rpact: Confirmatory Adaptive Clinical Trial Design and Analysis

Summary

This R Markdown document provides examples for designing trials with survival endpoints using rpact.

1 Introduction

The examples in this document are not intended to replace the official rpact documentation and help pages but rather to supplement them. They also only cover a selection of all rpact features.

General convention: In rpact, arguments containing the index “2” always refer to the control group, “1” refer to the intervention group, and treatment effects compare treatment versus control.

First, load the rpact package

library(rpact)
packageVersion("rpact") # version should be version 2.0.5 or later
## [1] '3.1.1'

2 Overview of relevant rpact functions

The sample size calculation for a group-sequential trial with a survival endpoints is performed in two steps:

  1. Define the (abstract) group-sequential design using the function getDesignGroupSequential(). For details regarding this step, see the R markdown file “Defining group-sequential boundaries with rpact”. This step can be omitted if the trial has no interim analyses.

  2. Calculate sample size for the survival endpoint by feeding the abstract design into the function getSampleSizeSurvival(). This step is described in detail below.

Other relevant rpact functions for survival are:

  • getPowerSurvival(): This function is the analogue to getSampleSizeSurvival() for the calculation of power rather than the sample size.
  • getEventProbabilities(): Calculates the probability of an event depending on the time and type of accrual, follow-up time, and survival distribution. This is useful for aligning interim analyses for different time-to-event endpoints.
  • getSimulationSurvival(): This function simulates group-sequential trials. For example, it allows to assess the power of trials with delayed treatment effects or to assess the data-dependent variability of the timing of interim analyses even if the protocol assumptions are perfectly fulfilled. It also allows to simulate hypothetical datasets for trials stopped early.

This document describes all functions mentioned above except for trial simulation (getSimulationSurvival()) which is described in the document Simulation-based design of group-sequential trials with a survival endpoint with rpact.

However, before describing the functions themselves, the document describes how survival functions, drop-out, and accrual can be specified in rpact which is required for all of these functions.

3 Specifying survival distributions in rpact

rpact allows to specify survival distributions with exponential, piecewise exponential, and Weibull distributions. Exponential and piecewise exponential distributions are described below.

Weibull distributions are specified in the same way as exponential distributions except that an additional scale parameter kappa needs to be provided which is 1 for the exponential distribution. Note that the parameters shape and scale in the standard R functions for the Weibull distribution in the stats-library (such as dweibull) are equivalent to kappa and 1/lambda, respectively, in rpact.

3.1 Exponential survival distributions

3.1.1 Event probability at a specific time point known

  • The time point is given by argument eventTime.
  • The probability of an event (i.e., 1 minus survival function) in the control group is given by argument pi2.
  • The probability of an event in the intervention arm can either be provided explicitly through argument pi1 or, alternatively, implicitly by specifying the target hazard ratio hazardRatio.

Example: If the intervention is expected to improve survival at 24 months from 70% to 80%, this would be expressed through arguments eventTime = 24, pi2 = 0.3, pi1 = 0.2.

3.1.2 Exponential parameter \(\lambda\) known

  • The constant hazard function in the control arm can be provided as argument lambda2.
  • The hazard in the intervention arm can either be provided explicitly through the argument lambda1 or, alternatively, implicitly by specifying the target hazard ratio hazardRatio.

3.1.3 Median survival known

Medians cannot be specified directly. However, one can exploit that the hazard rate \(\lambda\) of the exponential distribution is equal to \(\lambda = \log(2)/\text{median}\).

Example: If the intervention is expected to improve the median survival from 60 to 75 months, this would be expressed through the arguments lambda2 = log(2)/60, lambda1 = log(2)/75. Alternatively, it could be specified via lambda2 = log(2)/60, hazardRatio = 0.8 (as the hazard ratio is 60/75 = 0.8).

3.2 Weibull

Weibull distributions are specified in the same way as exponential distributions except that an additional scale parameter kappa needs to be provided which is 1 for the exponential distribution.

