lrschoenfeld: Schoenfeld Method for Log-Rank Test Sample Size Calculation

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

Schoenfeld Method for Log-Rank Test Sample Size Calculation

Description

Obtains the sample size and study duration by calibrating the number of events calculated using the Schoenfeld formula under the proportional hazards assumption.

Usage

lrschoenfeld(
  beta = 0.2,
  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_,
  hazardRatioH0 = 1,
  allocationRatioPlanned = 1,
  accrualTime = 0L,
  accrualIntensity = NA_real_,
  piecewiseSurvivalTime = 0L,
  stratumFraction = 1L,
  hazardRatio = NA_real_,
  lambda2 = NA_real_,
  gamma1 = 0L,
  gamma2 = 0L,
  followupTime = NA_real_,
  fixedFollowup = 0L,
  interval = as.numeric(c(0.001, 240)),
  spendingTime = NA_real_,
  rounding = 1L,
  calibrate = 1L,
  maxNumberOfIterations = 10000L,
  maxNumberOfRawDatasetsPerStage = 0L,
  seed = NA_integer_
)

Arguments

beta	Type II error. Defaults to 0.2.
kMax	The maximum number of stages.
informationRates	The information rates in terms of number of events for the conventional log-rank test and in terms of the actual information for weighted log-rank tests. 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.
hazardRatioH0	Hazard ratio under the null hypothesis for the active treatment versus control. Defaults to 1 for superiority test.
allocationRatioPlanned	Allocation ratio for the active treatment versus control. Defaults to 1 for equal randomization.
accrualTime	A vector that specifies the starting time of piecewise Poisson enrollment time intervals. Must start with 0, e.g., c(0, 3) breaks the time axis into 2 accrual intervals: [0, 3) and [3, Inf).
accrualIntensity	A vector of accrual intensities. One for each accrual time interval.
piecewiseSurvivalTime	A vector that specifies the starting time of piecewise exponential survival time intervals. Must start with 0, e.g., c(0, 6) breaks the time axis into 2 event intervals: [0, 6) and [6, Inf). Defaults to 0 for exponential distribution.
stratumFraction	A vector of stratum fractions that sum to 1. Defaults to 1 for no stratification.
hazardRatio	Hazard ratio under the alternative hypothesis for the active treatment versus control.
lambda2	A vector of hazard rates for the event in each analysis time interval by stratum for the control group.
gamma1	The hazard rate for exponential dropout, a vector of hazard rates for piecewise exponential dropout applicable for all strata, or a vector of hazard rates for dropout in each analysis time interval by stratum for the active treatment group.
gamma2	The hazard rate for exponential dropout, a vector of hazard rates for piecewise exponential dropout applicable for all strata, or a vector of hazard rates for dropout in each analysis time interval by stratum for the control group.
followupTime	Follow-up time for the last enrolled subject.
fixedFollowup	Whether a fixed follow-up design is used. Defaults to 0 for variable follow-up.
interval	The interval to search for the solution of followupTime. Defaults to c(0.001, 240).
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.
rounding	Whether to round up sample size and events. Defaults to 1 for sample size rounding.
calibrate	Whether to use simulations to calibrate the number of events calculated using the Schoenfeld formula.
maxNumberOfIterations	The number of simulation iterations. Defaults to 10000.
maxNumberOfRawDatasetsPerStage	The number of raw datasets per stage to extract.
seed	The seed to reproduce the simulation results. The seed from the environment will be used if left unspecified.

Details

This function calculates the sample size and study duration by calibrating the number of events estimated using the Schoenfeld formula under the proportional hazards assumption, particularly when the hazard ratio is far away from one and/or the allocation between groups is unequal.

For a fixed design, the Schoenfeld formula for the required number of events is

D = \frac{(\Phi^{-1}(1-\alpha) + \Phi^{-1}(1-\beta))^2} {(\theta - \theta_0)^2 r(1-r)}

where D is the total number of events required, \alpha is the type I error rate, \beta is the type II error rate, r is the randomization probability for the active treatment group, \theta_0 and \theta are the log hazard ratios under the null and alternative hypotheses, respectively.

The function first computes the number of events using the Schoenfeld formula. If calibrate is set to 1, the function uses simulations to calibrate the number of events, accounting for scenarios where the Schoenfeld formula may be inaccurate (e.g., when allocation is unequal or the hazard ratio is extreme).

Let D_{schoenfeld} be the number of events calculated by the Schoenfeld formula, and D_{calibrated} be the calibrated number of events. The calibrated number of events is calculated as #'

D_{\text{calibrated}} = \frac{\left\{\Phi^{-1}(1-\alpha) + \Phi^{-1}(1-\beta)\right\}^2} {\left\{\Phi^{-1}(1-\alpha) + \Phi^{-1}(1-\beta_{\text{schoenfeld}})\right\}^2} D_{\text{schoenfeld}}

where \beta_{schoenfeld} is the empirical type II error estimated via simulation.

A second round of simulation is performed to obtain the empirical power using the calibrated number of events.

Value

A list of two components:

• analyticalResults: An S3 class lrpower object for the asymptotic power.

• simulationResults: An S3 class lrsim object for the empirical power.

Author(s)

Kaifeng Lu, [e-mail]

Examples

(lr1 <- lrschoenfeld(
  beta = 0.1, kMax = 2, alpha = 0.025,
  hazardRatioH0 = 1, allocationRatioPlanned = 1,
  accrualIntensity = 20, hazardRatio = 0.3,
  lambda2 = 1.9/12,
  gamma1 = -log(1-0.1)/24, gamma2 = -log(1-0.1)/24,
  fixedFollowup = 0, rounding = 1,
  calibrate = 0, maxNumberOfIterations = 1000,
  seed = 12345))

(lr2 <- lrschoenfeld(
  beta = 0.1, kMax = 2, alpha = 0.025,
  hazardRatioH0 = 1, allocationRatioPlanned = 1,
  accrualIntensity = 20, hazardRatio = 0.3,
  lambda2 = 1.9/12,
  gamma1 = -log(1-0.1)/24, gamma2 = -log(1-0.1)/24,
  fixedFollowup = 0, rounding = 1,
  calibrate = 1, maxNumberOfIterations = 1000,
  seed = 12345))

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