getSimulationSurvival: Get Simulation Survival

Description

Returns the analysis times, power, stopping probabilities, conditional power, and expected sample size for testing the hazard ratio in a two treatment groups survival design.

Usage

getSimulationSurvival(
  design = NULL,
  ...,
  thetaH0 = 1,
  directionUpper = TRUE,
  pi1 = NA_real_,
  pi2 = NA_real_,
  lambda1 = NA_real_,
  lambda2 = NA_real_,
  median1 = NA_real_,
  median2 = NA_real_,
  hazardRatio = NA_real_,
  kappa = 1,
  piecewiseSurvivalTime = NA_real_,
  allocation1 = 1,
  allocation2 = 1,
  eventTime = 12L,
  accrualTime = c(0L, 12L),
  accrualIntensity = 0.1,
  accrualIntensityType = c("auto", "absolute", "relative"),
  dropoutRate1 = 0,
  dropoutRate2 = 0,
  dropoutTime = 12L,
  maxNumberOfSubjects = NA_real_,
  plannedEvents = NA_real_,
  minNumberOfEventsPerStage = NA_real_,
  maxNumberOfEventsPerStage = NA_real_,
  conditionalPower = NA_real_,
  thetaH1 = NA_real_,
  maxNumberOfIterations = 1000L,
  maxNumberOfRawDatasetsPerStage = 0,
  longTimeSimulationAllowed = FALSE,
  seed = NA_real_,
  showStatistics = FALSE
)

Arguments

design

The trial design. If no trial design is specified, a fixed sample size design is used. In this case, Type I error rate alpha, Type II error rate beta, twoSidedPower, and sided can be directly entered as argument where necessary.

...

Ensures that all arguments (starting from the "...") are to be named and that a warning will be displayed if unknown arguments are passed.

thetaH0

The null hypothesis value, default is 0 for the normal and the binary case (testing means and rates, respectively), it is 1 for the survival case (testing the hazard ratio). For non-inferiority designs, thetaH0 is the non-inferiority bound. That is, in case of (one-sided) testing of

means: a value != 0 (or a value != 1 for testing the mean ratio) can be specified.
rates: a value != 0 (or a value != 1 for testing the risk ratio pi1 / pi2) can be specified.
survival data: a bound for testing H0: hazard ratio = thetaH0 != 1 can be specified.

For testing a rate in one sample, a value thetaH0 in (0, 1) has to be specified for defining the null hypothesis H0: pi = thetaH0.

directionUpper

Specifies the direction of the alternative, only applicable for one-sided testing; default is TRUE which means that larger values of the test statistics yield smaller p-values.

pi1

A numeric value or vector that represents the assumed event rate in the treatment group, default is seq(0.2, 0.5, 0.1) (power calculations and simulations) or seq(0.4, 0.6, 0.1) (sample size calculations).

pi2

A numeric value that represents the assumed event rate in the control group, default is 0.2.

lambda1

The assumed hazard rate in the treatment group, there is no default. lambda1 can also be used to define piecewise exponentially distributed survival times (see details).

lambda2

The assumed hazard rate in the reference group, there is no default. lambda2 can also be used to define piecewise exponentially distributed survival times (see details).

median1

The assumed median survival time in the treatment group, there is no default.

median2

The assumed median survival time in the reference group, there is no default.

hazardRatio

The vector of hazard ratios under consideration. If the event or hazard rates in both treatment groups are defined, the hazard ratio needs not to be specified as it is calculated, there is no default.

kappa

A numeric value > 0. A kappa != 1 will be used for the specification of the shape of the Weibull distribution. Default is 1, i.e., the exponential survival distribution is used instead of the Weibull distribution. Note that the Weibull distribution cannot be used for the piecewise definition of the survival time distribution, i.e., only piecewiselambda (as a single value) and kappa can be specified. This function is equivalent to pweibull(t, shape = kappa, scale = 1 / lambda) of the stats package, i.e., the scale parameter is 1 / 'hazard rate'. For example, getPiecewiseExponentialDistribution(time = 130, piecewiseLambda = 0.01, kappa = 4.2) and pweibull(q = 130, shape = 4.2, scale = 1 / 0.01) provide the sample result.

