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Group Sequential Design for Two-Sample Wilcoxon Test

Source: R/getDesignMeans.R
getDesignWilcoxon.Rd

Obtains the power given sample size or obtains the sample size given power for a group sequential design for two-sample Wilcoxon test.

Usage

getDesignWilcoxon(
  beta = NA_real_,
  n = NA_real_,
  pLarger = 0.6,
  allocationRatioPlanned = 1,
  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.

pLarger

The probability that a randomly chosen sample from the treatment group is larger than a randomly chosen sample from the control group under the alternative hypothesis.

allocationRatioPlanned

Allocation ratio for the active treatment versus control. Defaults to 1 for equal randomization.

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 designWilcoxon 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).

    • numberOfSubjects: The maximum number of subjects.

    • expectedNumberOfSubjectsH1: The expected number of subjects under H1.

    • expectedNumberOfSubjectsH0: The expected number of subjects under H0.

    • pLarger: The probability that a randomly chosen sample from the treatment group is larger than a randomly chosen sample from the control group 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.

    • efficacyPLarger: The efficacy boundaries on the proportion of pairs of samples from the two treatment groups with the sample from the treatment group greater than that from the control group.

    • futilityPLarger: The futility boundaries on the proportion of pairs of samples from the two treatment groups with the sample from the treatment group greater than that from the control group.

    • 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.

    • allocationRatioPlanned: Allocation ratio for the active treatment versus control.

    • rounding: Whether to round up sample size.

Details

The Mann-Whitney U test is a non-parametric test for the difference in location between two independent groups. It is also known as the Wilcoxon rank-sum test. The test is based on the ranks of the data rather than the actual values, making it robust to outliers and non-normal distributions. The test statistic is the number of times a randomly chosen sample from the treatment group is larger than a randomly chosen sample from the control group, i.e., $$W_{XY} = \sum_{i=1}^{n_1}\sum_{j=1}^{n_2} I(X_i > Y_j)$$ where \(X_i\) and \(Y_j\) are the samples from the treatment and control groups, respectively. The test is often used when the data do not meet the assumptions of the t-test, such as non-normality or unequal variances. The test is also applicable to ordinal data. The test is one-sided, meaning that it only tests whether the treatment group is larger than the control group. Asymptotically, $$\frac{W_{XY} - n_1 n_2/2}{\sqrt{n_1 n_2 (n+1)/12}} \sim N(0,1) \quad \text{under} H_0$$ where \(n_1\) and \(n_2\) are the sample sizes of the treatment and control groups, respectively, and \(n=n_1+n_2\). Let \(\theta = P(X > Y)\), and \(\hat{\theta} = \frac{1}{nm}W_{XY}\). It follows that $$\sqrt{n}(\hat{\theta} - 1/2) \sim N\left(0, \frac{1}{12 r(1-r)}\right) \quad \text{under} H_0$$ where \(r = n_1/(n_1+n_2)\) is the randomization probability for the active treatment group. This formulation allows for group sequential testing with futility stopping and efficacy stopping.

Author

Kaifeng Lu, kaifenglu@gmail.com

Examples


# Example 1: fixed design
(design1 <- getDesignWilcoxon(
  beta = 0.1, n = NA,
  pLarger = pnorm((8 - 2)/sqrt(2*25^2)), alpha = 0.025))
#>                                                                              
#> Fixed design for two-sample Wilcoxon test                                    
#> Probability of observations in treatment larger than those in control: 0.567 
#> Overall power: 0.9002, overall alpha (1-sided): 0.025                        
#> Drift parameter: 3.243, inflation factor: 1                                  
#> Information: 2316                                                            
#> Number of subjects: 772                                                      
#> Allocation ratio: 1                                                          
#>                                                                              
#>                                   
#> Efficacy boundary (Z)       1.960 
#> Efficacy boundary (pLarger) 0.5407
#> Efficacy boundary (p)       0.0250

# Example 2: group sequential design
(design2 <- getDesignWilcoxon(
  beta = 0.1, n = NA,
  pLarger = pnorm((8 - 2)/sqrt(2*25^2)), alpha = 0.025,
  kMax = 3, typeAlphaSpending = "sfOF"))
#>                                                                                   
#> Group-sequential design with 3 stages for two-sample Wilcoxon test                
#> Probability of observations in treatment larger than those in control: 0.567      
#> Overall power: 0.9001, overall alpha (1-sided): 0.025                             
#> Drift parameter: 3.261, inflation factor: 1.012                                   
#> Maximum information: 2343, expected under H1: 1878.94, expected under H0: 2338.19 
#> Maximum # subjects: 781, expected under H1: 626.3, expected under H0: 779.4       
#> Allocation ratio: 1                                                               
#> Alpha spending: Lan-DeMets O'Brien-Fleming, beta spending: None                   
#>                                                                                   
#>                             Stage 1 Stage 2 Stage 3
#> Information rate            0.333   0.667   1.000  
#> Efficacy boundary (Z)       3.713   2.510   1.993  
#> Cumulative rejection        0.0335  0.5613  0.9001 
#> Cumulative alpha spent      0.0001  0.0061  0.0250 
#> Number of subjects          260.0   521.0   781.0  
#> Efficacy boundary (pLarger) 0.6329  0.5635  0.5412 
#> Efficacy boundary (p)       0.0001  0.0060  0.0231 
#> Information                 780.00  1563.00 2343.00