getPowerSurvival {rpact}

Get Power Survival

Description

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

Usage

getPowerSurvival(
  design = NULL,
  ...,
  typeOfComputation = c("Schoenfeld", "Freedman", "HsiehFreedman"),
  thetaH0 = 1,
  directionUpper = NA,
  pi1 = NA_real_,
  pi2 = NA_real_,
  lambda1 = NA_real_,
  lambda2 = NA_real_,
  median1 = NA_real_,
  median2 = NA_real_,
  kappa = 1,
  hazardRatio = NA_real_,
  piecewiseSurvivalTime = NA_real_,
  allocationRatioPlanned = 1,
  eventTime = 12,
  accrualTime = c(0, 12),
  accrualIntensity = 0.1,
  accrualIntensityType = c("auto", "absolute", "relative"),
  maxNumberOfSubjects = NA_real_,
  maxNumberOfEvents = NA_real_,
  dropoutRate1 = 0,
  dropoutRate2 = 0,
  dropoutTime = 12
)

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.

typeOfComputation

Three options are available: "Schoenfeld", "Freedman", "HsiehFreedman", the default is "Schoenfeld". For details, see Hsieh (Statistics in Medicine, 1992). For non-inferiority testing (i.e., thetaH0 != 1), only Schoenfeld's formula can be used.

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.

• count data: a bound for testing H0: lambda1 / lambda2 = 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

Logical. 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). Must be a positive numeric of length 1.

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). Must be a positive numeric of length 1.

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. Must be a positive numeric of length 1.

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.

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. Must be a positive numeric of length 1.

piecewiseSurvivalTime

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

allocationRatioPlanned

The planned allocation ratio n1 / n2 for a two treatment groups design, default is 1. For multi-arm designs, it is the allocation ratio relating the active arm(s) to the control. For simulating means and rates for a two treatment groups design, it can be a vector of length kMax, the number of stages. It can be a vector of length kMax, too, for multi-arm and enrichment designs. In these cases, a change of allocating subjects to treatment groups over the stages can be assessed. Note that internally allocationRatioPlanned is treated as a vector of length kMax, not a scalar.

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 numeric 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".

maxNumberOfSubjects

maxNumberOfSubjects > 0 needs to be specified. If accrual time and accrual intensity are specified, this will be calculated. Must be a positive integer of length 1.

maxNumberOfEvents

maxNumberOfEvents > 0 is the maximum number of events, it determines the power of the test and needs to be specified.

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.

Details

At given design the function calculates the power, stopping probabilities, and expected sample size at given number of events and number of subjects. It also calculates 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, an allocation ratio = n1 / n2 can be specified where n1 and n2 are the number of subjects in the two treatment groups.

The formula of Kim & Tsiatis (Biometrics, 1990) is used to calculate the expected number of events under the alternative (see also Lakatos & Lan, Statistics in Medicine, 1992). These formulas are generalized to piecewise survival times and non-constant piecewise accrual over time.

Value

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

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.

See Also

Other power functions: getPowerCounts(), getPowerMeans(), getPowerRates()

Examples

# Fixed sample size with minimum required definitions, pi1 = c(0.4,0.5,0.5) and 
# pi2 = 0.2 at event time 12, accrual time 12 and follow-up time 6 as default 
getPowerSurvival(maxNumberOfEvents = 40, maxNumberOfSubjects = 200)
## Not run: 
# Four stage O'Brien & Fleming group sequential design with minimum required 
# definitions, pi1 = c(0.4,0.5,0.5) and pi2 = 0.2 at event time 12, 
# accrual time 12 and follow-up time 6 as default  
getPowerSurvival(design = getDesignGroupSequential(kMax = 4), 
    maxNumberOfEvents = 40, maxNumberOfSubjects = 200)

# For fixed sample design, determine necessary accrual time if 200 subjects and 
# 30 subjects per time unit can be recruited 
getPowerSurvival(maxNumberOfEvents = 40, accrualTime = c(0), 
    accrualIntensity = 30, maxNumberOfSubjects = 200)

# Determine necessary accrual time if 200 subjects and if the first 6 time units 
# 20 subjects per time unit can be recruited, then 30 subjects per time unit 
getPowerSurvival(maxNumberOfEvents = 40, accrualTime = c(0, 6), 
    accrualIntensity = c(20, 30), maxNumberOfSubjects = 200)

# Determine maximum number of Subjects if the first 6 time units 20 subjects per 
# time unit can be recruited, and after 10 time units 30 subjects per time unit
getPowerSurvival(maxNumberOfEvents = 40, accrualTime = c(0, 6, 10), 
    accrualIntensity = c(20, 30))

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

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

# Specify effect size for a two-stage group 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
getPowerSurvival(design = getDesignGroupSequential(kMax = 2), pi1 = 0.2, pi2 = 0.3, 
    eventTime = 24, maxNumberOfEvents = 40, maxNumberOfSubjects = 200, 
    directionUpper = FALSE)

# 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
getPowerSurvival(design = getDesignGroupSequential(kMax = 2), hazardRatio = 0.5, 
    pi2 = 0.3, eventTime = 24, maxNumberOfEvents = 40, maxNumberOfSubjects = 200, 
    directionUpper = FALSE)

# 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
getPowerSurvival(design = getDesignGroupSequential(kMax = 2), hazardRatio = 0.5, 
    lambda2 = 0.02, maxNumberOfEvents = 40, maxNumberOfSubjects = 200, 
    directionUpper = FALSE) 

# Specification of piecewise exponential survival time and hazard ratios  
getPowerSurvival(design = getDesignGroupSequential(kMax = 2), 
    piecewiseSurvivalTime = c(0, 5, 10), lambda2 = c(0.01,0.02,0.04), 
    hazardRatio = c(1.5, 1.8, 2),  maxNumberOfEvents = 40, maxNumberOfSubjects = 200)

# Specification of piecewise exponential survival time as list and hazard ratios 
pws <- list(
    "0 - <5"  = 0.01,
    "5 - <10" = 0.02,
    ">=10"    = 0.04)
getPowerSurvival(design = getDesignGroupSequential(kMax = 2), 
    piecewiseSurvivalTime = pws, hazardRatio = c(1.5, 1.8, 2), 
    maxNumberOfEvents = 40, maxNumberOfSubjects = 200)

# Specification of piecewise exponential survival time for both treatment arms  
getPowerSurvival(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),  maxNumberOfEvents = 40, maxNumberOfSubjects = 200)

# Specification of piecewise exponential survival time as a list
pws <- list(
    "0 - <5"  = 0.01,
    "5 - <10" = 0.02,
    ">=10"    = 0.04)
getPowerSurvival(design = getDesignGroupSequential(kMax = 2), 
    piecewiseSurvivalTime = pws, hazardRatio = c(1.5, 1.8, 2), 
    maxNumberOfEvents = 40, maxNumberOfSubjects = 200)

# Specify effect size based on median survival times
getPowerSurvival(median1 = 5, median2 = 3, 
    maxNumberOfEvents = 40, maxNumberOfSubjects = 200, directionUpper = FALSE)

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

## End(Not run)