rpact provides the functions getSimulationMeans() (continuous endpoints), getSimulationRates() (binary endpoints) and getSimulationSurvival() (time-to-event endpoints) for simulation of group sequential trials with adaptive SSR.
For trials with adaptive SSR, the design can be created with the functions getDesignInverseNormal() or getDesignFisher(). The sample size is re-calculated based on the target conditional power (argument conditionalPower). Conditional power is by default evaluated at the observed estimates for the parameters. If the evaluation of conditional power at other parameter values is desired, they can be provided as arguments thetaH1 (for getSimulationMeans() and getSimulationSurvival()), or pi1H1 and pi2H1 (for getSimulationRates()). For continuous endpoints, by default, conditional power is evaluated at the standard deviation stDev under which the trial is simulated. In rpact 3.0, a variable stDevH1 can be entered to specify the standard deviation that is used for the sample size recalculation.
For the functions getSimulationMeans() and getSimulationRates() (but not in getSimulationSurvival()), the SSR function can be optionally modified using the argument calcSubjectsFunction() (see the respective help pages and the code below for details and examples).
In this vignette, we present an example of the use of these functions for a trial with a binary endpoint. For this, we will use the constraint promizing zone approach as described in Hsiao et al 2019.
For this vignette, additionally to rpact itself and ggplot2, we use the packages ggpubr and dplyr.
We first calculate the sample sizes (per treatment group) for the corresponding fixed designs with 90% power:
# fixed design powered for delta of 13%
ssMin <- getSampleSizeRates(pi1 = 0.33, pi2 = 0.2, alpha = 0.025, beta = 0.1)
(Nmin <- ceiling(ssMin$numberOfSubjects1)) [,1]
[1,] 241# fixed design powered for delta of 10%
ssMax <- getSampleSizeRates(pi1 = 0.30, pi2 = 0.2, alpha = 0.025, beta = 0.1)
(Nmax <- ceiling(ssMax$numberOfSubjects1)) [,1]
[1,] 392Assume that the sponsor is unwilling to make an up-front commitment for a trial with Nmax = 392 subjects per treatment group but that they are willing to provide an up-front commitment for a trial with Nmin = 241 subjects per treatment group. If the results at an interim analysis with an unblinded SSR look “promising”, the sponsor would then be willing to commit funding for up to 392 subjects per treatment group in total.
To help the sponsor, we investigate two designs with an interim analysis for SSR after 120 subjects per treatment group:
To combine the two stages for both designs, we use an inverse normal combination test with optimal weights for the minimal final group size (practically, this is a design with equal weights) and no provision for early stopping for neither efficacy nor futility at interim:
# group size at interim
(n1 <- floor(Nmin / 2)) [,1]
[1,] 120# inverse normal design with possible rejection of H0 only at the final analysis
designIN <- getDesignInverseNormal(
typeOfDesign = "noEarlyEfficacy",
kMax = 2,
alpha = 0.025
)
kable(summary(designIN))Sequential analysis with a maximum of 2 looks (inverse normal combination test design)
No early efficacy stop design, one-sided overall significance level 2.5%, power 80%, undefined endpoint, inflation factor 1, ASN H1 1, ASN H01 1, ASN H0 1.
| Stage | 1 | 2 |
|---|---|---|
| Information rate | 50% | 100% |
| Efficacy boundary (z-value scale) | Inf | 1.960 |
| Stage levels (one-sided) | 0 | 0.0250 |
| Cumulative alpha spent | 0 | 0.0250 |
| Overall power | 0 | 0.8000 |
It is straightforward to simulate the test characteristics from this design using the function getSimulationRates(). plannedSubjects refers to the cumulated sample sizes over the two stages in both treatment groups. If conditionalPower is specified, minNumberOfSubjectsPerStage and maxNumberOfSubjectsPerStage must be specified. They refer to the minimum and maximum overall sample sizes per stage (the first element is the first stage sample size), respectively. If pi1H1 and/or pi2H1 are not specified, the observed (simulated) rates at interim are used for the SSR.
# Design with sample size re-estimated to get conditional power of 0.9 at
# pi1H1 = 0.3, pi2H1 = 0.2 [minimum effect size]
# (evaluate at most interesting values for pi1)
simCpower <- getSimulationRates(designIN,
pi1 = c(0.2, 0.3, 0.33), pi2 = 0.2,
# cumulative overall sample size
plannedSubjects = 2 * c(n1, Nmin),
conditionalPower = 0.9,
# stage-wise minimal overall sample size
minNumberOfSubjectsPerStage = 2 * c(n1, (Nmin - n1)),
# stage-wise maximal overall sample size
maxNumberOfSubjectsPerStage = 2 * c(n1, (Nmax - n1)),
pi1H1 = 0.3, pi2H1 = 0.2,
maxNumberOfIterations = 1000,
seed = 12345
)
kable(simCpower, showStatistics = FALSE)Simulation of rates (inverse normal combination test design)
Design parameters
User defined parameters
Default parameters
Results
Legend
As described in Hsiao et al. 2019, this method chooses the sample size according to the following rules:
Choose the second stage size \(N^*\) between \(Nmin\) and \(Nmax\) such that the conditional power is \(cp_{max}\) for the minimal effect size we want to detect (i.e., ORR of 20% vs. 30%). We chose \(cp_{max} = 0.90\) here as in the original publication.