piecewiseSurvivalTime

A vector that specifies the time intervals for the piecewise definition of the exponential survival time cumulative distribution function (for details see getPiecewiseSurvivalTime).

allocation1

The number how many subjects are assigned to treatment 1 in a subsequent order, default is 1

allocation2

The number how many subjects are assigned to treatment 2 in a subsequent order, default is 1

eventTime

The assumed time under which the event rates are calculated, default is 12.

accrualTime

The assumed accrual time intervals for the study, default is c(0, 12) (for details see getAccrualTime).

accrualIntensity

A vector of accrual intensities, default is the relative intensity 0.1 (for details see getAccrualTime).

accrualIntensityType

A character value specifying the accrual intensity input type. Must be one of "auto", "absolute", or "relative"; default is "auto", i.e., if all values are < 1 the type is "relative", otherwise it is "absolute".

dropoutRate1

The assumed drop-out rate in the treatment group, default is 0.

dropoutRate2

The assumed drop-out rate in the control group, default is 0.

dropoutTime

The assumed time for drop-out rates in the control and the treatment group, default is 12.

maxNumberOfSubjects

maxNumberOfSubjects > 0 needs to be specified. If accrual time and accrual intensity is specified, this will be calculated.

plannedEvents

plannedEvents is a vector of length kMax (the number of stages of the design) that determines the number of cumulated (overall) events in survival designs when the interim stages are planned. For two treatment arms, it is the number of events for both treatment arms. For multi-arm designs, plannedEvents refers to the overall number of events for the selected arms plus control.

minNumberOfEventsPerStage

When performing a data driven sample size recalculation, the vector minNumberOfEventsPerStage with length kMax determines the minimum number of events per stage (i.e., not cumulated), the first element is not taken into account.

maxNumberOfEventsPerStage

When performing a data driven sample size recalculation, the vector maxNumberOfEventsPerStage with length kMax determines the maximum number of events per stage (i.e., not cumulated), the first element is not taken into account.

conditionalPower

If conditionalPower together with minNumberOfSubjectsPerStage and maxNumberOfSubjectsPerStage (or minNumberOfEventsPerStage and maxNumberOfEventsPerStage for survival designs) is specified, a sample size recalculation based on the specified conditional power is performed. It is defined as the power for the subsequent stage given the current data. By default, the conditional power will be calculated under the observed effect size. Optionally, you can also specify thetaH1 and stDevH1 (for simulating means), pi1H1 and pi2H1 (for simulating rates), or thetaH1 (for simulating hazard ratios) as parameters under which it is calculated and the sample size recalculation is performed.

thetaH1

If specified, the value of the alternative under which the conditional power or sample size recalculation calculation is performed.

maxNumberOfIterations

The number of simulation iterations, default is 1000.

maxNumberOfRawDatasetsPerStage

The number of raw datasets per stage that shall be extracted and saved as data.frame, default is 0. getRawData can be used to get the extracted raw data from the object.

longTimeSimulationAllowed

Logical that indicates whether long time simulations that consumes more than 30 seconds are allowed or not, default is FALSE.

seed

The seed to reproduce the simulation, default is a random seed.

showStatistics

If TRUE, summary statistics of the simulated data are displayed for the print command, otherwise the output is suppressed, default is FALSE.

Value

Returns a SimulationResults object. The following generics (R generic functions) are available for this object:

names to obtain the field names,
print to print the object,
summary to display a summary of the object,
plot to plot the object,
as.data.frame to coerce the object to a data.frame,
as.matrix to coerce the object to a matrix.

Piecewise survival time

The first element of the vector piecewiseSurvivalTime must be equal to 0. piecewiseSurvivalTime can also be a list that combines the definition of the time intervals and hazard rates in the reference group. The definition of the survival time in the treatment group is obtained by the specification of the hazard ratio (see examples for details).

