R/getDesignSurvivals.R
lrschoenfeld.RdObtains the sample size and study duration by calibrating the number of events calculated using the Schoenfeld formula under the proportional hazards assumption. The initial event target can be used directly or calibrated with simulation.
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_,
futilityCP = NA_real_,
futilityHR = 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 = FALSE,
spendingTime = NA_real_,
rounding = TRUE,
simulate = FALSE,
calibrate = FALSE,
maxNumberOfIterations = 10000L,
maxNumberOfRawDatasetsPerStage = 0L,
seed = NA_integer_
)Type II error (one minus the target power). Defaults to 0.2.
The maximum number of stages.
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.
Indicators of whether efficacy stopping is allowed
at each stage. Defaults to TRUE if left unspecified.
Indicators of whether futility stopping is allowed
at each stage. Defaults to TRUE if left unspecified.
Upper boundaries on the z-test statistic scale for stopping for efficacy.
The significance level. Defaults to 0.025.
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".
The parameter value for the alpha spending.
Corresponds to \(\Delta\) for "WT", \(\rho\) for "sfKD",
and \(\gamma\) for "sfHSD".
The user defined alpha spending. Cumulative alpha spent up to each stage.
Lower boundaries on the z-test statistic scale
for stopping for futility at stages 1, ..., kMax-1. Defaults to
rep(-8, kMax-1) if left unspecified. The futility bounds are
non-binding for the calculation of critical values.
A vector of length kMax - 1 for the futility
bounds on the conditional power scale.
A vector of length kMax - 1 for the futility
bounds on the hazard ratio scale.
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".
The parameter value for the beta spending.
Corresponds to \(\rho\) for "sfKD", and
\(\gamma\) for "sfHSD".
The user defined beta spending. Cumulative beta spent up to each stage.
Hazard ratio under the null hypothesis for the active treatment versus control. Defaults to 1 for superiority test.
Allocation ratio for the active treatment versus control. Defaults to 1 for equal randomization.
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 \([3, \infty)\).
A vector of accrual intensities. One for each accrual time interval.
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 \([6, \infty)\).
Defaults to 0 for exponential distribution.
A vector of stratum fractions that sum to 1. Defaults to 1 for no stratification.
Hazard ratio under the alternative hypothesis for the active treatment versus control.
A vector of hazard rates for the event in each analysis time interval by stratum for the control group.
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.
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.
Follow-up time for the last enrolled subject.
Whether a fixed follow-up design is used.
Defaults to FALSE for variable follow-up.
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.
Whether to round up sample size and events. Defaults to 1 for sample size rounding.
Whether to use simulation to estimate empirical power for the event target.
Whether to use simulation to calibrate the number of
events calculated using the Schoenfeld formula. If TRUE,
simulate is also treated as TRUE.
The number of simulation iterations. Defaults to 10000.
The number of raw datasets per stage to extract.
The seed to reproduce the simulation results.
A list of two components when simulate = TRUE:
analyticalResults: An S3 class lrpower object for
the asymptotic power.
simulationResults: An S3 class lrsim object for
the empirical power.
A list of one component for analyticalResults when
simulate = FALSE.
This function first calculates the required number of events using the group-sequential design and the Schoenfeld approximation. It then obtains the corresponding sample size and study duration from the accrual, event, and dropout assumptions. The optional simulation-based calibration is useful when the hazard ratio is far from one or the allocation is unequal, because the Schoenfeld approximation can then be inaccurate.
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.
Let \(D_{schoenfeld}\) be the initial number of events calculated
by the Schoenfeld formula, and \(D_{calibrated}\) be the calibrated
number of events. A simulation estimates the empirical power
\(p_{schoenfeld}\) at \(D_{schoenfeld}\). The calibrated number
of events is then calculated as
$$D_{\text{calibrated}} =
\frac{\left\{\Phi^{-1}(1-\alpha) + \Phi^{-1}(1-\beta)\right\}^2}
{\left\{\Phi^{-1}(1-\alpha) + \Phi^{-1}(p_{\text{schoenfeld}})\right\}^2}
D_{\text{schoenfeld}}$$
Here \(p_{schoenfeld}\) is the empirical rejection probability
estimated by the first simulation. If rounding = TRUE, the
calibrated total and cumulative stage event targets are rounded up or
to the nearest integer as appropriate.
A half-count correction is applied to the simulated rejection count if
needed so that empirical power values of exactly 0 or 1 do not produce
infinite normal quantiles.
A second simulation is performed to obtain empirical power using the
calibrated number of events. With simulate = FALSE, only the
analytical result is returned. With simulate = TRUE and
calibrate = FALSE, the simulation uses the initial target.
