Trial design with spending under NPH

Keaven M. Anderson

library(gsDesign)
library(gsDesign2)

Overview

This vignette covers how to implement designs for trials with spending assuming non-proportional hazards. We are primarily concerned with practical issues of implementation rather than design strategies, but we will not ignore design strategy.

Scenario for consideration

Here we set up enrollment, failure and dropout rates along with assumptions for enrollment duration and times of analyses. We assume there are 4 analysis (3 interim analyses + 1 final analysis) conducted 18, 24, 30, and 36 months after trial enrollment is opened.

n_analysis <- 4
analysis_time <- c(18, 24, 30, 36)

We assume there is a single stratum and enrollment targeted to last for 12 months. For the first 2 months, second 2 months, third 2 months and the remaining months, the relative enrollment rates are \(8:12:16:24\). These rates will be updated by a constant multiple at the time of design as we will note below.

enroll_rate <- define_enroll_rate(
  duration = c(2, 2, 2, 6),
  rate = c(8, 12, 16, 24)
)

enroll_rate |>
  gt::gt() |>
  gt::tab_header(title = "Planned Relative Enrollment Rates")

Planned Relative Enrollment Rates
stratum	duration	rate
All	2	8
All	2	12
All	2	16
All	6	24

We assume a hazard ratio (HR) of 0.9 for the first 3 months 0.6 thereafter. We also assume the the control time-to-event follows a piecewise exponential distribution with a median of 8 month for the first 3 months and 14 months thereafter.

fail_rate <- define_fail_rate(
  duration = c(3, 100),
  fail_rate = log(2) / c(8, 14),
  hr = c(.9, .6),
  dropout_rate = .001
)

fail_rate |>
  gt::gt() |>
  gt::tab_header(title = "Table of Failure Rate Assumptions")

Table of Failure Rate Assumptions
stratum	duration	fail_rate	dropout_rate	hr
All	3	0.08664340	0.001	0.9
All	100	0.04951051	0.001	0.6

Fixed design with no interim analysis

We can derive power for the above enrollment rates and failure rates as follows:

fixed_design_ahr(
  enroll_rate = enroll_rate,
  fail_rate = fail_rate,
  power = NULL,
  ratio = 1,
  study_duration = 36,
  event = NULL
) |> summary()
#> # A tibble: 1 × 7
#>   Design                   N Events  Time Bound alpha Power
#>   <chr>                <dbl>  <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 Average hazard ratio   216   151.    36  1.96 0.025 0.656

We now compute sample size and then translate from a continuous sample size to an integer sample size.

fixed_design <- fixed_design_ahr(
  enroll_rate = enroll_rate,
  fail_rate = fail_rate,
  power = .9,
  ratio = 1,
  study_duration = 36,
  event = NULL
) |> to_integer()

fixed_design$analysis
#> # A tibble: 1 × 7
#>   design     n event  time bound alpha power
#>   <chr>  <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 ahr      410  287.  36.0  1.96 0.025 0.901

Group sequential design

We now consider a group sequential design with bounds derived using spending functions. We target the interim analysis for 24 months and the final analysis for 36 months. Spending for both efficacy and futility is based on the proportion of events expected at each analysis divided by the total expected events at the final analysis.

gs <- gs_design_ahr(
  enroll_rate = enroll_rate,
  fail_rate = fail_rate,
  info_frac = NULL,
  analysis_time = c(24, 36),
  upper = gs_spending_bound,
  lower = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
  lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.1, param = NULL, timing = NULL),
  h1_spending = TRUE
) |> to_integer()

gs |>
  summary() |>
  gt::gt()

Bound	Z	~HR at bound	Nominal p	Alternate hypothesis	Null hypothesis
Analysis: 1 Time: 23.9 N: 434 Event: 232 AHR: 0.71 Information fraction: 0.76
Futility	1.04	0.8709	0.1486	0.0582	0.8514
Efficacy	2.31	0.7361	0.0104	0.6235	0.0104
Analysis: 2 Time: 35.8 N: 434 Event: 303 AHR: 0.68 Information fraction: 1
Futility	1.94	0.7986	0.0260	0.0988	0.9723
Efficacy	2.02	0.7920	0.0219	0.8998	0.0244