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Simulation Workflow for OLE Phase

Lei Shi

2026-05-06

Source: vignettes/OLE_simulation_workflow.Rmd
OLE_simulation_workflow.Rmd

OLE study

This vignette demonstrates Monte Carlo simulation for evaluating the difference-in-differences (DID) estimators proposed in Zhou et al. (2024) for the open-label extension (OLE) phase.

1. Simulate a dataset for OLE study 

# Initialize an empty data list
set.seed(2023)

data_matrix_list <- list()
ntrial <- 3

# Specify the significance level alpha
alpha <- 0.05

# Specify the true effect size at the end of the OLE study
true_effect_long <- 0

# Specify column names
covariates_col_name <- c("x1", "x2", "x3", "x4", "x5")
outcome_col_name <- c("y1", "y2", "y3", "y4")
treatment_col_name <- "A"
trial_status_col_name <- "S"
# Sequentially adding in datasets
for (trial_iter in 1:ntrial) {
  # simulate 300 sample
  normal <- copula::normalCopula(param = c(0.8), dim = 4, dispstr = "ar1")

  # ========== generate internal covariates =============
  X_int <- simulate_X_copula(
    n = 200,
    p = 4,
    cp = normal, # copula
    margins = c("binom", "binom", "binom", "exp"), # specify marginal distributions
    paramMargins = list(
      list(size = 1, prob = 0.7), # specify parameters for marginals
      list(size = 1, prob = 0.9),
      list(size = 1, prob = 0.3),
      list(rate = 1 / 10)
    )
  )

  X_int$x4 <- round(X_int$x4) + 1
  X_int$x5 <- 30 + 10 * X_int$x1 + (7) * X_int$x2 + (-6) * X_int$x3 + (-0.5) * X_int$x4 + rnorm(200, mean = 0, sd = 10)

  varnames <- c("1", paste0("x", 1:5))

  # ============ generate external covariates ==============
  X_ext <- simulate_X_copula(
    n = 100,
    p = 4,
    cp = normal, # copula
    margins = c("binom", "binom", "binom", "exp"), # specify marginal distributions
    paramMargins = list(
      list(size = 1, prob = 0.7), # specify parameters for marginals
      list(size = 1, prob = 0.9),
      list(size = 1, prob = 0.3),
      list(rate = 1 / 10)
    )
  )

  X_ext$x4 <- round(X_ext$x4) + 1
  X_ext$x5 <- 50 + 10 * X_ext$x1 + (2) * X_ext$x2 + (-1) * X_ext$x3 + (-0.3) * X_ext$x4 + rnorm(100, mean = 0, sd = 10)

  varnames <- c("1", paste0("x", 1:5))

  # ============ Specify outcome models ==============
  model_form_x_t1 <- setNames(c(10.0, 0.05, -1.5, -1.0, -0.2, -0.1), varnames) # 1.5*A, sigma = 4.0
  model_form_x_t2 <- setNames(c(6.0, 0.5, -0.5, -1.0, -0.3, -0.06), varnames) # 1.8*A, sigma = 4.0
  model_form_x_t3 <- setNames(c(5.0, 1.9, 1.4, -1.3, -0.4, -0.15), varnames) # 1.6*A, sigma = 4.0
  model_form_x_t4 <- setNames(c(1.2, 1.0, 2.0, -0.5, -0.4, -0.10), varnames) # 2.5*A, sigma = 5.0

  outcome_model_specs <- list(
    list(
      effect = 0, model_form_x = model_form_x_t1, # from data: true_effect = 1.5
      noise_mean = 0, noise_sd = 4
    ), # model form for the first time point, given by model_form_x_t1
    list(
      effect = 0, model_form_x = model_form_x_t2, # from data: true_effect = 1.8
      noise_mean = 0, noise_sd = 4
    ), # model form for the second time point, given by model_form_x_t2
    list(
      effect = 0, model_form_x = model_form_x_t3, # from data: true_effect = 1.6
      noise_mean = 0, noise_sd = 4
    ), # model form for the third time point, given by model_form_x_t3
    list(
      effect = true_effect_long, model_form_x = model_form_x_t4, # from data: true_effect = 2.5
      noise_mean = 0, noise_sd = 4
    ) # model form for the fourth time point, given by model_form_x_t4
  )

