Objective
The goal of gsDesign2 is to enable fixed or group sequential design under nonproportional hazards. Piecewise constant enrollment, failure rates and dropout rates for a stratified population are available to enable highly flexible enrollment, timetoevent and timetodropout assumptions. Substantial flexibility on top of what is in the gsDesign package is intended for selecting boundaries. Comments on usability and features are encouraged as this is still a young package.
Installation
Install the released version of gsDesign2 from CRAN:
install.packages("gsDesign2")
Or install the development version from GitHub with:
remotes::install_github("Merck/gsDesign2")
Use cases
Step 1: specifying enrollment and failure rates
This is a basic example which shows you how to solve a common problem. We assume there is a 4 month delay in treatment effect. Specifically, we assume a hazard ratio of 1 for 4 months and 0.6 thereafter. For this example we assume an exponential failure rate and low exponential dropout rate. The enroll_rate
specification indicates an expected enrollment duration of 12 months with exponential interarrival times.
library(gsDesign2)
# Basic example
# Constant enrollment over 12 months
# Rate will be adjusted later by gsDesign2 NPH to get sample size
enroll_rate < define_enroll_rate(duration = 12, rate = 1)
# 12 month median exponential failure rate in control
# 4 month delay in effect with HR=0.6 after
# Low exponential dropout rate
median_surv < 12
fail_rate < define_fail_rate(
duration = c(4, Inf),
fail_rate = log(2) / median_surv,
hr = c(1, .6),
dropout_rate = .001
)
The resulting failure rate specification is the following table. As many rows and strata as needed can be specified to approximate whatever patterns you wish.
fail_rate > gt::gt()
stratum  duration  fail_rate  dropout_rate  hr 

All  4  0.05776227  0.001  1.0 
All  Inf  0.05776227  0.001  0.6 
Step 2: derive a fixed design with no interim analyses
Computing a fixed sample size design with 2.5% onesided Type I error and 90% power. We specify a trial duration of 36 months with analysis_time
. Enrollment duration is the sum of enroll_rate$duration
. We used fixed_design()
since there is a single analysis:
fd < fixed_design_ahr(
enroll_rate = enroll_rate,
fail_rate = fail_rate,
alpha = 0.025,
power = 0.9,
study_duration = 36,
ratio = 1 # Experimental/control randomization ratio
)
The input enrollment rates have now been scaled to achieve power:
fd$enroll_rate > gt::gt()
stratum  duration  rate 

All  12  35.05288 
The failure and dropout rates remain unchanged from what was input. The summary is obtained below. The columns are:

Design
: sample size derivation method. 
N
: sample size; generally you will round up to an even number. 
Event
: generally you will round up. 
Bound
: Z value for efficacy; this is the inverse normal from 1  alpha. 
alpha
: 1sided alpha level for testing. 
Power
: power corresponding to enrollment, failure rate, and trial targeted events.
Fixed Design under AHR Method^{1}  
Design  N  Events  Time  Bound  alpha  Power 

Average hazard ratio  420.6346  311.0028  36  1.959964  0.025  0.9 
^{1} Power computed with average hazard ratio method. 
Step 3: group sequential design
We provide a simple example for a group sequential design that demonstrates a couple of features not available in the gsDesign package. The first is specifying analysis times by calendar time rather than information fraction. The second is not having an efficacy and futility bound at each analysis. This is in addition to having methods for nonproportional hazards as demonstrated in the fixed design above and again here.
We use an O’BrienFleming spending function to derive our efficacy bounds at 24 and 36 months. For futility, we simply require a nominally significant trend in the wrong direction (p < 0.1) after 8 months, a trend in favor of experimental treatment after 14 months (Z > 0) and no bound later (Z = − ∞). Thus, we have two efficacy analyses and two separate, earlier futility analysis. Power is set to 80% due to the somewhat aggressive futility bounds that are used for safety (analysis 1 half way through enrollment) and proof of concept (analysis 2). Such aggressive futility bounds may be desirable when a previous proof of concept for experimental treatment has not been established; essentially, this becomes a Phase II/III design with an interim evaluation of appropriate efficacy trends before completing the trial.
gsd < gs_design_ahr(
enroll_rate = enroll_rate,
fail_rate = fail_rate,
ratio = 1,
alpha = 0.025,
beta = 0.2, # 80% power; enables aggressive futility bound specified
analysis_time = c(8, 14, 24, 36),
binding = FALSE, # Nonbinding futility bound
upper = gs_spending_bound, # Use spending bound for efficacy; total_spend is normally alpha
upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025),
test_upper = c(FALSE, FALSE, TRUE, TRUE), # Only test efficacy after 1st analysis
lower = gs_b, # Fixed Zvalues will be provided for futility bound
lpar = c(qnorm(0.1), 0, Inf, Inf)
)
Now we summarize the derived design. The summary table is further described in the vignette summarize group sequential designs in gt tables. Note that the design trend in favor of experimental treatment is very minor at 8 months due to the delayed effect assumption used (see AHR at analysis 1 in table). The design trend at 16 months is somewhat more favorable when we are looking for HR < 1 (favoring experimental treatment) for a proof of concept. Actual bounds and timing selected for a trial are situation dependent, but we hope the suggestions here are provocative for what might be considered.
Bound summary for AHR design  
AHR approximations of ~HR at bound  
Bound  Z  Nominal p^{1}  ~HR at bound^{2}  Cumulative boundary crossing probability  

Alternate hypothesis  Null hypothesis  
Analysis: 1 Time: 8 N: 279.1 Event: 53.2 AHR: 0.91 Information fraction: 0.17  
Futility  1.28  0.9000  1.4210  0.0539  0.1000 
Analysis: 2 Time: 14 N: 418.7 Event: 137.2 AHR: 0.82 Information fraction: 0.44  
Futility  0.00  0.5000  1.0000  0.1451  0.5091 
Analysis: 3 Time: 24 N: 418.7 Event: 238.4 AHR: 0.72 Information fraction: 0.76  
Efficacy  2.30  0.0106  0.7421  0.5582  0.0106 
Analysis: 4 Time: 36 N: 418.7 Event: 309.5 AHR: 0.69 Information fraction: 1  
Efficacy  2.02  0.0219  0.7951  0.8000  ^{3} 0.0244 
^{1} Onesided pvalue for experimental vs control treatment. Value < 0.5 favors experimental, > 0.5 favors control.  
^{2} Approximate hazard ratio to cross bound.  
^{3} Cumulative alpha for final analysis (0.0244) is less than the full alpha (0.025) when the futility bound is nonbinding. The smaller value subtracts the probability of crossing a futility bound before crossing an efficacy bound at a later analysis (0.025  0.0006 = 0.0244) under the null hypothesis. 