library(SimDesign)
library(SimNPH)
library(parallel)

cl <- makeCluster(8)
clusterEvalQ(cl, {
  library(SimDesign)
  library(SimNPH)
  library(parallel)
})

A simple scenario with fixed followup

This is a simple example on how to run a simulation of analyses of datasets with a delayed treatment effect using the max-combo and the log-rank test. The

Setting up the Scenarios

Setting up the simulations to be run. createDesign creates a tibble with every combination of the parameters. Each line corresponds to one simulation to be run.

The function generate_delayed_effect needs the columns: n_trt: number of patients in the treatment arm, n_ctrl: number of patients in the control arm, delay: delay until onset of treatment effect, hazard_ctrl: hazard under control and before onset of treatment effect, hazart_trt: hazard under treatment, t_max: maximum time of observation.

An example design tibble with all parameters filled out can be created with desing_skeleton_delayed_effect. Use the function to output an example function call that you can copy and modify as needed, or assign the result to a variable to obtain a design tibble with some default parameters.

By default this will create a simulation design skeleton for simulations of 50 patients in each arm, a constant hazard of 0.2 under control and a hazard of 0.02 under treatment after the effect onset varying from 0 to 10.

N_sim <- 100

Assumptions <- assumptions_delayed_effect()
#> expand.grid(
#>   delay=m2d(seq(0, 10, by=2)), # delay of 0, 1, ..., 10 months
#>   hazard_ctrl=m2r(24),         # median survival control of 24 months
#>   hazard_trt=m2r(36),          # median survival treatment of 36 months
#>   random_withdrawal=m2r(120)   # median time to random withdrawal 10 years
#> )

Options <- design_fixed_followup()
#> expand.grid(
#>   n_trt=150,         # 150 patients in the treatment arm
#>   n_ctrl=150,        # 150 patients in the control arm
#>   followup=m2d(24),  # followup 2 years
#>   recruitment=m2d(6) # recruitment time 6 months
#> 
#> )

Design <- merge(Assumptions, Options, by=NULL)

knitr::kable(Design)
delay hazard_ctrl hazard_trt random_withdrawal n_trt n_ctrl followup recruitment
0.000 0.0009489 0.0006326 0.0001898 150 150 730.5 182.625
60.875 0.0009489 0.0006326 0.0001898 150 150 730.5 182.625
121.750 0.0009489 0.0006326 0.0001898 150 150 730.5 182.625
182.625 0.0009489 0.0006326 0.0001898 150 150 730.5 182.625
243.500 0.0009489 0.0006326 0.0001898 150 150 730.5 182.625
304.375 0.0009489 0.0006326 0.0001898 150 150 730.5 182.625

Defining the ‘Generate’ funcion

Define the data generating ‘generate’ function, that beside simulating the time to event also applies the different censoring processes.

my_generator <- function(condition, fixed_objects=NULL){
  generate_delayed_effect(condition, fixed_objects) |>
    recruitment_uniform(condition$recruitment) |>
    random_censoring_exp(condition$random_withdrawal) |>
    admin_censoring_time(condition$followup)
}

Defining the ‘Summarise’ function

Next, we need to specify a summary function that computes the desired operating characteristics for each simulation scenario, and each analysis method. In the example below, we use a summary function that computes the power (as we consider scenarios under the alternative in the example) of the log-rank test and the max-combo test across simulated scenarios. For each scenario the function just averages the number of times the computed p-value is below the significance level obtain the power.

The results object contains the results of all replications of the corresponding method for each row of the Design object. In this example results$p contains all N_sim p-values returned by the analyse_maxcombo or analyse_logrank functions respectively. The Summary will contain columns with the rejection rate and some other summary statistics for both methods.

alpha <- 0.05

Summarise <- create_summarise_function(
  maxcombo = summarise_test(alpha),
  logrank  = summarise_test(alpha)
)

Putting it all together

Now we put it all together: in Design we give the scenarios defined before. We want to run 100 replications for each scenario. We want to generate data using the generate_delayed_effect function using the parameters from Design and analyse each replication of each scenario with the two functions analyse_logrank and analyse_maxcombo. The output should be summarised with the Summarise function defined before and the simulations should be run in parallel.

res <- runSimulation(
  Design,
  replications = N_sim,
  generate = my_generator,
  analyse = list(
    logrank  = analyse_logrank(),
    maxcombo = analyse_maxcombo()
  ),
  summarise = Summarise,
  cl = cl,
  save=FALSE
)
#> 
#> Number of parallel clusters in use: 8
#> 
#> Simulation complete. Total execution time: 12.66s

Finally we select the interesting columns from the output. Since all other parameters are the same for each scenario we just select delay. And we are interested in the rejection rate of the tests.

res |> 
  subset(select=c("delay", "maxcombo.rejection_0.05", "logrank.rejection_0.05")) |>
  knitr::kable()
delay maxcombo.rejection_0.05 logrank.rejection_0.05
0.000 0.47 0.48
60.875 0.41 0.42
121.750 0.38 0.32
182.625 0.21 0.17
243.500 0.27 0.20
304.375 0.18 0.12

A scenario with an interim analysis

In this scenario we extend the scenario from above to include a fixed followup as well as an interim analysis after a fixed number of events. For this we will define additional analyse functions.

