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

Planning
Means
This document provides examples for designing trials with continuous endpoints using rpact.
Author

Dietmar Volz Marcel Wolbers Gernot Wassmer Friedrich Pahlke

Published

February 16, 2024

Introduction

These examples 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 3.0 or later
[1] '4.0.0'

Sample size calculation for a superiority trial without interim analyses

The sample size for a trial with continuous endpoints can be calculated using the function getSampleSizeMeans(). This function is fully documented in the relevant help page (?getSampleSizeMeans). Some examples are provided below.

getSampleSizeMeans() requires that the mean difference between the two arms is larger under the alternative than under the null hypothesis. For superiority trials, this implies that rpact requires that the targeted mean difference is >0 under the alternative hypothesis. If this is not the case, the function produces an error message. To circumvent this and power for a negative mean difference, one can simply switch the two arms (leading to a positive mean difference) as the situation is perfectly symmetric.

By default, getSampleSizeMeans() tests hypotheses about the mean difference. rpact also supports testing hypotheses about mean ratios if the argument meanRatio is set to TRUE but this will not be discussed further in this document.

By default, rpact uses sample size formulas for the \(t\)-test, i.e., it assumes that the standard deviation in the two groups is equal but unknown and estimated from the data. If sample size calculations for the \(z\)-test are desired, one can set the argument normalApproximation to TRUE but this is usually not recommended.

# Example of a standard trial:
# - targeted mean difference is 10 (alternative = 10)
# - standard deviation in both arms is assumed to be 24 (stDev = 24)
# - two-sided test (sided = 2), Type I error 0.05 (alpha = 0.05) and power 80%
# - (beta = 0.2)
sampleSizeResult <- getSampleSizeMeans(
    alternative = 10, stDev = 24, sided = 2,
    alpha = 0.05, beta = 0.2
)
kable(sampleSizeResult)
stages alternative meanRatio thetaH0 normalApproximation stDev groups allocationRatioPlanned nFixed nFixed1 nFixed2 criticalValuesEffectScaleLower criticalValuesEffectScaleUpper criticalValuesPValueScale
1 10 FALSE 0 FALSE 24 2 1 182.7789 91.38944 91.38944 -7.00557 7.00557 0.05

The generic summary() function produces the output

kable(summary(sampleSizeResult))
Warning in is.na(parameterValues): is.na() auf Nicht-(Liste oder Vektor) des
Typs 'environment' angewendet
object NA NA NA NA NA NA NA NA NA NA NA NA NA
1 10 FALSE 0 FALSE 24 2 1 182.7789 91.38944 91.38944 -7.00557 7.00557 0.05

As per the output above, the required total sample size for the trial is 183 and the critical value corresponds to a minimal detectable mean difference of approximately 7.01.

Unequal randomization between the treatment groups can be defind with allocationRatioPlanned, for example,

# Extension of standard trial:
# - 2(intervention):1(control) randomization (allocationRatioPlanned = 2)
kable(summary(getSampleSizeMeans(
    alternative = 10, stDev = 24,
    allocationRatioPlanned = 2, sided = 2, alpha = 0.05, beta = 0.2
)))
Warning in is.na(parameterValues): is.na() auf Nicht-(Liste oder Vektor) des
Typs 'environment' angewendet
object NA NA NA NA NA NA NA NA NA NA NA NA NA
1 10 FALSE 0 FALSE 24 2 2 205.3814 136.921 68.46048 -7.004498 7.004498 0.05

Power for a given sample size can be calculated using the function getPowerMeans() which has the same arguments as getSampleSizeMeans() except that the maximum total sample is given (maxNumberOfSubjects) instead of the Type II error (beta).

# Calculate power for the 2:1 rendomized trial with total sample size 206
# (as above) assuming a larger difference of 12
powerResult <- getPowerMeans(
    alternative = 12, stDev = 24, sided = 2,
    allocationRatioPlanned = 2, maxNumberOfSubjects = 206, alpha = 0.05
)
kable(powerResult)
stages alternative meanRatio thetaH0 normalApproximation stDev groups allocationRatioPlanned directionUpper effect maxNumberOfSubjects overallReject nFixed nFixed1 nFixed2 criticalValuesEffectScaleLower criticalValuesEffectScaleUpper criticalValuesPValueScale
1 12 FALSE 0 FALSE 24 2 2 NA 12 206 0.920291 206 137.3333 68.66667 -6.993847 6.993847 0.05

The calculated power is provided in the output as “Overall reject” and is 0.92 for the example alternative = 12.

The summary() function produces

kable(summary(powerResult))
Warning in is.na(parameterValues): is.na() auf Nicht-(Liste oder Vektor) des
Typs 'environment' angewendet
object NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
1 12 FALSE 0 FALSE 24 2 2 NA 12 206 0.920291 206 137.3333 68.66667 -6.993847 6.993847 0.05

getPowerMeans() (as well as getSampleSizeMeans()) can also be called with a vector argument for the mean difference under the alternative H1 (alternative). This is illustrated below via a plot of power depending on these values. For examples of all available plots, see the R Markdown document How to create admirable plots with rpact.