3.3 Piecewise exponential survival

3.3.1 Piecewise constant hazard rate \(\lambda\) in each interval known

Example: If the survival function in the control arm is assumed to be 0.03 (events/time-unit of follow-up) for the first 6 months, 0.06 for months 6-12, and 0.02 from month 12 onwards, this can be specified using the argument piecewiseSurvivalTime as follows:

piecewiseSurvivalTime = list(
    "0 - <6" = 0.03,
    "6 - <12" = 0.06,
    ">= 12" = 0.02)

Alternatively, the same distribution can be specified by giving the start times of each interval as argument piecewiseSurvivalTime and the actual hazard rate in that interval as argument lambda2. I.e., the relevant arguments for this example would be:

piecewiseSurvivalTime = c(0,6,12), lambda2 = c(0.03,0.06,0.02)

For the intervention arm, one could either explicitly specify the hazard rate in the same time intervals through the argument lambda1 or, more conveniently, specify the survival function in the intervention arm indirectly via the target hazard ratio (argument hazardRatio).

Note: The sample size calculation functions discussed in this document assume that the proportional hazards assumption holds, i.e., that lambda1 and lambda2 are proportional if both are provided. Otherwise, the function call to getSampleSizeSurvival() gives an error. For situations with non-proportional hazards, please see the separate document which discusses the simulation tool using the function getSimulationSurvival().

3.3.2 Survival function at different time points known

Piecewise exponential distributions are useful to approximate survival functions. In principle, it is possible to approximate any distribution function well with a piecewise exponential distribution if suitably narrow intervals are chosen.

Assume that the survival function is \(S(t_1)\) at time \(t_1\) and \(S(t_2)\) at time \(t_2\) with \(t_2>t_1\) and that the hazard rate in the interval \([t_1,t_2]\) is constant. Then, the hazard rate in the interval can be derived as \((\log(S(t_1))-\log(S(t_2)))/(t_2-t_1)\).

Example: Assume that it is known that the survival function is 1, 0.9, 0.7, and 0.5 at months 0, 12, 24, 48 in the control arm. Then, an interpolating piecewise exponential distribution can be derived as follows:

t <- c(0, 12, 24, 48) # time points at which survival function is known 
                      # (must include 0)
S <- c(1, 0.9, 0.7, 0.5) # Survival function at timepoints t
# derive hazard in each intervals per the formula above
lambda <- -diff(log(S))/diff(t)
# Define parameters for piecewise exponential distribution in the control arm 
# interpolating the targeted survival values
# (code for lambda1 below assumes that the hazard afer month 48 is 
#  identical to the hazard in interval [24,48])
piecewiseSurvivalTime <- t
lambda2 <- c(lambda,tail(lambda,n = 1))
lambda2 # print hazard rates 
## [1] 0.008780043 0.020942869 0.014019677 0.014019677

Appendix: Random numbers, cumulative distribution, and quantiles for the piecewise exponential distribution

rpact also provides general functionality for the piecewise exponential distribution, see ?getPiecewiseExponentialDistribution for details. Below is some example code which shows that the derivation of the piecewise exponential distribution in the previous example is correct.

# plot the piecewise exponential survival distribution from the example above
tp <- seq(0,72,by = 0.01) 
plot(tp,
    1 - getPiecewiseExponentialDistribution(time = tp,
        piecewiseSurvivalTime = piecewiseSurvivalTime,
        piecewiseLambda = lambda2),
    type = "l", xlab = "Time (months)", ylab = "Survival function S(t)",
        lwd = 2, ylim = c(0,1),axes = F,
    main = "Piecewise exponential distribution for the example")
axis(1,at = seq(0,72,by = 12)); axis(2,at = seq(0,1,by = 0.1))
abline(v = seq(0,72,by = 12),h = seq(0,1,by = 0.1),col = gray(0.9))

# Calculate median survival for the example (which should give 48 months here)
getPiecewiseExponentialQuantile(0.5, piecewiseSurvivalTime = piecewiseSurvivalTime,
     piecewiseLambda = lambda2)
## [1] 48

4 Specifying dropout in rpact

rpact models dropout with an independent exponentially distributed variable. Dropout is specified by giving the probability of dropout at a specific time point. For example, an annual (12-monthly) dropout probability of 5% in both treatment arms can be specified through the following arguments:

dropoutRate1 = 0.05, dropoutRate2 = 0.05, dropoutTime = 12

5 Specifying accrual in rpact

rpact allows to specify arbitrarily complex recruitment scenarios. Some examples are provided below.