Staggered patient entry

accrualTime is the time period of subjects' accrual in a study. It can be a value that defines the end of accrual or a vector. In this case, accrualTime can be used to define a non-constant accrual over time. For this, accrualTime is a vector that defines the accrual intervals. The first element of accrualTime must be equal to 0 and, additionally, accrualIntensity needs to be specified. accrualIntensity itself is a value or a vector (depending on the length of accrualtime) that defines the intensity how subjects enter the trial in the intervals defined through accrualTime.

accrualTime can also be a list that combines the definition of the accrual time and accrual intensity (see below and examples for details).

If the length of accrualTime and the length of accrualIntensity are the same (i.e., the end of accrual is undefined), maxNumberOfSubjects > 0 needs to be specified and the end of accrual is calculated. In that case, accrualIntensity is the number of subjects per time unit, i.e., the absolute accrual intensity.

If the length of accrualTime equals the length of accrualIntensity - 1 (i.e., the end of accrual is defined), maxNumberOfSubjects is calculated if the absolute accrual intensity is given. If all elements in accrualIntensity are smaller than 1, accrualIntensity defines the *relative* intensity how subjects enter the trial. For example, accrualIntensity = c(0.1, 0.2) specifies that in the second accrual interval the intensity is doubled as compared to the first accrual interval. The actual (absolute) accrual intensity is calculated for the calculated or given maxNumberOfSubjects. Note that the default is accrualIntensity = 0.1 meaning that the *absolute* accrual intensity will be calculated.

Simulation Data

The summary statistics "Simulated data" contains the following parameters: median [range]; mean +/-sd

$show(showStatistics = FALSE) or $setShowStatistics(FALSE) can be used to disable the output of the aggregated simulated data.

Example 1: simulationResults <- getSimulationSurvival(maxNumberOfSubjects = 100, plannedEvents = 30) simulationResults$show(showStatistics = FALSE)

Example 2: simulationResults <- getSimulationSurvival(maxNumberOfSubjects = 100, plannedEvents = 30) simulationResults$setShowStatistics(FALSE) simulationResults

getData can be used to get the aggregated simulated data from the object as data.frame. The data frame contains the following columns:

iterationNumber: The number of the simulation iteration.
stageNumber: The stage.
pi1: The assumed or derived event rate in the treatment group.
pi2: The assumed or derived event rate in the control group.
hazardRatio: The hazard ratio under consideration (if available).
analysisTime: The analysis time.
numberOfSubjects: The number of subjects under consideration when the (interim) analysis takes place.
eventsPerStage1: The observed number of events per stage in treatment group 1.
eventsPerStage2: The observed number of events per stage in treatment group 2.
eventsPerStage: The observed number of events per stage in both treatment groups.
rejectPerStage: 1 if null hypothesis can be rejected, 0 otherwise.
futilityPerStage: 1 if study should be stopped for futility, 0 otherwise.
eventsNotAchieved: 1 if number of events could not be reached with observed number of subjects, 0 otherwise.
testStatistic: The test statistic that is used for the test decision, depends on which design was chosen (group sequential, inverse normal, or Fisher combination test)'
logRankStatistic: Z-score statistic which corresponds to a one-sided log-rank test at considered stage.
hazardRatioEstimateLR: The estimated hazard ratio, derived from the log-rank statistic.
trialStop: TRUE if study should be stopped for efficacy or futility or final stage, FALSE otherwise.
conditionalPowerAchieved: The conditional power for the subsequent stage of the trial for selected sample size and effect. The effect is either estimated from the data or can be user defined with thetaH1.

