Step 1: Define design paramaters

To run simulations for a group sequential design, several design characteristics are required. The following code creates a design for an unstratified 2-arm trial with equal randomization. Enrollment is targeted to last for 12 months at a constant enrollment rate. The control arm is specified as exponential with a median of 10 months. The experimental arm distribution has a piecewise exponential distribution with a delay of 3 months with no benefit relative to control (HR = 1) followed by a hazard ratio of 0.6 thereafter. Additionally, there is an exponential dropout rate of 0.001 per month (or unit of time). The set up of these parameters is similar to the vignette Simulate Fixed Designs with Ease via sim_fixed_n. The total sample size is derived for 90% power.

stratum <- data.frame(stratum = "All", p = 1)
block <- rep(c("experimental", "control"), 2)
# enrollment rate will be updated later, 
# multiplied by a constant to get targeted power
enroll_rate <- data.frame(stratum = "All", rate = 1, duration = 12)
fail_rate <- data.frame(stratum = "All",
                        duration = c(3, Inf), fail_rate = log(2) / 10, 
                        hr = c(1, 0.6), dropout_rate = 0.001)
# Derive design using the average hazard ratio method
x <- gs_design_ahr(enroll_rate = enroll_rate, fail_rate = fail_rate,
                   analysis_time = c(12, 24, 36), alpha = 0.025, beta = 0.1,
                   # spending function for upper bound
                   upper = gs_spending_bound, 
                   upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025),
                   # Fixed lower bound
                   lower = gs_b,
                   lpar = rep(-Inf, 3)) |> to_integer()

sample_size <- x$analysis$n |> max()
event <- x$analysis$event
eff_bound <- x$bound$z[x$bound$bound == "upper"]

cat(paste("The total sample size is ", sample_size, "\n", sep = ''))

## The total sample size is 362

cat("The number of events at IA1, IA2 and FA are:", event, "\n")

## The number of events at IA1, IA2 and FA are: 106 227 287

cat("The efficacy bounds at IA1, IA2 and FA are:", round(eff_bound, 3), "\n")

## The efficacy bounds at IA1, IA2 and FA are: 3.508 2.269 2.023

cat("Targeted analysis times:", round(x$analysis$time, 1), "\n")

## Targeted analysis times: 12 24 35.9

Now we get the updated planned enrollment rate from the design to achieve the above targeted sample size.

enroll_rate <- x$enroll_rate
enroll_rate

##   stratum     rate duration
## 1     All 30.16667       12

There are additional parameters required for a group sequential design simulation as demonstrated below. One is the testing method. We focus on logrank here. Users can change these to other tests of interest; in comments below we demonstrate logrank and modestly weighted logrank test (Magirr and Burman (2019)) and Fleming-Harrington (Harrington and Fleming (1982)) tests; how to set up group sequential designs for these alternate tests is beyond the scope of this article. More testing methods are available at reference page of simtrial.

# Example for logrank
weight <- fh(rho = 0, gamma = 0)
test <- wlr
# Example for Modestly Weighted Logrank Test (Magirr-Burman)
# weight <- mb(delay = Inf, w_max = 2)
# Example for Fleming-Harrington(0, 0.5)
# weight <- fh(rho = 0, gamma = 0.5)

The final step is to specify how when data is cut off for each analysis in the group sequential design. The create_cut() function includes 5 options for the analysis cutoff:

1. planned calendar time,
2. targeted events,
3. maximum time extension to reach targeted events,
4. planned minimum time after the previous analysis, and
5. minimal follow-up time after specified enrollment fraction.

More details and examples are available at the help page.

A straightforward method for cutting analyses is based on events. For instance, the following code specifies 2 interim analyses and 1 final analysis cut when 106, 227, 287 events occur. In this event-driven approach, there is no need to update the efficacy boundary.

ia1_cut <- create_cut(target_event_overall = event[1])
ia2_cut <- create_cut(target_event_overall = event[2])
fa_cut <- create_cut(target_event_overall = event[3])

cut <- list(ia1 = ia1_cut, ia2 = ia2_cut, fa = fa_cut)

Users can set more complex cutting. For example,

• The first interim analysis occurs at the targeted IA 1 analyusis time or when at least the IA 1 targeted events are observed, which is later. However, if the target number of events is not reached, we will wait a maximum of 16 months from start of enrollment.
• The second interim analysis is targeted to take place at the targeted time or targeted events for IA 2, whichever is later. Additionally, this interim analysis will be scheduled at least 10 months after the first interim analysis, but no later than 28 months from start of enrollment.
• The final analysis is set for the targeted final analysis time or targeted events at the final analysis, whichever is later. It must be at least 6 months after IA 2.

Please keep in mind that if the cut is not event-driven, boundary updates are necessary; this is not covered here. You can find more information about the boundary updates in the boundary update vignette. In this vignette, we use the event-driven cut from above for illustrative purposes.

ia1_cut <- create_cut(
  planned_calendar_time = round(x$analysis$time[1]), 
  target_event_overall = x$analysis$event[1],
  max_extension_for_target_event = 16)

ia2_cut <- create_cut(
  planned_calendar_time = round(x$analysis$time[2]),
  target_event_overall = x$analysis$event[2],
  min_time_after_previous_analysis = 10, 
  max_extension_for_target_event = 28)

fa_cut <- create_cut(
  planned_calendar_time = round(x$analysis$time[3]),
  min_time_after_previous_analysis = 6,
  target_event_overall = x$analysis$event[3])

cut <- list(ia1 = ia1_cut, ia2 = ia2_cut, fa = fa_cut)

Step 2: Run sim_gs_n()

Now that we have set up the design characteristics in Step 1, we can proceed to run sim_gs_n() for a specified number of simulations. This function automatically utilizes a parallel computing backend, which helps reduce the running time.

n_sim <- 100 # Number of simulated trials
sim_res <- sim_gs_n(
  n_sim = n_sim,
  sample_size = sample_size, stratum = stratum, block = block,
  enroll_rate = enroll_rate, fail_rate = fail_rate,
  test = test, weight = weight, cut = cut)

The output of sim_gs_n is a data frame with one row per simulation per analysis. We show results for the first 2 simulated trials here. The estimate column is the sum(0 - E) for the logrank test; the se column is its standard error estimated under the null hypothesis. The z column is the test statistic for the logrank test (estimate / se). The info and info0 columns are the information at the current analysis under the alternate and null hypotheses, respectively.

sim_res |> head(n = 6) |> gt() |> tab_header("Overview Each Simulation results") |>
  fmt_number(columns = c(5, 8:12), decimals = 2)

Step 3: Summarize simulations

With the 100 simulations provided, users can summarize the simulated power and compare it to the target power of 90% as follows:

sim_res |>
  left_join(data.frame(analysis = 1:3, eff_bound = eff_bound)) |>
  group_by(analysis) |>
  summarize(`Mean time` = mean(cut_date), `sd(time)` = sd(cut_date), `Simulated power` = mean(z >= eff_bound)) |>
  ungroup() |>
  mutate(`Asymptotic power` = x$bound$probability[x$bound$bound == "upper"]) |>
  gt() |>
  tab_header("Summary of 100 simulations") |> 
  fmt_number(columns = 2, decimals = 1) |>
  fmt_number(columns = 3:5, decimals = 2)