With calibrate = TRUE, the simulation uses the calibrated target.
When simulate = FALSE, the function returns the number of events
calculated using the Schoenfeld formula and the corresponding sample size
and study duration without simulation results for the empirical power.
When simulate = TRUE, the function returns the event target used
for the simulation and the corresponding sample size and study duration
along with simulation results for empirical power. The target is the
Schoenfeld target when calibrate = FALSE and the calibrated target
when calibrate = TRUE.
(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 = FALSE, rounding = TRUE,
simulate = TRUE, calibrate = FALSE,
maxNumberOfIterations = 1000,
seed = 12345))
#> $analyticalResults
#>
#> Group-sequential design with 2 stages for log-rank test
#> Overall power: 0.9085, overall significance level (1-sided): 0.025
#> Maximum # events: 30, expected # events: 26
#> Maximum # dropouts: 1.4, expected # dropouts: 1.2
#> Maximum # subjects: 78, expected # subjects: 78
#> Maximum information: 7.5, expected information: 6.51
#> Total study duration: 7.2, expected study duration: 6.4
#> Accrual duration: 3.9, follow-up duration: 3.3, fixed follow-up: FALSE
#> Allocation ratio: 1
#> Alpha spending: Lan-DeMets O'Brien-Fleming, beta spending: None
#>
#> Stage 1 Stage 2
#> Information rate 0.500 1.000
#> Efficacy boundary (Z) 2.963 1.969
#> Cumulative rejection 0.2640 0.9085
#> Cumulative alpha spent 0.0015 0.0250
#> Number of events 15.0 30.0
#> Number of dropouts 0.7 1.4
#> Number of subjects 78.0 78.0
#> Analysis time 4.2 7.2
#> Efficacy boundary (HR) 0.217 0.487
#> Efficacy boundary (p) 0.0015 0.0245
#> Information 3.75 7.50
#> HR 0.300 0.300
#>
#> $simulationResults
#>
#> Group-sequential design with 2 stages for log-rank test
#> Empirical power: 0.878
#> Expected # events: 26.8
#> Expected # dropouts: 1.2
#> Expected # subjects: 77.5
#> Expected study duration: 6.5
#> n: 78, fixed follow-up: FALSE
#> Number of simulations: 1000
#>
#> Stage 1 Stage 2
#> Cumulative rejection 0.2140 0.8780
#> Cumulative futility 0.0000 0.1220
#> Number of events 15.0 30.0
#> Number of dropouts 0.7 1.4
#> Number of subjects 75.6 78.0
#> Analysis time 4.2 7.1
#>
(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 = FALSE, rounding = TRUE,
simulate = TRUE, calibrate = TRUE,
maxNumberOfIterations = 1000,
seed = 12345))
#> $analyticalResults
#>
#> Group-sequential design with 2 stages for log-rank test
#> Overall power: 0.9321, overall significance level (1-sided): 0.025
#> Maximum # events: 33, expected # events: 27.7
#> Maximum # dropouts: 1.5, expected # dropouts: 1.3
#> Maximum # subjects: 78, expected # subjects: 78
#> Maximum information: 8.25, expected information: 6.92
#> Total study duration: 8, expected study duration: 6.8
#> Accrual duration: 3.9, follow-up duration: 4.1, fixed follow-up: FALSE
#> Allocation ratio: 1
#> Alpha spending: Lan-DeMets O'Brien-Fleming, beta spending: None
#>
#> Stage 1 Stage 2
#> Information rate 0.515 1.000
#> Efficacy boundary (Z) 2.913 1.970
#> Cumulative rejection 0.3333 0.9321
#> Cumulative alpha spent 0.0018 0.0250
#> Number of events 17.0 33.0
#> Number of dropouts 0.8 1.5
#> Number of subjects 78.0 78.0
#> Analysis time 4.5 8.0
#> Efficacy boundary (HR) 0.243 0.504
#> Efficacy boundary (p) 0.0018 0.0244
#> Information 4.25 8.25
#> HR 0.300 0.300
#>
#> $simulationResults
#>
#> Group-sequential design with 2 stages for log-rank test
#> Empirical power: 0.915
#> Expected # events: 28.3
#> Expected # dropouts: 1.3
#> Expected # subjects: 77.7
#> Expected study duration: 6.9
#> n: 78, fixed follow-up: FALSE
#> Number of simulations: 1000
#>
#> Stage 1 Stage 2
#> Cumulative rejection 0.2950 0.9150
#> Cumulative futility 0.0000 0.0850
#> Number of events 17.0 33.0
#> Number of dropouts 0.8 1.5
#> Number of subjects 77.1 78.0
#> Analysis time 4.5 7.9
#>