Raw Data

getRawData can be used to get the simulated raw data from the object as data.frame. Note that getSimulationSurvival must called before with maxNumberOfRawDatasetsPerStage > 0. The data frame contains the following columns:

iterationNumber: The number of the simulation iteration.
stopStage: The stage of stopping.
subjectId: The subject id (increasing number 1, 2, 3, ...)
accrualTime: The accrual time, i.e., the time when the subject entered the trial.
treatmentGroup: The treatment group number (1 or 2).
survivalTime: The survival time of the subject.
dropoutTime: The dropout time of the subject (may be NA).
observationTime: The specific observation time.
timeUnderObservation: The time under observation is defined as follows: if (event == TRUE) timeUnderObservation <- survivalTime; else if (dropoutEvent == TRUE) timeUnderObservation <- dropoutTime; else timeUnderObservation <- observationTime - accrualTime;
event: TRUE if an event occurred; FALSE otherwise.
dropoutEvent: TRUE if an dropout event occurred; FALSE otherwise.

How to get help for generic functions

Click on the link of a generic in the list above to go directly to the help documentation of the rpact specific implementation of the generic. Note that you can use the R function methods to get all the methods of a generic and to identify the object specific name of it, e.g., use methods("plot") to get all the methods for the plot generic. There you can find, e.g., plot.AnalysisResults and obtain the specific help documentation linked above by typing ?plot.AnalysisResults.

Details

At given design the function simulates the power, stopping probabilities, conditional power, and expected sample size at given number of events, number of subjects, and parameter configuration. It also simulates the time when the required events are expected under the given assumptions (exponentially, piecewise exponentially, or Weibull distributed survival times and constant or non-constant piecewise accrual). Additionally, integers allocation1 and allocation2 can be specified that determine the number allocated to treatment group 1 and treatment group 2, respectively.

conditionalPower The definition of thetaH1 makes only sense if kMax > 1 and if conditionalPower, minNumberOfEventsPerStage, and maxNumberOfEventsPerStage are defined.

Note that numberOfSubjects, numberOfSubjects1, and numberOfSubjects2 in the output are expected number of subjects.

Examples

Run this code

# NOT RUN {
# Fixed sample size with minimum required definitions, pi1 = (0.3,0.4,0.5,0.6) and 
# pi2 = 0.3 at event time 12, and accrual time 24 
getSimulationSurvival(pi1 = seq(0.3,0.6,0.1), pi2 = 0.3, eventTime = 12, 
    accrualTime = 24, plannedEvents = 40, maxNumberOfSubjects = 200, 
    maxNumberOfIterations = 10)
# }
# NOT RUN {
# Increase number of simulation iterations 
getSimulationSurvival(pi1 = seq(0.3,0.6,0.1), pi2 = 0.3, eventTime = 12, 
    accrualTime = 24, plannedEvents = 40, maxNumberOfSubjects = 200, 
    maxNumberOfIterations = 50)

# Determine necessary accrual time with default settings if 200 subjects and 
# 30 subjects per time unit can be recruited 
getSimulationSurvival(plannedEvents = 40, accrualTime = 0, 
    accrualIntensity = 30, maxNumberOfSubjects = 200, maxNumberOfIterations = 50)

# Determine necessary accrual time with default settings if 200 subjects and 
# if the first 6 time units 20 subjects per time unit can be recruited, 
# then 30 subjects per time unit 
getSimulationSurvival(plannedEvents = 40, accrualTime = c(0, 6), 
    accrualIntensity = c(20, 30), maxNumberOfSubjects = 200, 
    maxNumberOfIterations = 50)

# Determine maximum number of Subjects with default settings if the first 
# 6 time units 20 subjects per time unit can be recruited, and after 
# 10 time units 30 subjects per time unit
getSimulationSurvival(plannedEvents = 40, accrualTime = c(0, 6, 10), 
    accrualIntensity = c(20, 30), maxNumberOfIterations = 50)

# Specify accrual time as a list
at <- list(
    "0 - <6"  = 20,
    "6 - Inf" = 30)
getSimulationSurvival(plannedEvents = 40, accrualTime = at, 
    maxNumberOfSubjects = 200, maxNumberOfIterations = 50)

# Specify accrual time as a list, if maximum number of subjects need to be calculated
at <- list(
    "0 - <6"   = 20,
    "6 - <=10" = 30)
getSimulationSurvival(plannedEvents = 40, accrualTime = at, maxNumberOfIterations = 50)

