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rpact: Confirmatory Adaptive Clinical Trial Design and Analysis

Designing Group Sequential Trials with Two Groups and a Survival Endpoint with rpact

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] '3.3.1'

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 = FALSE,
    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))