--- title: "Simulation Workflow for OLE Phase" author: "Lei Shi, Matthew Secrest" date: "`r Sys.Date()`" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{Simulation Workflow for OLE Phase} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- ```{r, message=FALSE, warning=FALSE, include=FALSE} knitr::opts_chunk$set( collapse = TRUE, comment = "#>" ) ``` ```{r setup, message=FALSE, warning=FALSE, include=FALSE} library(rdborrow) ``` ## OLE study This vignette demonstrates Monte Carlo simulation for evaluating the difference-in-differences (DID) estimators proposed in [Zhou et al. (2024)](https://doi.org/10.1080/10543406.2024.2444222) for the open-label extension (OLE) phase. The simulated data has four outcome columns (`y1`, `y2`, `y3`, `y4`) and a crossover point `T_cross = 2`. This means: - **`y1`, `y2`** (columns 1 to `T_cross`): placebo-controlled phase (Period I). Both RCT controls and external controls are untreated. These outcomes serve as negative controls for bias correction. - **`y3`, `y4`** (columns `T_cross + 1` to end): open-label extension phase (Period II). RCT controls have switched to treatment; external controls remain untreated. Treatment effects are estimated here. In other words, `T_cross` is the last index of the placebo-controlled phase in `outcome_col_name`. ```{r} head(SyntheticData) ``` ### 1. Simulate a dataset for OLE study ```{r} # Initialize an empty data list set.seed(2023) data_matrix_list <- list() # a small ntrial keeps this vignette fast to build; use hundreds to # thousands of trials for stable operating characteristic estimates ntrial <- 20 # 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" ``` ```{r} # 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 ```{r} 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" ) method_IPW_DID <- did_ec_ipw( ps_formula = "S ~ x1 + x2 + x3 + x4 + x5", trt_formula = "A ~ x1 + x2 + x3 + x4 + x5", bootstrap = 50, bootstrap_ci_type = "perc" ) method_AIPW_DID <- did_ec_aipw( ps_formula = "S ~ x1 + x2 + x3 + x4 + x5", trt_formula = "A ~ x1 + x2 + x3 + x4 + x5", outcome_formula = model_form_mu, bootstrap = 50, bootstrap_ci_type = "perc" ) method_OR_DID <- did_ec_or( outcome_formula_ext = model_form_mu, outcome_formula_rct_ctrl = model_form_mu, outcome_formula_rct_trt = model_form_mu, bootstrap = 50, bootstrap_ci_type = "perc" ) method_obj_list <- list( method_IPW_DID, method_AIPW_DID, method_OR_DID ) ``` ```{r} # 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" ) ) ``` ```{r message=FALSE, warning=FALSE} simulation_report <- run_simulation(simulation_OLE_obj, quiet = TRUE) ``` ```{r} simulation_report ``` ## 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](https://doi.org/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](https://doi.org/10.1080/10543406.2025.2489283).