# Specify effect size for a two-stage group sequential design with 
# O'Brien & Fleming boundaries. Effect size is based on event rates  
# at specified event time, directionUpper = FALSE needs to be specified
# because it should be shown that hazard ratio < 1
getSimulationSurvival(design = getDesignGroupSequential(kMax = 2), 
    pi1 = 0.2, pi2 = 0.3, eventTime = 24, plannedEvents = c(20, 40), 
    maxNumberOfSubjects = 200, directionUpper = FALSE, maxNumberOfIterations = 50)

# As above, but with a three-stage O'Brien and Fleming design with 
# specified information rates, note that planned events consists of integer values
d3 <- getDesignGroupSequential(informationRates = c(0.4, 0.7, 1))
getSimulationSurvival(design = d3, pi1 = 0.2, pi2 = 0.3, eventTime = 24, 
    plannedEvents = round(d3$informationRates * 40), 
    maxNumberOfSubjects = 200, directionUpper = FALSE, 
    maxNumberOfIterations = 50)

# Effect size is based on event rate at specified event time for the reference 
# group and hazard ratio, directionUpper = FALSE needs to be specified because 
# it should be shown that hazard ratio < 1
getSimulationSurvival(design = getDesignGroupSequential(kMax = 2), hazardRatio = 0.5, 
    pi2 = 0.3, eventTime = 24, plannedEvents = c(20, 40), maxNumberOfSubjects = 200, 
    directionUpper = FALSE, maxNumberOfIterations = 50)

# Effect size is based on hazard rate for the reference group and 
# hazard ratio, directionUpper = FALSE needs to be specified because 
# it should be shown that hazard ratio < 1
getSimulationSurvival(design = getDesignGroupSequential(kMax = 2), 
    hazardRatio = 0.5, lambda2 = 0.02, plannedEvents = c(20, 40), 
    maxNumberOfSubjects = 200, directionUpper = FALSE, 
    maxNumberOfIterations = 50) 

# Specification of piecewise exponential survival time and hazard ratios, 
# note that in getSimulationSurvival only on hazard ratio is used
# in the case that the survival time is piecewise expoential
getSimulationSurvival(design = getDesignGroupSequential(kMax = 2), 
    piecewiseSurvivalTime = c(0, 5, 10), lambda2 = c(0.01, 0.02, 0.04), 
    hazardRatio = 1.5, plannedEvents = c(20, 40), maxNumberOfSubjects = 200, 
    maxNumberOfIterations = 50)

pws <- list(
    "0 - <5"  = 0.01,
    "5 - <10" = 0.02,
    ">=10"    = 0.04)
getSimulationSurvival(design = getDesignGroupSequential(kMax = 2), 
    piecewiseSurvivalTime = pws, hazardRatio = c(1.5), 
    plannedEvents = c(20, 40), maxNumberOfSubjects = 200, 
    maxNumberOfIterations = 50)

# Specification of piecewise exponential survival time for both treatment arms  
getSimulationSurvival(design = getDesignGroupSequential(kMax = 2), 
    piecewiseSurvivalTime = c(0, 5, 10), lambda2 = c(0.01, 0.02, 0.04), 
    lambda1 = c(0.015, 0.03, 0.06), plannedEvents = c(20, 40), 
    maxNumberOfSubjects = 200, maxNumberOfIterations = 50)

# Specification of piecewise exponential survival time as a list, 
# note that in getSimulationSurvival only on hazard ratio 
# (not a vector) can be used
pws <- list(
    "0 - <5"  = 0.01,
    "5 - <10" = 0.02,
    ">=10"    = 0.04)
getSimulationSurvival(design = getDesignGroupSequential(kMax = 2), 
    piecewiseSurvivalTime = pws, hazardRatio = 1.5, 
    plannedEvents = c(20, 40), maxNumberOfSubjects = 200, 
    maxNumberOfIterations = 50)

