getDesignSlopeDiff: Group Sequential Design for Two-Sample Slope Difference

In lrstat: Power and Sample Size Calculation for Non-Proportional Hazards and Beyond

Group Sequential Design for Two-Sample Slope Difference

Description

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

Usage

getDesignSlopeDiff(
  beta = NA_real_,
  n = NA_real_,
  slopeDiffH0 = 0,
  slopeDiff = 0.5,
  stDev = 1,
  stDevCovariate = 1,
  allocationRatioPlanned = 1,
  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.
slopeDiffH0	The slope difference under the null hypothesis. Defaults to 0.
slopeDiff	The slope difference under the alternative hypothesis.
stDev	The standard deviation of the residual.
stDevCovariate	The standard deviation of the covariate.
allocationRatioPlanned	Allocation ratio for the active treatment versus control. 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.
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

We assume a simple linear regression of the form

y_{g,i} = \alpha_g + \beta_g x_{g,i} + \epsilon_{g,i}

for treatment group g, where \epsilon_{g,i} is the residual error for subject i in group g, which is assumed to be normally distributed with mean 0 and standard deviation \sigma_\epsilon. The covariate x_{g,i} is assumed to be normally distributed with mean 0 and standard deviation \sigma_x. Since

\hat{\beta}_g = \frac{\sum_{i=1}^{n_g} (x_{g,i}-\bar{x}_g) y_{g,i}} {\sum_{i=1}^{n_g}(x_{g,i}-\bar{x}_g)^2}

where n_g is the sample size for group g, it follows that

\hat{\beta}_g \sim N(\beta_g, \frac{\sigma_\epsilon^2}{\sum_{i=1}^{n_g}(x_{g,i}-\bar{x}_g)^2}).

The slope difference is defined as

\hat{\beta}_1 - \hat{\beta}_2

where \hat{\beta}_1 and \hat{\beta}_2 are the estimated slopes for treatment groups 1 and 2, respectively. Since the variance of \hat{\beta}_g is

\frac{\sigma_\epsilon^2}{n_g \sigma_x^2}

we have

\hat{\beta}_1 - \hat{\beta}_2 \sim N(\beta_1 - \beta_2, (n_1^{-1} + n_2^{-1})\sigma_\epsilon^2/\sigma_x^2)

which can be used to calculate the power and sample size for the group sequential design.

Value

An S3 class designSlopeDiff 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.

  • slopeDiffH0: The slope difference under the null hypothesis.

  • slopeDiff: The slope difference under the alternative hypothesis.

  • stDev: The standard deviation of the residual.

  • stDevCovariate: The standard deviation of the covariate.

• 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.

  • efficacySlopeDiff: The efficacy boundaries on the slope difference scale.

  • futilitySlopeDiff: The futility boundaries on the slope difference 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.

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

  • 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.

  • rounding: Whether to round up sample size.

Author(s)

Kaifeng Lu, [e-mail]

Examples

(design1 <- getDesignSlopeDiff(
  beta = 0.1, n = NA, slopeDiff = -0.5,
  stDev = 10, stDevCovariate = 6,
  normalApproximation = FALSE, alpha = 0.025))

lrstat documentation built on Aug. 8, 2025, 7:36 p.m.