5.1 Absolute accrual intensity and accrual duration known

Example 1: A constant accrual intensity of 24 subjects/months over 30 months (i.e., a maximum sample size of 24*30 = 720 subjects) can be specified through the arguments below:

accrualTime = c(0, 30), accrualIntensity = 24

Example 2: A flexible accrual intensity of 20 subjects/months in the first 6 months, 25 subjects/months for months 6-12, and 30 subjects per month for months 12-24 can be specified through either of the two equivalent options below:

Option 1: List-based definition:

accrualTime = list(
    "0 - <6" = 20,
    "6 - <12" = 25,
    "12- <24" = 30)

Option 2: Vector-based definition (note that the length of the accrualTime vector is 1 larger than the length of the accrualIntensity as the end time of the accrual period is also included):

accrualTime = c(0, 6, 12, 24), accrualIntensity = c(20, 25, 30)

5.2 Absolute accrual intensity known, accrual duration unknown

Example 1: A constant accrual intensity of 24 subjects/months with unspecified end of recruitment is specified through the arguments below:

accrualTime = 0, accrualIntensity = 24 
# Note: accrualTime is the start of accrual which must be explicitly set to 0

Example 2: A flexible accrual intensity of 20 subjects/months in the first 6 months, 25 subjects/months from month 6-12, and 30 subjects thereafter can be specified through either of the two equivalent options below:

Option 1: List-based definition:

accrualTime = list(
    "0 - <6" = 20,
    "6 - <12" = 25,
    ">= 12" = 30)

Option 2: Vector-based definition (note that the length of the accrualTime vector is equal to the length of the accrualIntensity vector as only the start time of the last accrual intensity is provided):

accrualTime = c(0, 6, 12), accrualIntensity = c(20, 25, 30)

6 Sample size calculation for superiority trials without interim analyses

Given the specification of the survival distributions, drop-out and accrual as described above, sample size calculations can be readily performed in rpact using the function getSampleSizeSurvival(). Importantly, in survival trials, the number of events (and not the sample size) determines the power of the trial. Hence, the choice of the sample size to reach this target number of events is based on study-specific trade-offs between the costs per recruited patient, study duration, and desired minimal follow-up duration.

In practice, a range of sample sizes need to be explored manually or via suitable graphs to find an optimal trade off. Given a plausible absolute accrual intensity (usually provided by operations), a simple approach in rpact is to use one or both of the two options below:

  1. Perform the calculation by trying multiple maximum sample sizes (argument maxNumberOfSubjects).
  2. It often makes sense to require a minimal follow-up time (argument followUpTime) for all patients in the trial at the time of the analysis. In this case, rpact automatically determines the maximum sample size.

6.1 Example - exponential survival, flexible accrual intensity, no interim analyses

  • Exponential PFS with a median PFS of 60 months in control (lambda2 = log(2)/60) and a target hazard ratio of 0.74 (hazardRatio = 0.74).
  • Log-rank test at the two-sided 5%-significance level (sided = 2, alpha = 0.05), power 80% (beta = 0.2).
  • Annual drop-out of 2.5% in both arms (dropoutRate1 = 0.025, dropoutRate2 = 0.025, dropoutTime = 12).
  • Recruitment is 42 patients/month from month 6 onwards after linear ramp up. (accrualTime = c(0,1,2,3,4,5,6), accrualIntensity = c(6,12,18,24,30,36,42))
  • Randomization ratio 1:1 (allocationRatioPlanned = 1). This is the default and is thus not explicitly set in the function call below.
  • Two sample size choices will be initially explored:
    • A fixed total sample size of 1200 (maxNumberOfSubjects = 1200).
    • Alternatively, the total sample size will be implicitly determined by specifying that every subject must have a minimal follow-up duration of at 12 months at the time of the analysis (followUpTime = 12).

