Group Sequential Design with Non-Constant Effect • gsDesign2

Objective

The goal of gsDesign2 is to enable fixed or group sequential design under non-proportional hazards. Piecewise constant enrollment, failure rates and dropout rates for a stratified population are available to enable highly flexible enrollment, time-to-event and time-to-dropout 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 inter-arrival 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% one-sided 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: 1-sided alpha level for testing.
Power: power corresponding to enrollment, failure rate, and trial targeted events.

fd |>
  summary() |>
  as_gt()

Design	N	Events	Time	Bound	alpha	Power
Fixed Design under AHR Method¹
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 non-proportional hazards as demonstrated in the fixed design above and again here.

We use an O’Brien-Fleming 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 = -\infty). 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, # Non-binding 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 Z-values 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.

gsd |>
  summary() |>
  as_gt()

Bound	Z	Nominal p¹	~HR at bound²	Cumulative boundary crossing probability
Bound summary for AHR design
AHR approximations of ~HR at bound
Bound	Z	Nominal p¹	~HR at bound²	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 One-sided p-value 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 non-binding. 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.