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Group Sequential Design for One-Sample Proportion

Source: R/getDesignProportions.R
getDesignOneProportion.Rd

Obtains the power given sample size or obtains the sample size given power for a group sequential design for one-sample proportion.

Usage

getDesignOneProportion(
  beta = NA_real_,
  n = NA_real_,
  piH0 = 0.1,
  pi = 0.2,
  nullVariance = TRUE,
  normalApproximation = TRUE,
  rounding = TRUE,
  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_
)

Arguments

beta

The type II error.

n

The total sample size.

piH0

The response probability under the null hypothesis.

pi

The response probability under the alternative hypothesis.

nullVariance

Whether to use the variance under the null or the variance under the alternative.

normalApproximation

The type of computation of the p-values. If TRUE, the normal approximation will be used, otherwise the calculations are performed with the binomial distribution. The exact calculation using the binomial distribution is only implemented for the fixed design.

rounding

Whether to round up sample size. Defaults to 1 for sample size rounding.

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.

Value

An S3 class designOneProportion 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 as well as for the binomial exact test in a fixed design.

    • 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).

    • numberOfSubjects: The maximum number of subjects.

    • expectedNumberOfSubjectsH1: The expected number of subjects under H1.

    • expectedNumberOfSubjectsH0: The expected number of subjects under H0.

    • piH0: The response probability under the null hypothesis.

    • pi: The response probability under the alternative hypothesis.

  • 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.

    • 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.

    • efficacyResponses: The efficacy boundaries on the number of responses scale.

    • futilityResponses: The futility boundaries on the number of responses scale.

    • numberOfSubjects: The number of subjects.

  • 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.

    • nullVariance: Whether to use the variance under the null or the empirical variance under the alternative.

    • normalApproximation: The type of computation of the p-values. If TRUE, the variance is assumed to be known, otherwise the calculations are performed with the binomial distribution.

    • rounding: Whether to round up sample size.

Author

Kaifeng Lu, kaifenglu@gmail.com

Examples


# Example 1: group sequential trial power calculation
(design1 <- getDesignOneProportion(
  beta = 0.2, n = NA, piH0 = 0.15, pi = 0.25,
  kMax = 3, alpha = 0.05, typeAlphaSpending = "sfOF"))
#>                                                                                   
#> Group-sequential design with 3 stages for one-sample proportion                   
#> Response probability under H0: 0.15, response probability under H1: 0.25          
#> Overall power: 0.8012, overall alpha (1-sided): 0.05                              
#> Drift parameter: 2.203, inflation factor: 1.001                                   
#> Maximum information: 485.33, expected under H1: 388.39, expected under H0: 482.56 
#> Maximum # subjects: 91, expected under H1: 72.8, expected under H0: 90.5          
#> Test statistic: z-test                                                            
#> Alpha spending: Lan-DeMets O'Brien-Fleming, beta spending: None                   
#>                                                                                   
#>                                 Stage 1 Stage 2 Stage 3
#> Information rate                0.330   0.670   1.000  
#> Efficacy boundary (Z)           3.220   2.133   1.696  
#> Cumulative rejection            0.0822  0.5209  0.8012 
#> Cumulative alpha spent          0.0006  0.0167  0.0500 
#> Number of subjects              30.0    61.0    91.0   
#> Efficacy boundary (# responses) 11      16      20     
#> Efficacy boundary (p)           0.0006  0.0165  0.0450 
#> Information                     160.00  325.33  485.33 

# Example 2: sample size calculation for one-sample binomial exact test
(design2 <- getDesignOneProportion(
  beta = 0.2, n = NA, piH0 = 0.15, pi = 0.25,
  normalApproximation = FALSE, alpha = 0.05))
#>                                                                              
#> Fixed design for one-sample proportion                                       
#> Response probability under H0: 0.15, response probability under H1: 0.25     
#> Overall power: 0.8097, overall alpha (1-sided): 0.05, attained alpha: 0.0355 
#> Drift parameter: 2.422, inflation factor: 1                                  
#> Information: 586.67                                                          
#> Number of subjects: 110                                                      
#> Test statistic: exact test                                                   
#>                                                                              
#>                                       
#> Efficacy boundary (Z)           1.651 
#> Efficacy boundary (# responses) 24    
#> Efficacy boundary (p)           0.0355