Based on this, the required number of events and timing of interim analyses for the fixed total sample size of 1200 can be determined as follows:

sampleSize1 <- getSampleSizeSurvival(sided = 2,alpha = 0.05,beta = 0.2,
    lambda2 = log(2)/60,hazardRatio = 0.74,
    dropoutRate1 = 0.025, dropoutRate2 = 0.025, dropoutTime = 12,
    accrualTime = c(0,1,2,3,4,5,6), 
    accrualIntensity = c(6,12,18,24,30,36,42),
    maxNumberOfSubjects = 1200)

sampleSize1
## Design plan parameters and output for survival data:
## 
## Design parameters:
##   Critical values                              : 1.96 
##   Two-sided power                              : FALSE 
##   Significance level                           : 0.0500 
##   Type II error rate                           : 0.2000 
##   Test                                         : two-sided 
## 
## User defined parameters:
##   lambda(2)                                    : 0.0116 
##   Hazard ratio                                 : 0.740 
##   Maximum number of subjects                   : 1200 
##   Accrual time                                 : 1.00, 2.00, 3.00, 4.00, 5.00, 6.00, 31.57 
##   Accrual intensity                            : 6, 12, 18, 24, 30, 36, 42 
##   Drop-out rate (1)                            : 0.025 
##   Drop-out rate (2)                            : 0.025 
## 
## Default parameters:
##   Theta H0                                     : 1 
##   Type of computation                          : Schoenfeld 
##   Planned allocation ratio                     : 1 
##   kappa                                        : 1 
##   Piecewise survival times                     : 0.00 
##   Drop-out time                                : 12.00 
## 
## Sample size and output:
##   Direction upper                              : FALSE 
##   median(1)                                    : 81.1 
##   median(2)                                    : 60.0 
##   lambda(1)                                    : 0.00855 
##   Number of events                             : 346.3 
##   Total accrual time                           : 31.57 
##   Follow up time                               : 21.54 
##   Number of events fixed                       : 346.3 
##   Number of subjects fixed                     : 1200.0 
##   Number of subjects fixed (1)                 : 600.0 
##   Number of subjects fixed (2)                 : 600.0 
##   Analysis times                               : 53.11 
##   Study duration                               : 53.11 
##   Lower critical values (treatment effect scale) : 1.234 
##   Upper critical values (treatment effect scale) : 0.810 
##   Local two-sided significance levels          : 0.0500 
## 
## Legend:
##   (i): values of treatment arm i

Thus, the required number of events is 347 and the MDD corresponds to an observed HR of .The 1200 subjects will be recruited over 31.57 months and the total study duration is 53.11 months. The user could now vary maxNumberSubject to further optimize the trade-off between sample size and study duration.

Alternatively, specifying a minimum follow-up duration of 12 months leads to the following result:

sampleSize2 <- getSampleSizeSurvival(sided = 2,alpha = 0.05,beta = 0.2,
    lambda2 = log(2)/60,hazardRatio = 0.74,
    dropoutRate1 = 0.025, dropoutRate2 = 0.025,dropoutTime = 12,
    accrualTime = c(0,1,2,3,4,5,6), 
    accrualIntensity = c(6,12,18,24,30,36,42),
    followUpTime = 12)
sampleSize2
## Design plan parameters and output for survival data:
## 
## Design parameters:
##   Critical values                              : 1.96 
##   Two-sided power                              : FALSE 
##   Significance level                           : 0.0500 
##   Type II error rate                           : 0.2000 
##   Test                                         : two-sided 
## 
## User defined parameters:
##   lambda(2)                                    : 0.0116 
##   Hazard ratio                                 : 0.740 
##   Accrual intensity                            : 6, 12, 18, 24, 30, 36, 42 
##   Follow up time                               : 12.00 
##   Drop-out rate (1)                            : 0.025 
##   Drop-out rate (2)                            : 0.025 
## 
## Default parameters:
##   Theta H0                                     : 1 
##   Type of computation                          : Schoenfeld 
##   Planned allocation ratio                     : 1 
##   kappa                                        : 1 
##   Piecewise survival times                     : 0.00 
##   Drop-out time                                : 12.00 
## 
## Sample size and output:
##   Direction upper                              : FALSE 
##   median(1)                                    : 81.1 
##   median(2)                                    : 60.0 
##   lambda(1)                                    : 0.00855 
##   Maximum number of subjects                   : 1433.7 
##   Number of events                             : 346.3 
##   Accrual time                                 : 1.00, 2.00, 3.00, 4.00, 5.00, 6.00, 37.13 
##   Total accrual time                           : 37.13 
##   Number of events fixed                       : 346.3 
##   Number of subjects fixed                     : 1433.7 
##   Number of subjects fixed (1)                 : 716.8 
##   Number of subjects fixed (2)                 : 716.8 
##   Analysis times                               : 49.13 
##   Study duration                               : 49.13 
##   Lower critical values (treatment effect scale) : 1.234 
##   Upper critical values (treatment effect scale) : 0.810 
##   Local two-sided significance levels          : 0.0500 
## 
## Legend:
##   (i): values of treatment arm i

This specification leads to a higher sample size of 1434 subjects which will be recruited over 37.13 months and the total study duration is 49.13 months.

