Free examples and use-cases: rpact vignettes
rpact: Confirmatory Adaptive Clinical Trial Design and Analysis
Comparing sample size and power calculation results for a group-sequential trial with a survival endpoint: rpact vs. gsDesign
Gernot Wassmer, Friedrich Pahlke, and Marcel Wolbers
Last change: 10 Januar, 2022
Summary
This R Markdown document provides an example that illustrates how to compare sample size and power calculation results of the two different R packages rpact and gsDesign.
1 The design
- 1:1 randomized
- Two-sided log-rank test; 80% power at the 5% significance level (or one-sided at 2.5%)
- Target HR for primary endpoint (PFS) is 0.75
- PFS in the control arm follows a piece-wise exponential distribution, with the hazard rate h(t) estimated using historical controls as follows:
- h(t) = 0.025 for t between 0 and 6 months;
- h(t) = 0.04 for t between 6 and 9 months;
- h(t) = 0.015 for t between 9 and 15 months;
- h(t) = 0.01 for t between 15 and 21 months;
- h(t) = 0.007 for t beyond 21 months.
- An annual dropout probability of 20%
- Interim analyses at 33% and 70% of total information
- Alpha-spending version of O’Brien-Fleming boundary for efficacy
- No futility interim
- 1405 subjects recruited in total
- Staggered recruitment:
- 15 pt/month during first 12 months;
- subsequently, increase of # of sites and ramp up of recruitment by +6 pt/month each month until a maximum of 45 pt/month
2 Calculation with gsDesign
library(gsDesign)
options(warn = -1) # avoid warnings generated by gsDesign
x <- gsSurv(k = 3,test.type = 1, alpha = 0.025, beta = 0.2,
timing = c(0.33,0.7), sfu = sfLDOF, # boundary
hr = 0.75,
lambdaC = c(0.025, 0.04, 0.015, 0.01, 0.007), # piecewise lambdas
S = c(6, 3, 6, 6), # piecewise survival times
eta = -log(1 - 0.2) / 12, # dropout
gamma = c(15, 21, 27, 33, 39, 45), # recruitment, pt no
R = c(12, 1, 1, 1, 1, (1405 - 300) / 45), # recruitment duration
minfup = NULL)
print(x,digits = 5) ## Time to event group sequential design with HR= 0.75
## Equal randomization: ratio=1
## One-sided group sequential design with
## 80 % power and 2.5 % Type I Error.
##
## Analysis N Z Nominal p Spend
## 1 128 3.73 0.0001 0.0001
## 2 271 2.44 0.0074 0.0073
## 3 386 2.00 0.0227 0.0176
## Total 0.0250
##
## ++ alpha spending:
## Lan-DeMets O'Brien-Fleming approximation spending function with none = 1.
##
## Boundary crossing probabilities and expected sample size
## assume any cross stops the trial
##
## Upper boundary (power or Type I Error)
## Analysis
## Theta 1 2 3 Total E{N}
## 0.0000 0.0001 0.0073 0.0176 0.025 385.0
## 0.1437 0.0175 0.4517 0.3309 0.800 329.1
## T n Events HR efficacy
## IA 1 26.78703 785.4162 127.3407 0.516
## IA 2 38.62360 1318.0620 270.1171 0.743
## Final 50.80093 1405.0000 385.8810 0.816
## Accrual rates:
## Stratum 1
## 0-12 15
## 12-13 21
## 13-14 27
## 14-15 33
## 15-16 39
## 16-40.56 45
## Control event rates (H1):
## Stratum 1
## 0-6 0.025
## 6-9 0.040
## 9-15 0.015
## 15-21 0.010
## 21-Inf 0.007
## Censoring rates:
## Stratum 1
## 0-6 0.0186
## 6-9 0.0186
## 9-15 0.0186
## 15-21 0.0186
## 21-Inf 0.01863 Calculation with rpact
3.1 Design
## [1] '3.2.1'design <- getDesignGroupSequential(
sided = 1, alpha = 0.025, beta = 0.2,
informationRates = c(0.33, 0.7, 1),
typeOfDesign = "asOF")
design## Design parameters and output of group sequential design:
##
## User defined parameters:
## Type of design : O'Brien & Fleming type alpha spending
## Information rates : 0.330, 0.700, 1.000
##
## Derived from user defined parameters:
## Maximum number of stages : 3
##
## Default parameters:
## Stages : 1, 2, 3
## Significance level : 0.0250
## Type II error rate : 0.2000
## Two-sided power : FALSE
## Test : one-sided
## Tolerance : 0.00000001
## Type of beta spending : none
##
## Output:
## Cumulative alpha spending : 0.00009549, 0.00738449, 0.02499999
## Critical values : 3.731, 2.440, 2.000
## Stage levels (one-sided) : 0.00009549, 0.00735097, 0.022744893.2 Sample size/timing of interim
piecewiseSurvivalTime <- list(
"0 - <6" = 0.025,
"6 - <9" = 0.04,
"9 - <15" = 0.015,
"15 - <21" = 0.01,
">= 21" = 0.007)
accrualTime <- list(
"0 - <12" = 15,
"12 - <13" = 21,
"13 - <14" = 27,
"14 - <15" = 33,