  # =========== generate trial data ============
  Data <- simulate_trial(X_int,
    X_ext,
    num_treated = 150,
    OLE_flag = TRUE,
    T_cross = 2,
    outcome_model_specs
  )


  data_matrix_list[[trial_iter]] <- Data
}

2 Bootstrap inference 

# Specify a list of methods to be tested
bootstrap_obj <- setup_bootstrap(
  replicates = 50,
  bootstrap_CI_type = "perc"
)

model_form_mu <- c(
  "y1 ~ x1 + x2 + x3 + x4 + x5",
  "y2 ~ x1 + x2 + x3 + x4 + x5",
  "y3 ~ x1 + x2 + x3 + x4 + x5",
  "y4 ~ x1 + x2 + x3 + x4 + x5"
)

## IPW - optimal weight
method_IPW_DID <- setup_method_DID(
  method_name = "IPW",
  bootstrap_flag = TRUE,
  bootstrap_obj = bootstrap_obj,
  model_form_piS = "S ~ x1 + x2 + x3 + x4 + x5",
  model_form_piA = "A ~ x1 + x2 + x3 + x4 + x5"
)

## AIPW - optimal weight
method_AIPW_DID <- setup_method_DID(
  method_name = "AIPW",
  bootstrap_flag = TRUE,
  bootstrap_obj = bootstrap_obj,
  model_form_piS = "S ~ x1 + x2 + x3 + x4 + x5",
  model_form_piA = "A ~ x1 + x2 + x3 + x4 + x5",
  model_form_mu0_ext = model_form_mu
)

## OR - optimal weight

method_OR_DID <- setup_method_DID(
  method_name = "OR",
  bootstrap_flag = TRUE,
  bootstrap_obj = bootstrap_obj,
  model_form_mu0_ext = model_form_mu,
  model_form_mu0_rct = model_form_mu,
  model_form_mu1_rct = model_form_mu
)

## method list
method_obj_list <- list(
  method_IPW_DID,
  method_AIPW_DID,
  method_OR_DID
)
# create a simulation object for primary analysis
simulation_OLE_obj <- setup_simulation_OLE(
  data_matrix_list = data_matrix_list, # two scenarios
  trial_status_col_name = trial_status_col_name,
  treatment_col_name = treatment_col_name,
  outcome_col_name = outcome_col_name,
  covariates_col_name = covariates_col_name,
  method_obj_list = method_obj_list,
  true_effect = true_effect_long,
  T_cross = 2,
  alpha = alpha,
  method_description = c(
    "IPW, DID",
    "AIPW, DID",
    "OR, DID"
  )
)
simulation_report <- run_simulation(simulation_OLE_obj, quiet = FALSE)
#> Null:  | method setting:  1 | data setting:  1 
#> Null:  | method setting:  1 | data setting:  2 
#> Null:  | method setting:  1 | data setting:  3 
#> Null:  | method setting:  2 | data setting:  1 
#> Null:  | method setting:  2 | data setting:  2 
#> Null:  | method setting:  2 | data setting:  3 
#> Null:  | method setting:  3 | data setting:  1 
#> Null:  | method setting:  3 | data setting:  2 
#> Null:  | method setting:  3 | data setting:  3
simulation_report # Type I error and Power
#>   method_description      bias  variance      mse coverage type_I_error
#> 1           IPW, DID 0.7608010 2.8208371 3.399655        1            0
#> 2          AIPW, DID 1.6168995 2.8917875 5.506151        1            0
#> 3            OR, DID 0.7888743 0.5824244 1.204747        1            0
# simulation hypo test

The above code generates a report for the simulation results.

References

  • Zhou X, Pang H, Drake C, Burger HU, Zhu J (2024). “Estimating treatment effect in randomized trial after control to treatment crossover using external controls.” Journal of Biopharmaceutical Statistics. doi: 10.1080/10543406.2024.2444222.
  • Shi L, Pang H, Chen C, Zhu J (2025). “rdborrow: an R package for causal inference incorporating external controls in randomized controlled trials with longitudinal outcomes.” Journal of Biopharmaceutical Statistics, 35(6), 1043-1066. doi: 10.1080/10543406.2025.2489283.