# Example: Calculate power for design with sample size 206 as above
# alternative values ranging from 5 to 15
powerResult <- getPowerMeans(
    alternative = 5:15, stDev = 24, sided = 2,
    allocationRatioPlanned = 2, maxNumberOfSubjects = 206, alpha = 0.05
)
plot(powerResult, type = 7) # one of several possible plots

Sample size calculation for a non-inferiority trial without interim analyses

The sample size calculation proceeds in the same fashion as for superiority trials except that the role of the null and the alternative hypothesis are reversed and the test is always one-sided. In this case, the non-inferiority margin \(\Delta\) corresponds to the treatment effect under the null hypothesis (thetaH0) which one aims to reject.

# Example: Non-inferiority trial with margin delta = 10, standard deviation = 14
# - One-sided alpha = 0.05, 1:1 randomization
# - H0: treatment difference <= -12 (i.e., = -12 for calculations, thetaH0 = -1)
#    vs. alternative H1: treatment difference = 0 (alternative = 0)
sampleSizeNoninf <- getSampleSizeMeans(
    thetaH0 = -12, alternative = 0,
    stDev = 14, alpha = 0.025, beta = 0.2, sided = 1
)
kable(sampleSizeNoninf)
stages alternative meanRatio thetaH0 normalApproximation stDev groups allocationRatioPlanned nFixed nFixed1 nFixed2 criticalValuesEffectScale
1 0 FALSE -12 FALSE 14 2 1 44.73721 22.3686 22.3686 -3.556151

Sample size calculation for group sequential designs

Sample size calculation for a group sequential trials 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.
  2. Calculate sample size for the continuous endpoint by feeding the abstract design into the function getSampleSizeMeans().

In general, rpact supports both one-sided and two-sided group sequential designs. However, if futility boundaries are specified, only one-sided tests are permitted. For simplicity, it is often preferred to use one-sided tests for group sequential designs (typically, with \(\alpha = 0.025\)).

R code for a simple example is provided below:

# Example: Group-sequential design with  O'Brien & Fleming type alpha-spending
# and one interim at 60% information
design <- getDesignGroupSequential(
    sided = 1, alpha = 0.025, beta = 0.2,
    informationRates = c(0.6, 1), typeOfDesign = "asOF"
)

# Trial assumes an effect size of 10 as above, a stDev = 24, and an allocation
# ratio of 2
sampleSizeResultGS <- getSampleSizeMeans(
    design,
    alternative = 10, stDev = 24, allocationRatioPlanned = 2
)
# Standard rpact output (sample size object only, not design object)
kable(sampleSizeResultGS)
stages alternative meanRatio thetaH0 normalApproximation stDev groups allocationRatioPlanned maxNumberOfSubjects maxNumberOfSubjects1 maxNumberOfSubjects2 numberOfSubjects numberOfSubjects1 numberOfSubjects2 rejectPerStage earlyStop expectedNumberOfSubjectsH0 expectedNumberOfSubjectsH01 expectedNumberOfSubjectsH1 criticalValuesEffectScale
1 10 FALSE 0 FALSE 24 2 2 207.1351 138.09 69.04502 124.2810 82.85402 41.42701 0.3123193 0.3123193 206.8195 202.3981 181.2581 12.392731
2 10 FALSE 0 FALSE 24 2 2 207.1351 138.09 69.04502 207.1351 138.09003 69.04502 0.4876807 0.3123193 206.8195 202.3981 181.2581 7.049874 
# Summary rpact output for sample size object
kable(summary(sampleSizeResultGS))
Warning in is.na(parameterValues): is.na() auf Nicht-(Liste oder Vektor) des
Typs 'environment' angewendet
object NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
1 10 FALSE 0 FALSE 24 2 2 207.1351 138.09 69.04502 124.2810 82.85402 41.42701 0.3123193 0.3123193 206.8195 202.3981 181.2581 12.392731
2 10 FALSE 0 FALSE 24 2 2 207.1351 138.09 69.04502 207.1351 138.09003 69.04502 0.4876807 0.3123193 206.8195 202.3981 181.2581 7.049874

System: rpact 4.0.0, R version 4.3.3 (2024-02-29 ucrt), platform: x86_64-w64-mingw32

To cite R in publications use:

R Core Team (2024). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/. To cite package ‘rpact’ in publications use:

Wassmer G, Pahlke F (2024). rpact: Confirmatory Adaptive Clinical Trial Design and Analysis. R package version 4.0.0, https://www.rpact.com, https://github.com/rpact-com/rpact, https://rpact-com.github.io/rpact/, https://www.rpact.org.