getDesignWilcoxon {lrstat}R Documentation

Group Sequential Design for Two-Sample Wilcoxon Test

Description

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.

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.

Value

An S3 class designWilcoxon object with three components:

Author(s)

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))

# 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"))


[Package lrstat version 0.2.15 Index]