First we extend the Design to include a column with the number of events after which an interim analysis should be done.

Options <- design_group_sequential()
#> expand.grid(
#>   n_trt=200,          # 200 patients in the treatment arm
#>   n_ctrl=200,         # 200 patients in the control arm
#>   followup=m2d(48),   # maximum followup time 4 years
#>   recruitment=m2d(6), # recruitment time 6 months
#>   interim_events=150, # interim analysis after 150 events
#>   final_events=300    # final analysis after 300 events
#> )

Design <- merge(Assumptions, Options, by=NULL)

knitr::kable(Design)
delay hazard_ctrl hazard_trt random_withdrawal n_trt n_ctrl followup recruitment interim_events final_events
0.000 0.0009489 0.0006326 0.0001898 200 200 1461 182.625 150 300
60.875 0.0009489 0.0006326 0.0001898 200 200 1461 182.625 150 300
121.750 0.0009489 0.0006326 0.0001898 200 200 1461 182.625 150 300
182.625 0.0009489 0.0006326 0.0001898 200 200 1461 182.625 150 300
243.500 0.0009489 0.0006326 0.0001898 200 200 1461 182.625 150 300
304.375 0.0009489 0.0006326 0.0001898 200 200 1461 182.625 150 300

‘Analyse’ functions with an interim analysis

The analyse_group_sequential function allows to combine two or more analyse functions to create an analysis function corresponding to a group sequential design. The arguments are the times or events after which the analyses are done, the nominal alpha at each stage and the analyse functions to be used at each stage.

## O'Brien-Fleming Bounds for GSD with interim analysis at information time 1/2
nominal_alpha <- ldbounds::ldBounds(c(0.5,1))$nom.alpha

clusterExport(cl, "nominal_alpha")

Analyse <-  list(
  logrank_seq  = analyse_group_sequential(
    followup = c(condition$interim_events, condition$final_events),
    followup_type = c("event", "event"),
    alpha = nominal_alpha,
    analyse_functions = analyse_logrank()
  ),
  maxcombo_seq = analyse_group_sequential(
    followup = c(condition$interim_events, condition$final_events),
    followup_type = c("event", "event"),
    alpha = nominal_alpha,
    analyse_functions = analyse_maxcombo()
  )
)

A ‘Summarise’ function for the more complex scenario

The output of the function created with analyse_group_sequential contains additional columns. rejected_at_stage includes the stage at which the null was first rejected or Inf if the null was not rejected, N_pat and N_evt contain the number of patients recruited and the number of events observed before the null was rejected and followup contains the time after study start at which the last analysis was done.

The results object also includes the results returned by each stage in results_stages, but here we only use the overall test-decision.

Summarise <- create_summarise_function(
  maxcombo_seq = summarise_group_sequential(),
  logrank_seq = summarise_group_sequential()
)

Putting it all together

The call to runSimulation looks almost the same as above but now the additional columns we defined in our Summarise functions are included in the result.

res <- runSimulation(
  Design,
  replications = N_sim,
  generate = my_generator,
  analyse = Analyse,
  summarise = Summarise,
  cl = cl,
  save=FALSE
)
#> 
#> Number of parallel clusters in use: 8
#> 
#> Simulation complete. Total execution time: 28.55s

In the case of a group sequential design we are also interested in the average running time of the study in terms of patients recruited, number of events and running time of the study.

res |>
  subset(select=c(
    "delay", 
    "maxcombo_seq.rejection", "logrank_seq.rejection",
    "maxcombo_seq.n_pat", "logrank_seq.n_pat",
    "maxcombo_seq.n_evt", "logrank_seq.n_evt",
    "maxcombo_seq.followup", "logrank_seq.followup"
    )) |>
  knitr::kable()
delay maxcombo_seq.rejection logrank_seq.rejection maxcombo_seq.n_pat logrank_seq.n_pat maxcombo_seq.n_evt logrank_seq.n_evt maxcombo_seq.followup logrank_seq.followup
0.000 0.85 0.88 400 400 218.77 215.51 1323.78 1295.83
60.875 0.80 0.78 400 400 223.06 224.74 1337.55 1350.15
121.750 0.73 0.72 400 400 222.32 226.26 1313.66 1344.21
182.625 0.72 0.66 400 400 233.85 234.60 1406.49 1413.30
243.500 0.60 0.59 400 400 235.39 237.80 1406.64 1426.79
304.375 0.54 0.51 400 400 239.42 242.40 1426.88 1450.40

Estimation

Calculating the true values of the summary statistics

To evaluate the performance of an estimator, we first compute the values of some true summary statistics to which the estimates will be compared. The most relevant true summary statistics can be computed by a convenience function for each scenario. Just pipe the Design data.frame to the function and the values of the statistics are added as columns.