If such a sample size \(N^*\) does not exist, then proceed as follows:
To simulate the CPZ design in rpact, we can use the function getSimulationRates() again. However, the situation is more complicated because we need to re-define the sample size recalculation rule using the argument calcSubjectsFunction (see the help page ?getSimulationRates for more information regarding calcSubjectsFunction()):
# CPZ design (evaluate at the most interesting values for pi1)
# home-made SSR function
myCPZSampleSizeCalculationFunction <- function(..., stage,
plannedSubjects,
conditionalPower,
minNumberOfSubjectsPerStage,
maxNumberOfSubjectsPerStage,
conditionalCriticalValue,
overallRate) {
rateUnderH0 <- (overallRate[1] + overallRate[2]) / 2
# function adapted from example in ?getSimulationRates
calculateStageSubjects <- function(cp) {
2 * (max(0, conditionalCriticalValue *
sqrt(2 * rateUnderH0 * (1 - rateUnderH0)) +
stats::qnorm(cp) * sqrt(overallRate[1] *
(1 - overallRate[1]) + overallRate[2] * (1 - overallRate[2]))))^2 /
(max(1e-12, (overallRate[1] - overallRate[2])))^2
}
# Calculate sample size required to reach maximum desired conditional power
# cp_max (provided as argument conditionalPower)
stageSubjectsCPmax <- calculateStageSubjects(cp = conditionalPower)
# Calculate sample size required to reach minimum desired conditional power
# cp_min (**manually set for this example to 0.8**)
stageSubjectsCPmin <- calculateStageSubjects(cp = 0.8)
# Define stageSubjects
stageSubjects <- ceiling(min(max(
minNumberOfSubjectsPerStage[stage],
stageSubjectsCPmax
), maxNumberOfSubjectsPerStage[stage]))
# Set stageSubjects to minimal sample size in case minimum conditional power
# cannot be reached with available sample size
if (stageSubjectsCPmin > maxNumberOfSubjectsPerStage[stage]) {
stageSubjects <- minNumberOfSubjectsPerStage[stage]
}
# return sample size
return(stageSubjects)
}
# Now simulate the CPZ design
simCPZ <- getSimulationRates(designIN,
pi1 = c(0.2, 0.3, 0.33), pi2 = 0.2,
plannedSubjects = 2 * c(n1, Nmin), # cumulative overall sample size
conditionalPower = 0.9,
# stage-wise minimal overall sample size
minNumberOfSubjectsPerStage = 2 * c(n1, (Nmin - n1)),
# stage-wise maximal overall sample size
maxNumberOfSubjectsPerStage = 2 * c(n1, (Nmax - n1)),
pi1H1 = 0.3, pi2H1 = 0.2,
calcSubjectsFunction = myCPZSampleSizeCalculationFunction,
maxNumberOfIterations = 1000,
seed = 12345
)
kable(simCPZ, showStatistics = FALSE)Simulation of rates (inverse normal combination test design)
Design parameters
User defined parameters
Default parameters
Results
Legend
We first use the aggregated data from the two simulations to compare the dependence of the re-calculated sample size and the corresponding conditional power on the interim Z-score between the two designs. For this, we use the function getData(), the summarize() command of the dplyr package and plot it with ggplot2. Note that for this illustration we summaries over all values of pi1. This makes sense because we used a fixed pi1H1 and pi2H1 for both sample size simulation methods.
# aggregate data across simulation runs for both simulations and extract Z-score,
# conditionalPower, and totalSampleSize1 (per group)
aggSimCpower <- getData(simCpower)
sumCpower <- aggSimCpower %>%
group_by(iterationNumber) %>%
summarise(
design = "SS re-calculation for cp = 90%",
Z1 = testStatistic[1], conditionalPower = conditionalPowerAchieved[2],
totalSampleSize1 = (numberOfSubjects[1] + numberOfSubjects[2]) / 2
) %>%
arrange(Z1) %>%
filter(Z1 > 0, Z1 < 5)
aggSimCPZ <- getData(simCPZ)
sumCPZ <- aggSimCPZ %>%
group_by(iterationNumber) %>%
summarise(
design = "Constrained promising zone (CPZ)",
Z1 = testStatistic[1], conditionalPower = conditionalPowerAchieved[2],
totalSampleSize1 = (numberOfSubjects[1] + numberOfSubjects[2]) / 2
) %>%
arrange(Z1) %>%
filter(Z1 > 0, Z1 < 5)
sumBoth <- rbind(sumCpower, sumCPZ)
# Plot it
plot1 <- ggplot(aes(Z1, conditionalPower,
col = design,
group = design
), data = sumBoth) +
geom_line(aes(linetype = design), lwd = 1.2) +
scale_x_continuous(name = "Z-score at interim analysis") +
scale_y_continuous(
breaks = seq(0, 1, by = 0.1),
name = "Conditional power at re-calculated sample size"
) +
scale_color_manual(values = c("#d7191c", "#fdae61"))
plot2 <- ggplot(aes(Z1, totalSampleSize1,
col = design,
group = design
), data = sumBoth) +
geom_line(aes(linetype = design), lwd = 1.2) +
scale_x_continuous(name = "Z-score at interim analysis") +
scale_y_continuous(name = "Re-calculated final sample size (per group)") +
scale_color_manual(values = c("#d7191c", "#fdae61"))
ggarrange(plot1, plot2, common.legend = TRUE, legend = "bottom")
To compare the two designs across a wider range of parameters, we re-simulate the designs for a finer grid of assumed ORR in the intervention group and then visualize the results.
