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Group Sequential Design for Equivalence in Mean Ratio in 2x2 Crossover

Source: R/getDesignMeans.R
getDesignMeanRatioXOEquiv.Rd

Obtains the power given sample size or obtains the sample size given power for a group sequential design for equivalence mean ratio in 2x2 crossover.

Usage

getDesignMeanRatioXOEquiv(
  beta = NA_real_,
  n = NA_real_,
  meanRatioLower = NA_real_,
  meanRatioUpper = NA_real_,
  meanRatio = 1,
  CV = 1,
  allocationRatioPlanned = 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.

meanRatioLower

The lower equivalence limit of mean ratio.

meanRatioUpper

The upper equivalence limit of mean ratio.

meanRatio

The mean ratio under the alternative hypothesis.

CV

The coefficient of variation.

allocationRatioPlanned

Allocation ratio for sequence A/B versus sequence B/A. Defaults to 1 for equal randomization.

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 designMeanRatioEquiv 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.

    • 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.

    • meanRatioLower: The lower equivalence limit of mean ratio.

    • meanRatioUpper: The upper equivalence limit of mean ratio.

    • meanRatio: The mean ratio under the alternative hypothesis.

    • CV: The coefficient of variation.

  • 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.

    • efficacyMeanRatioLower: The efficacy boundaries on the mean ratio scale for the one-sided null hypothesis on the lower equivalence limit.

    • efficacyMeanRatioUpper: The efficacy boundaries on the mean ratio 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.

    • allocationRatioPlanned: Allocation ratio for sequence A/B versus sequence B/A.

    • 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 <- getDesignMeanRatioXOEquiv(
  beta = 0.1, n = NA, meanRatioLower = 0.8, meanRatioUpper = 1.25,
  meanRatio = 1, CV = 0.35,
  kMax = 4, alpha = 0.05, typeAlphaSpending = "sfOF"))
#>                                                                                      
#> Group-sequential design with 4 stages for equivalence in mean ratio in 2x2 crossover 
#> Lower limit for mean ratio: 0.8, upper limit for mean ratio: 1.25                    
#> Mean ratio under H1: 1, coefficient of variation: 0.35                               
#> Overall power: 0.9033, overall alpha (1-sided): 0.05, attained alpha: 0.05           
#> Maximum information: 224.99, expected under H1: 189.53, expected under H0: 223.67    
#> Maximum # subjects: 52, expected under H1: 43.8, expected under H0: 51.7             
#> Sequence allocation ratio: 1                                                         
#> 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  0.6303  0.9033 
#> Cumulative alpha for each 1-sided test 0.0001  0.0056  0.0236  0.0500 
#> Cumulative alpha spent                 0.0000  0.0000  0.0236  0.0500 
#> Number of subjects                     13.0    26.0    39.0    52.0   
#> Boundary for lower limit (mean ratio)  1.319   1.016   0.934   0.897  
#> Boundary for upper limit (mean ratio)  0.758   0.984   1.070   1.115  
#> Boundary for each 1-sided test (p)     0.0001  0.0055  0.0219  0.0427 
#> Information                            56.25   112.50  168.75  224.99 

# Example 2: sample size calculation for t-test
(design2 <- getDesignMeanRatioXOEquiv(
  beta = 0.1, n = NA, meanRatioLower = 0.8, meanRatioUpper = 1.25,
  meanRatio = 1, CV = 0.35,
  normalApproximation = FALSE, alpha = 0.05))
#>                                                                            
#> Fixed design for equivalence in mean ratio in 2x2 crossover                
#> Lower limit for mean ratio: 0.8, upper limit for mean ratio: 1.25          
#> Mean ratio under H1: 1, coefficient of variation: 0.35                     
#> Overall power: 0.9024, overall alpha (1-sided): 0.05, attained alpha: 0.05 
#> Information: 224.99                                                        
#> Number of subjects: 52                                                     
#> Sequence allocation ratio: 1                                               
#>                                                                            
#>                                             
#> Boundary for each 1-sided test (t)    1.676 
#> Boundary for lower limit (mean ratio) 0.895 
#> Boundary for upper limit (mean ratio) 1.118 
#> Boundary for each 1-sided test (p)    0.0500