Options <- design_fixed_followup()
#> expand.grid(
#>   n_trt=150,         # 150 patients in the treatment arm
#>   n_ctrl=150,        # 150 patients in the control arm
#>   followup=m2d(24),  # followup 2 years
#>   recruitment=m2d(6) # recruitment time 6 months
#> 
#> )

Design <- merge(Assumptions, Options, by=NULL)

Design <- Design |> 
  true_summary_statistics_delayed_effect(cutoff_stats = 20)

knitr::kable(Design)
delay hazard_ctrl hazard_trt random_withdrawal n_trt n_ctrl followup recruitment median_survival_trt median_survival_ctrl rmst_trt_20 rmst_ctrl_20 gAHR_20 AHR_20 AHRoc_20 AHRoc_robust_20
0.000 0.0009489 0.0006326 0.0001898 150 150 730.5 182.625 1095.7500 730.5 19.87402 19.81142 0.6666667 0.6666667 0.6666667 0.6666667
60.875 0.0009489 0.0006326 0.0001898 150 150 730.5 182.625 1065.3125 730.5 19.81142 19.81142 1.0000000 1.0000000 1.0000000 1.0000000
121.750 0.0009489 0.0006326 0.0001898 150 150 730.5 182.625 1034.8750 730.5 19.81142 19.81142 1.0000000 1.0000000 1.0000000 1.0000000
182.625 0.0009489 0.0006326 0.0001898 150 150 730.5 182.625 1004.4375 730.5 19.81142 19.81142 1.0000000 1.0000000 1.0000000 1.0000000
243.500 0.0009489 0.0006326 0.0001898 150 150 730.5 182.625 974.0000 730.5 19.81142 19.81142 1.0000000 1.0000000 1.0000000 1.0000000
304.375 0.0009489 0.0006326 0.0001898 150 150 730.5 182.625 943.5625 730.5 19.81142 19.81142 1.0000000 1.0000000 1.0000000 1.0000000

Defining the Summarise function

In the Summarise function the true value against which the estimator should be compared has to be specified. If coverage and average width of the confidence intervals should be estimated, the CI bounds should also be specified.

The arguments to the functions are left un-evaluated and are later evaluated in the results and condition datasets respectively. So any expressions using variables from results can be used for the estimated value and the CI bounds and expressions using variables from condition can be used in the argument for the real value.

In this case we want to compare the hazard ratio estimated by the Cox model to the geometric average hazard ratio as well as to the hazard ratio after onset of treatment, calculated from the two respective columns of the Design data.frame.

Note that one name can be used twice to summarise the output of one analysis method two times, like in this case, comparing it to two different summary statistics.

Summarise <- create_summarise_function(
    coxph=summarise_estimator(hr, gAHR_20, hr_lower, hr_upper, name="gAHR"),
    coxph=summarise_estimator(hr, hazard_trt/hazard_ctrl, hr_lower, hr_upper, name="HR")
  )

Putting it all together

Analyse <- list(
  coxph = analyse_coxph()
)

res <- runSimulation(
  Design,
  replications = N_sim,
  generate = my_generator,
  analyse = Analyse,
  summarise = Summarise,
  cl = cl,
  save=FALSE
)
#> 
#> Number of parallel clusters in use: 8
#> 
#> Simulation complete. Total execution time: 0.95s
res |>
  subset(select=c(
    "delay", "coxph.HR.bias", "coxph.gAHR.bias", "coxph.HR.mse", 
    "coxph.gAHR.mse", "coxph.HR.coverage", "coxph.gAHR.coverage"
  )) |> 
  knitr::kable()
delay coxph.HR.bias coxph.gAHR.bias coxph.HR.mse coxph.gAHR.mse coxph.HR.coverage coxph.gAHR.coverage
0.000 0.0012685 0.0012685 0.0140868 0.0140868 0.96 0.96
60.875 0.0620345 -0.2712988 0.0208090 0.0905637 0.95 0.58
121.750 0.0891195 -0.2442138 0.0277615 0.0794596 0.94 0.64
182.625 0.1441876 -0.1891457 0.0455052 0.0604912 0.83 0.72
243.500 0.1762438 -0.1570895 0.0562535 0.0498688 0.79 0.82
304.375 0.2133027 -0.1200306 0.0726039 0.0415132 0.72 0.87