# Simulate designs again over a range of parameters and plot them
pi1Seq <- seq(0.2, 0.4, by = 0.01)
simCpowerLong <- getSimulationRates(designIN,
pi1 = pi1Seq, pi2 = 0.2,
plannedSubjects = c(2 * n1, 2 * Nmin),
conditionalPower = 0.9,
minNumberOfSubjectsPerStage = c(2 * n1, 2 * (Nmin - n1)),
maxNumberOfSubjectsPerStage = c(2 * n1, 2 * (Nmax - n1)),
pi1H1 = 0.3, pi2H1 = 0.2,
maxNumberOfIterations = 10000,
seed = 12345
)
plot(simCpowerLong, type = 6)
simCPZLong <- getSimulationRates(designIN,
pi1 = pi1Seq, pi2 = 0.2,
# cumulative overall sample size
plannedSubjects = 2 * c(n1, Nmin),
conditionalPower = 0.9,
# stage-wise minimal overall sample size
minNumberOfSubjectsPerStage = 2 * c(n1, (Nmin - n1)),
# stage-wise maximal overall sample size
maxNumberOfSubjectsPerStage = 2 * c(n1, (Nmax - n1)),
pi1H1 = 0.3, pi2H1 = 0.2,
calcSubjectsFunction = myCPZSampleSizeCalculationFunction,
maxNumberOfIterations = 10000,
seed = 12345
)
plot(simCPZLong, type = 6)
# Pool datasets from simulations (and fixed designs)
simCpowerData <- with(simCpowerLong,
data.frame(
design = "SS re-calculation for cp = 90%",
pi1 = pi1, pi2 = pi2, effect = effect, power = overallReject,
expectedNumberOfSubjects1 = expectedNumberOfSubjects / 2
),
stringsAsFactors = F
)
simCPZData <- with(simCPZLong,
data.frame(
design = "Constrained promising zone (CPZ)",
pi1 = pi1, pi2 = pi2, effect = effect, power = overallReject,
expectedNumberOfSubjects1 = expectedNumberOfSubjects / 2
),
stringsAsFactors = F
)
simFixed241 <- with(
simCPZData,
data.frame(
design = "Fixed (n = 241)",
pi1 = pi1, pi2 = pi2, effect = effect,
power = getPowerRates(
pi1 = pi1, pi2 = 0.2, maxNumberOfSubjects = 2 * 241,
alpha = 0.025, sided = 1
)$overallReject,
expectedNumberOfSubjects1 = 241, stringsAsFactors = F
)
)
simFixed392 <- with(
simCPZData,
data.frame(
design = "Fixed (n = 392)",
pi1 = pi1, pi2 = pi2, effect = effect,
power = getPowerRates(
pi1 = pi1, pi2 = 0.2, maxNumberOfSubjects = 2 * 392,
alpha = 0.025, sided = 1
)$overallReject,
expectedNumberOfSubjects1 = 392, stringsAsFactors = F
)
)
simdata <- rbind(simCpowerData, simCPZData, simFixed241, simFixed392)
simdata$design <- factor(simdata$design,
levels = c(
"Fixed (n = 241)", "Fixed (n = 392)",
"SS re-calculation for cp = 90%", "Constrained promising zone (CPZ)"
)
)
# Plot difference in ORR vs power
ggplot(aes(effect, power, col = design), data = simdata) +
geom_line(lwd = 1.2) +
scale_x_continuous(name = "Difference in ORR") +
scale_y_continuous(breaks = seq(0, 1, by = 0.1), name = "Power") +
geom_vline(xintercept = c(0.1, 0.13), color = "black") +
scale_color_manual(values = c("#2c7bb6", "#abd9e9", "#fdae61", "#d7191c"))
# Plot difference in ORR vs expected sample size
ggplot(aes(effect, expectedNumberOfSubjects1, col = design), data = simdata) +
geom_line(lwd = 1.2) +
scale_x_continuous(name = "Difference in ORR") +
scale_y_continuous(name = "Expected sample size (per group)") +
scale_color_manual(values = c("#2c7bb6", "#abd9e9", "#fdae61", "#d7191c"))
System: rpact 3.5.1, R version 4.3.2 (2023-10-31 ucrt), platform: x86_64-w64-mingw32
To cite R in publications use:
R Core Team (2023). 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 3.5.1, https://www.rpact.com, https://github.com/rpact-com/rpact, https://rpact-com.github.io/rpact/, https://www.rpact.org.