Author: Yujie Zhao and Keaven Anderson

# Simulate Fixed Designs with Ease via sim_fixed_n

#### Yujie Zhao and Keaven Anderson

    library(gsDesign2)
    library(simtrial)
    library(dplyr)
    library(gt)

    set.seed(2027)

The `sim_fixed_n()` function simulates a two-arm trial with a single endpoint, accounting for time-varying enrollment, hazard ratios, and failure and dropout rates.

While there are limitations, there are advantages of calling `sim_fixed_n()` directly:

- It is simple, which allows for a single function call to perform an arbitrary number of simulations.
- It automatically implements a parallel computating backend to reduce running time.
- It offers up to 5 options for determining the data cutoff for analysis through the `timing_type` parameter.

If people are interested in more complicated simulations, please refer to the vignette Custom Fixed Design Simulations: A Tutorial on Writing Code from the Ground Up.

The process for simulating via `sim_fixed_n()` is outlined in Steps 1 to 3 below.

# Step 1: Define design parameters

To run simulations for a fixed design, several design characteristics may be used. Depending on the data cutoff for analysis option, different inputs may be required. The following lines of code specify an unstratified 2-arm trial with equal randomization. The simulation is repeated 2 times. Enrollment is targeted to last for 12 months at a constant enrollment rate. The median for the control arm is 10 months, with a delayed effect during the first 3 months followed by a hazard ratio of 0.7 thereafter. There is an exponential dropout rate of 0.001 over time.

    n_sim <- 100
    total_duration <- 36
    stratum <- data.frame(stratum = "All", p = 1)
    block <- rep(c("experimental", "control"), 2)

    enroll_rate <- data.frame(stratum = "All", rate = 12, duration = 500 / 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)

We specify sample size and targeted event count based on the `fixed_design_ahr()` function and the above options. The following design computes the sample size and targeted event counts for 85% power. In this approach, users can obtain the sample size and targeted events from the output of `x`, specifically by using `sample_size <- x$analysis$n` and `target_event <- x$analysis$event`.

    x <- fixed_design_ahr(enroll_rate = enroll_rate, fail_rate = fail_rate,
                          alpha = 0.025, power = 0.85, ratio = 1,
                          study_duration = total_duration) |> to_integer()
    x |> summary() |> gt() |>
      tab_header(title = "Sample Size and Targeted Events Based on AHR Method",
                 subtitle = "Fixed Design with 85% Power, One-sided 2.5% Type I error") |>
      fmt_number(columns = c(4, 5, 7), decimals = 2)

Sample Size and Targeted Events Based on AHR Method
Fixed Design with 85% Power, One-sided 2.5% Type I error
Design | N | Events | Time | AHR | Bound | alpha | Power
--- | --- | --- | --- | --- | --- | --- | ---
Average hazard ratio | 516 | 295 | 36.01 | 0.70 | 1.959964 | 0.03 | 0.8504588

Now we set the derived targeted sample size, enrollment rate, and event count from the above.

    sample_size <- x$analysis$n
    target_event <- x$analysis$event
    enroll_rate <- x$enroll_rate

# Step 2: Run `sim_fixed_n()`

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

The `timing_type` specifies one or more of the following cutoffs for setting the time for analysis:

- `timing_type = 1`: The planned study duration.
- `timing_type = 2`: The time until target event count has been observed.
- `timing_type = 3`: The planned minimum follow-up period after enrollment completion.
- `timing_type = 4`: The maximum of planned study duration and time to observe the targeted event count (i.e., using `timing_type = 1` and `timing_type = 2` together).
- `timing_type = 5`: The maximum of time to observe the targeted event count and minimum follow-up following enrollment completion (i.e., using `timing_type = 2` and `timing_type = 3` together).

The `rho_gamma` argument is a data frame containing the variables `rho` and `gamma`, both of which should be greater than or equal to zero, to specify one Fleming-Harrington weighted log-rank test per row. For instance, setting `rho = 0` and `gamma = 0` yields the standard unweighted log-rank test, while `rho = 0 and gamma = 0.5` provides the weighted Fleming-Harrington (0, 0.5) log-rank test. If you are interested in tests other than the Fleming-Harrington weighted log-rank test, please refer to the vignette at “Articles”/“Simulate fixed/group sequential designs”/“Custom Fixed Design Simulations: A Tutorial on Writing Code from the Ground Up”.

    sim_res <- sim_fixed_n(
      n_sim = 2, # only use 2 simulations for initial run
      sample_size = sample_size,
      block = block,
      stratum = stratum,
      target_event = target_event,
      total_duration = total_duration,
      enroll_rate = enroll_rate,
      fail_rate = fail_rate,
      timing_type = 1:5,
      rho_gamma = data.frame(rho = 0, gamma = 0))