# Specification of piecewise exponential survival time and delayed effect 
# (response after 5 time units)  
getSimulationSurvival(design = getDesignGroupSequential(kMax = 2), 
    piecewiseSurvivalTime = c(0, 5, 10), lambda2 = c(0.01, 0.02, 0.04), 
    lambda1 = c(0.01, 0.02, 0.06), plannedEvents = c(20, 40), 
    maxNumberOfSubjects = 200, maxNumberOfIterations = 50)

# Specify effect size based on median survival times
getSimulationSurvival(median1 = 5, median2 = 3, plannedEvents = 40, 
    maxNumberOfSubjects = 200, directionUpper = FALSE, 
    maxNumberOfIterations = 50)

# Specify effect size based on median survival 
# times of Weibull distribtion with kappa = 2
getSimulationSurvival(median1 = 5, median2 = 3, kappa = 2, 
    plannedEvents = 40, maxNumberOfSubjects = 200, 
    directionUpper = FALSE, maxNumberOfIterations = 50)

# Perform recalculation of number of events based on conditional power for a 
# three-stage design with inverse normal combination test, where the conditional power 
# is calculated under the specified effect size thetaH1 = 1.3 and up to a four-fold 
# increase in originally planned sample size (number of events) is allowed
# Note that the first value in minNumberOfEventsPerStage and 
# maxNumberOfEventsPerStage is arbitrary, i.e., it has no effect.
dIN <- getDesignInverseNormal(informationRates = c(0.4, 0.7, 1))

resultsWithSSR1 <- getSimulationSurvival(design = dIN, 
    hazardRatio = seq(1, 1.6, 0.1), 
    pi2 = 0.3, conditionalPower = 0.8, thetaH1 = 1.3, 
    plannedEvents = c(58, 102, 146), 
    minNumberOfEventsPerStage = c(NA, 44, 44), 
    maxNumberOfEventsPerStage = 4 * c(NA, 44, 44),
    maxNumberOfSubjects = 800, maxNumberOfIterations = 50)
resultsWithSSR1

# If thetaH1 is unspecified, the observed hazard ratio estimate 
# (calculated from the log-rank statistic) is used for performing the 
# recalculation of the number of events
resultsWithSSR2 <- getSimulationSurvival(design = dIN, 
    hazardRatio = seq(1, 1.6, 0.1), 
    pi2 = 0.3, conditionalPower = 0.8, plannedEvents = c(58, 102, 146), 
    minNumberOfEventsPerStage = c(NA, 44, 44), 
    maxNumberOfEventsPerStage = 4 * c(NA, 44, 44),
    maxNumberOfSubjects = 800, maxNumberOfIterations = 50)
resultsWithSSR2

# Compare it with design without event size recalculation
resultsWithoutSSR <- getSimulationSurvival(design = dIN, 
    hazardRatio = seq(1, 1.6, 0.1), pi2 = 0.3, 
    plannedEvents = c(58, 102, 145), maxNumberOfSubjects = 800, 
    maxNumberOfIterations = 50)
resultsWithoutSSR$overallReject
resultsWithSSR1$overallReject
resultsWithSSR2$overallReject

# Confirm that event size racalcuation increases the Type I error rate, 
# i.e., you have to use the combination test 
dGS <- getDesignGroupSequential(informationRates = c(0.4, 0.7, 1))
resultsWithSSRGS <- getSimulationSurvival(design = dGS, hazardRatio = seq(1), 
    pi2 = 0.3, conditionalPower = 0.8, plannedEvents = c(58, 102, 145), 
    minNumberOfEventsPerStage = c(NA, 44, 44), 
    maxNumberOfEventsPerStage = 4 * c(NA, 44, 44),
    maxNumberOfSubjects = 800, maxNumberOfIterations = 50)
resultsWithSSRGS$overallReject

# Set seed to get reproduceable results
identical(
    getSimulationSurvival(plannedEvents = 40, maxNumberOfSubjects = 200, 
        seed = 99)$analysisTime,
    getSimulationSurvival(plannedEvents = 40, maxNumberOfSubjects = 200, 
        seed = 99)$analysisTime
)
# }
# NOT RUN {
# }

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