With the generic summary() function the two calculations are summarized as follows:

summary(sampleSize1) 
## Sample size calculation for a survival endpoint
## 
## Fixed sample analysis, significance level 5% (two-sided).
## The sample size was calculated for a two-sample logrank test, 
## H0: hazard ratio = 1, H1: hazard ratio = 0.74, control lambda(2) = 0.012, 
## number of subjects = 1200, accrual time = c(1, 2, 3, 4, 5, 6, 31.571), 
## dropout rate(1) = 0.025, dropout rate(2) = 0.025, dropout time = 12, power 80%.
## 
## Stage                                      Fixed 
## Efficacy boundary (z-value scale)          1.960 
## Number of subjects                        1200.0 
## Number of events                             347 
## Expected study duration                     53.1 
## Two-sided local significance level        0.0500 
## Efficacy boundary (t)              1.234 - 0.810 
## 
## Legend:
##   (t): treatment effect scale
summary(sampleSize2)
## Sample size calculation for a survival endpoint
## 
## Fixed sample analysis, significance level 5% (two-sided).
## The sample size was calculated for a two-sample logrank test, 
## H0: hazard ratio = 1, H1: hazard ratio = 0.74, control lambda(2) = 0.012, 
## accrual time = c(1, 2, 3, 4, 5, 6, 37.135), dropout rate(1) = 0.025, 
## dropout rate(2) = 0.025, dropout time = 12, power 80%.
## 
## Stage                                      Fixed 
## Efficacy boundary (z-value scale)          1.960 
## Number of subjects                        1433.7 
## Number of events                             347 
## Expected study duration                     49.1 
## Two-sided local significance level        0.0500 
## Efficacy boundary (t)              1.234 - 0.810 
## 
## Legend:
##   (t): treatment effect scale

To further explore the possible trade-offs, one could visualize recruitment and study durations for a range of sample sizes as illustrated in the code below:

# set up data frame which contains sample sizes and corresponding durations
sampleSizeDuration <- data.frame(
    maxNumberSubjects = seq(600,1800,by = 50),
    accrualTime = NA,
    studyDuration = NA)

# calculate recruitment and study duration for each sample size
for (i in 1:nrow(sampleSizeDuration)) { 
    sampleSizeResult <- getSampleSizeSurvival(
        sided = 2,alpha = 0.05,beta = 0.2,
        lambda2 = log(2)/60,hazardRatio = 0.74,
        dropoutRate1 = 0.025, dropoutRate2 = 0.025,dropoutTime = 12,
        accrualTime = c(0,1,2,3,4,5,6), 
        accrualIntensity = c(6,12,18,24,30,36,42),
        maxNumberOfSubjects = sampleSizeDuration$maxNumberSubjects[i])
    sampleSizeDuration$accrualTime[i] <- sampleSizeResult$totalAccrualTime
    sampleSizeDuration$studyDuration[i] <- sampleSizeResult$maxStudyDuration
}

# plot result
plot(sampleSizeDuration$maxNumberSubjects,
    sampleSizeDuration$studyDuration,type = "l",
    xlab = "Total sample size",
    ylab = "Duration (months)",
    main = "Recruitment and study duration vs sample size",
    ylim = c(0,max(sampleSizeDuration$studyDuration)),
    col = "blue",lwd = 1.5)
lines(sampleSizeDuration$maxNumberSubjects,
    sampleSizeDuration$accrualTime,col = "red",lwd = 1.5)
legend(x = 1000,y = 100,
    legend = c("Study duration under H1",
        "Recruitment duration"),
    col = c("blue","red"),lty = 1,lwd = 1.5)