The output of `sim_fixed_n` is a data frame with one row per simulated dataset per cutoff specified in `timing_type`, per test statistic specified in `rho_gamma`. Here we have just run 2 simulated trials and see how the different cutoffs vary for the 2 trial instances.

    sim_res |>
      gt() |>
      tab_header("Tests for Each Simulation Result", subtitle = "Logrank Test for Different Analysis Cutoffs") |>
      fmt_number(columns = c(4, 5, 7), decimals = 2)

Tests for Each Simulation Result
Logrank Test for Different Analysis Cutoffs
method | parameter | estimate | se | z | event | ln_hr | cut | duration | sim
--- | --- | --- | --- | --- | --- | --- | --- | --- | ---
WLR | FH(rho=0, gamma=0) | -31.52010 | 8.38 | 3.76 | 287 | −0.45 | Planned duration | 36.00000 | 1
WLR | FH(rho=0, gamma=0) | -32.12426 | 8.49 | 3.78 | 295 | −0.45 | Targeted events | 36.66316 | 1
WLR | FH(rho=0, gamma=0) | -29.13247 | 8.25 | 3.53 | 278 | −0.43 | Minimum follow-up | 35.20859 | 1
WLR | FH(rho=0, gamma=0) | -32.12426 | 8.49 | 3.78 | 295 | −0.45 | Max(planned duration, event cut) | 36.66316 | 1
WLR | FH(rho=0, gamma=0) | -32.12426 | 8.49 | 3.78 | 295 | −0.45 | Max(min follow-up, event cut) | 36.66316 | 1
WLR | FH(rho=0, gamma=0) | -31.88270 | 8.00 | 3.99 | 268 | −0.50 | Planned duration | 36.00000 | 2
WLR | FH(rho=0, gamma=0) | -38.09785 | 8.40 | 4.54 | 295 | −0.54 | Targeted events | 38.32488 | 2
WLR | FH(rho=0, gamma=0) | -34.29092 | 8.23 | 4.17 | 283 | −0.51 | Minimum follow-up | 37.47843 | 2
WLR | FH(rho=0, gamma=0) | -38.09785 | 8.40 | 4.54 | 295 | −0.54 | Max(planned duration, event cut) | 38.32488 | 2
WLR | FH(rho=0, gamma=0) | -38.09785 | 8.40 | 4.54 | 295 | −0.54 | Max(min follow-up, event cut) | 38.32488 | 2

# Step 3: Summarize simulations

Now we run 100 simulated trials and summarize the results by how data is cutoff for analysis.

    sim_res <- sim_fixed_n(
      n_sim = n_sim,
      sample_size = sample_size,
      block = block, stratum = stratum,
      target_event = target_event,
      total_duration = total_duration,
      enroll_rate = enroll_rate,
      fail_rate = fail_rate,
      timing_type = 1:5,
      rho_gamma = data.frame(rho = 0, gamma = 0))

With the 100 simulations provided, users can summarize the simulated power and compare it to the targeted 85% power. All cutoff methods approximate the targeted power well and have a similar average duration and mean number of events.

    sim_res |>
      group_by(cut) |>
      summarize(`Simulated Power` = mean(z > qnorm(1 - 0.025)),
                `Mean events` = mean(event),
                `Mean duration` = mean(duration)) |>
      mutate(`Sample size` = sample_size,
             `Targeted events` = target_event) |>
      gt() |>
      tab_header(title = "Summary of 100 simulations by 5 different analysis cutoff methods",
                 subtitle = "Tested by logrank") |>
      fmt_number(columns = c(2:4), decimals = 2)

Summary of 100 simulations by 5 different analysis cutoff methods
Tested by logrank
cut | Simulated Power | Mean events | Mean duration | Sample size | Targeted events
--- | --- | --- | --- | --- | ---
Max(min follow-up, event cut) | 0.85 | 299.35 | 36.39 | 516 | 295
Max(planned duration, event cut) | 0.85 | 301.54 | 36.59 | 516 | 295
Minimum follow-up | 0.84 | 291.57 | 35.78 | 516 | 295
Planned duration | 0.84 | 294.74 | 36.00 | 516 | 295
Targeted events | 0.85 | 295.00 | 36.00 | 516 | 295

We can also do things like summarize distribution of event counts at the planned study duration. We can see the event count varies a fair amount.

    hist(sim_res$event[sim_res$cut == "Planned duration"],
         breaks = 10,
         main = "Distribution of Event Counts at Planned Study Duration",
         xlab = "Event Count at Targeted Trial Duration")

We also evaluate the distribution of the trial duration when analysis is performed when the targeted events are achieved.

    plot(density(sim_res$duration[sim_res$cut == "Targeted events"]),
         main = "Trial Duration Smoothed Density",
         xlab = "Trial duration when Targeted Event Count is Observed")