.025,
  futilityBounds = NULL,
  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.

symmetricBounds

If TRUE, futility bounds are set to the negative of efficacy bounds at each analysis (subject to futilityStopping). If FALSE, futility bounds are determined by futilityBounds or beta spending.

astar

The overall futility stopping probability under the null hypothesis.

futilityBounds

A vector of length kMax for the futility stopping boundaries on the Z-scale under the null hypothesis. Defaults to rep(-8, kMax) if left unspecified. The futility bounds are non-binding for the calculation of critical values.

typeBetaSpending

The type of beta spending function for determining futility bounds under the null hypothesis when futilityBounds is not provided. The same types as typeAlphaSpending are allowed, except that "none" corresponds to no futility stopping under the null hypothesis.

parameterBetaSpending

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

userBetaSpending

A vector of length kMax for the user defined beta spending function when typeBetaSpending == "user". The last element must be equal to astar and the vector must be increasing.

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.

    • astar: The overall futility stopping probability under the null hypothesis.

    • 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.

Details

The futility stopping boundaries under the null hypothesis are non-binding. The function determines efficacy and futility bounds based on the inputs provided, following a clear priority order.

Efficacy bounds: If criticalValues are supplied, they take precedence and all alpha-spending parameters are ignored. Otherwise, efficacy bounds are derived from the specified alpha-spending function.

Futility bounds: Futility inputs are evaluated in the following order of priority:

  1. If futilityBounds are provided, they override beta-spending parameters.

  2. If futilityBounds are not specified, futility bounds are computed using the beta-spending approach with astar being the maximum futility stopping probability under the null hypothesis. If typeBetaSpending == "none", then there is no futility stopping under the null hypothesis.

If symmetricBounds = TRUE, the futility bounds are set to -efficacyBounds and beta-spending inputs are ignored.

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


(design1 <- getDesign2(
  beta = 0.149, theta = -log(0.65),
  kMax = 2, informationRates = c(0.87, 1),
  alpha = 0.004, typeAlphaSpending = "sfOF",
  astar = 0.1, typeBetaSpending = "sfHSD",
  parameterBetaSpending = -8))
#>                                                                             
#> Group-sequential design with 2 stages                                       
#> theta: 0.431, maximum information: 74.5                                     
#> Overall power: 0.851, overall alpha (1-sided): 0.004, attained alpha: 0.004 
#> Drift parameter: 3.718, inflation factor: 1.014                             
#> Expected information under H1: 56.39, expected information under H0: 0.56   
#> Alpha spending: Lan-DeMets O'Brien-Fleming, beta spending: HSD(gamma = -8)  
#>                                                                             
#>                               Stage 1 Stage 2
#> Information rate              0.870   1.000  
#> Efficacy boundary (Z)         2.873   2.706  
#> Futility boundary (Z)         -2.873  -2.706 
#> Cumulative rejection          0.7240  0.8510 
#> Cumulative futility           0.0000  0.0000 
#> Cumulative alpha spent        0.0020  0.0040 
#> Efficacy boundary (theta)     0.357   0.314  
#> Futility boundary (theta)     -0.357  -0.314 
#> Efficacy boundary (p)         0.0020  0.0034 
#> Futility boundary (p)         0.9980  0.9966 
#> Information                   64.81   74.50  
#> Cumulative rejection under H0 0.0020  0.0040 
#> Cumulative futility under H0  0.0020  0.0040