"15 - <16" = 39,
">= 16" = 45)
y <- getPowerSurvival(
design = design, typeOfComputation = "Schoenfeld",
thetaH0 = 1, directionUpper = FALSE,
dropoutRate1 = 0.2, dropoutRate2 = 0.2, dropoutTime = 12,
allocationRatioPlanned = 1,
accrualTime = accrualTime,
piecewiseSurvivalTime = piecewiseSurvivalTime,
hazardRatio = 0.75,
maxNumberOfEvents = x$n.I[3],
maxNumberOfSubjects = 1405)
y## Design plan parameters and output for survival data:
##
## Design parameters:
## Information rates : 0.330, 0.700, 1.000
## Critical values : 3.731, 2.440, 2.000
## Futility bounds (non-binding) : -Inf, -Inf
## Cumulative alpha spending : 0.00009549, 0.00738449, 0.02499999
## Local one-sided significance levels : 0.00009549, 0.00735097, 0.02274489
## Significance level : 0.0250
## Test : one-sided
##
## User defined parameters:
## Direction upper : FALSE
## lambda(2) : 0.025, 0.040, 0.015, 0.010, 0.007
## Hazard ratio : 0.750
## Maximum number of subjects : 1405
## Maximum number of events : 385.9
## Accrual time : 12.00, 13.00, 14.00, 15.00, 16.00, 40.56
## Accrual intensity : 15, 21, 27, 33, 39, 45
## Piecewise survival times : 0, 6, 9, 15, 21
## Drop-out rate (1) : 0.200
## Drop-out rate (2) : 0.200
##
## Default parameters:
## Theta H0 : 1
## Type of computation : Schoenfeld
## Planned allocation ratio : 1
## kappa : 1
## Drop-out time : 12.00
##
## Sample size and output:
## lambda(1) : 0.01875, 0.03000, 0.01125, 0.00750, 0.00525
## Total accrual time : 40.56
## Follow up time : 10.25
## Expected number of events : 328.9
## Overall reject : 0.8009
## Reject per stage [1] : 0.01754
## Reject per stage [2] : 0.45262
## Reject per stage [3] : 0.33070
## Early stop : 0.4702
## Analysis times [1] : 26.79
## Analysis times [2] : 38.62
## Analysis times [3] : 50.80
## Expected study duration : 44.87
## Maximal study duration : 50.80
## Cumulative number of events [1] : 127.3
## Cumulative number of events [2] : 270.1
## Cumulative number of events [3] : 385.9
## Number of subjects [1] : 785.4
## Number of subjects [2] : 1318.1
## Number of subjects [3] : 1405.0
## Expected number of subjects : 1354.8
## Critical values (treatment effect scale) [1] : 0.516
## Critical values (treatment effect scale) [2] : 0.743
## Critical values (treatment effect scale) [3] : 0.816
##
## Legend:
## (i): values of treatment arm i
## [k]: values at stage k4 Comparison: analysis time of rpact vs. gsDesign
Absolute differences:
timeDiff <- as.data.frame(sprintf("%.5f", (x$T - y$analysisTime)))
rownames(timeDiff) <- c("Stage 1", "Stage 2", "Stage 3")
colnames(timeDiff) <- "Difference analysis time"
timeDiff## Difference analysis time
## Stage 1 -0.00000
## Stage 2 0.00004
## Stage 3 -0.000115 Remark
Obviously, there is a difference in the calculation of the necessary number of events which are, in rpact, calculated as
(qnorm(0.975) + qnorm(0.8))^2 / log(0.75)^2 * 4 *
getDesignCharacteristics(getDesignGroupSequential(sided = 1, alpha = 0.025,
kMax = 3, typeOfDesign = "asOF", informationRates = c(0.33,0.7,1)))$inflationFactor## [1] 385.0479which is slightly different to the maximum number of events in gsDesign which is
## [1] 385.881Therefore, running
getSampleSizeSurvival(
design = design, typeOfComputation = "Schoenfeld",
thetaH0 = 1,
dropoutRate1 = 0.2, dropoutRate2 = 0.2, dropoutTime = 12,
allocationRatioPlanned = 1,
accrualTime = accrualTime,
piecewiseSurvivalTime = piecewiseSurvivalTime,
hazardRatio = 0.75,
maxNumberOfSubjects = 1405)$analysisTime## [,1]
## [1,] 26.76183
## [2,] 38.57834
## [3,] 50.63114is not exactly equal to getPowerSurvival from above. This, however, has absolutely no consequences in practice but explains the slight differences in rpact and gsDesign.
System: rpact 3.2.1, R version 4.1.2 (2021-11-01), platform: x86_64-w64-mingw32
To cite R in publications use:
R Core Team (2021). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/.
To cite package ‘rpact’ in publications use:
Gernot Wassmer and Friedrich Pahlke (2022). rpact: Confirmatory Adaptive Clinical Trial Design and Analysis. R package version 3.2.1. https://CRAN.R-project.org/package=rpact
This work by Gernot Wassmer and Friedrich Pahlke is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.