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Group Sequential Design for Equivalence in Paired Mean Difference

Source: R/getDesignMeans.R
getDesignPairedMeanDiffEquiv.Rd

Obtains the power given sample size or obtains the sample size given power for a group sequential design for equivalence in paired mean difference.

Usage

getDesignPairedMeanDiffEquiv(
  beta = NA_real_,
  n = NA_real_,
  pairedDiffLower = NA_real_,
  pairedDiffUpper = NA_real_,
  pairedDiff = 0,
  stDev = 1,
  normalApproximation = TRUE,
  rounding = TRUE,
  kMax = 1L,
  informationRates = NA_real_,
  alpha = 0.05,
  typeAlphaSpending = "sfOF",
  parameterAlphaSpending = NA_real_,
  userAlphaSpending = NA_real_,
  spendingTime = NA_real_
)

Arguments

beta

The type II error.

n

The total sample size.

pairedDiffLower

The lower equivalence limit of paired difference.

pairedDiffUpper

The upper equivalence limit of paired difference.

pairedDiff

The paired difference under the alternative hypothesis.

stDev

The standard deviation for paired difference.

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 t distribution. The exact calculation using the t 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.

alpha

The significance level for each of the two one-sided tests. Defaults to 0.05.

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.

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 designPairedMeanDiffEquiv object with three components:

  • overallResults: A data frame containing the following variables:

    • overallReject: The overall rejection probability.

    • alpha: The significance level for each of the two one-sided tests. Defaults to 0.05.

    • attainedAlpha: The attained significance level under H0.

    • kMax: The number of stages.

    • information: The maximum information.

    • expectedInformationH1: The expected information under H1.

    • expectedInformationH0: The expected information under H0.

    • numberOfSubjects: The maximum number of subjects.

    • expectedNumberOfSubjectsH1: The expected number of subjects under H1.

    • expectedNumberOfSubjectsH0: The expected number of subjects under H0.

    • pairedDiffLower: The lower equivalence limit of paired difference.

    • pairedDiffUpper: The upper equivalence limit of paired difference.

    • pairedDiff: The paired difference under the alternative hypothesis.

    • stDev: The standard deviation for paired difference.

  • byStageResults: A data frame containing the following variables:

    • informationRates: The information rates.

    • efficacyBounds: The efficacy boundaries on the Z-scale for each of the two one-sided tests.

    • rejectPerStage: The probability for efficacy stopping.

    • cumulativeRejection: The cumulative probability for efficacy stopping.

    • cumulativeAlphaSpent: The cumulative alpha for each of the two one-sided tests.

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

    • efficacyPairedDiffLower: The efficacy boundaries on the paired difference scale for the one-sided null hypothesis on the lower equivalence limit.

    • efficacyPairedDiffUpper: The efficacy boundaries on the paired difference scale for the one-sided null hypothesis on the upper equivalence limit.

    • efficacyP: The efficacy bounds on the p-value scale for each of the two one-sided tests.

    • information: The cumulative information.

    • 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.

    • spendingTime: The error spending time at each analysis.

    • 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 t distribution. The exact calculation using the t distribution is only implemented for the fixed design.

    • rounding: Whether to round up sample size.

Author

Kaifeng Lu, kaifenglu@gmail.com

Examples


# Example 1: group sequential trial power calculation
(design1 <- getDesignPairedMeanDiffEquiv(
  beta = 0.1, n = NA, pairedDiffLower = -1.3, pairedDiffUpper = 1.3,
  pairedDiff = 0, stDev = 2.2,
  kMax = 4, alpha = 0.05, typeAlphaSpending = "sfOF"))
#>                                                                                 
#> Group-sequential design with 4 stages for equivalence in paired mean difference 
#> Lower limit for paired difference: -1.3, upper limit for paired difference: 1.3 
#> Paired difference under H1: 0, standard deviation for paired difference: 2.2    
#> Overall power: 0.9024, overall alpha: 0.05, attained alpha: 0.05                
#> Maximum information: 6.61, expected under H1: 5.57, expected under H0: 6.57     
#> Maximum # subjects: 32, expected under H1: 27, expected under H0: 31.8          
#> Alpha spending: Lan-DeMets O'Brien-Fleming                                      
#>                                                                                 
#>                                        Stage 1 Stage 2 Stage 3 Stage 4
#> Information rate                       0.250   0.500   0.750   1.000  
#> Boundary for each 1-sided test (Z)     3.750   2.540   2.016   1.720  
#> Cumulative rejection                   0.0000  0.0002  0.6283  0.9024 
#> Cumulative alpha for each 1-sided test 0.0001  0.0056  0.0236  0.0500 
#> Cumulative alpha attained under H0     0.0000  0.0000  0.0235  0.0500 
#> Number of subjects                     8.0     16.0    24.0    32.0   
#> Boundary for lower limit (paired diff) 1.616   0.097   -0.395  -0.631 
#> Boundary for upper limit (paired diff) -1.616  -0.097  0.395   0.631  
#> Boundary for each 1-sided test (p)     0.0001  0.0055  0.0219  0.0427 
#> Information                            1.65    3.31    4.96    6.61   

# Example 2: sample size calculation for t-test
(design2 <- getDesignPairedMeanDiffEquiv(
  beta = 0.1, n = NA, pairedDiffLower = -1.3, pairedDiffUpper = 1.3,
  pairedDiff = 0, stDev = 2.2,
  normalApproximation = FALSE, alpha = 0.05))
#>                                                                                 
#> Fixed design for equivalence in paired mean difference                          
#> Lower limit for paired difference: -1.3, upper limit for paired difference: 1.3 
#> Paired difference under H1: 0, standard deviation for paired difference: 2.2    
#> Overall power: 0.9064, overall alpha: 0.05, attained alpha: 0.05                
#> Information: 6.82                                                               
#> Number of subjects: 33                                                          
#>                                                                                 
#>                                              
#> Boundary for each 1-sided test (t)     1.694 
#> Boundary for lower limit (paired diff) -0.651
#> Boundary for upper limit (paired diff) 0.651 
#> Boundary for each 1-sided test (p)     0.0500