Group Sequential Simulation with Completers Analysis

This vignette demonstrates how to simulate a group sequential design where an interim analysis is conducted based on a specific number of “completers” (subjects who have finished their follow-up).

Simulation Setup

We define a trial with the following parameters:

Sample Size: 200 patients.
Enrollment: Recruited over 12 months.
Follow-up: 2 years maximum follow-up per patient.
Interim Analysis: Conducted when 40% of patients (80 subjects) have completed their 2-year follow-up. The interim analysis includes all data available at that time (completers and partial follow-up).
Final Analysis: Conducted when all patients have completed follow-up (or dropped out). Includes all available data.

# Parameters
n_total <- 200
enroll_duration <- 12 # months
max_followup <- 12 # months (using months as time unit for clarity)

# Convert to years if rates are annual, but let's stick to consistent units.
# Let's say rates are per YEAR, so we convert time to years.
# Time unit: Year
n_total <- 200
enroll_duration <- 1 # 1 year
max_followup <- 1 # 1 year

enroll_rate <- data.frame(
  rate = n_total / enroll_duration, 
  duration = enroll_duration
)

fail_rate <- data.frame(
  treatment = c("Control", "Experimental"),
  rate = c(0.5, 0.35) # Events per year
)

dropout_rate <- data.frame(
  treatment = c("Control", "Experimental"),
  rate = c(0.05, 0.05),
  duration = c(100, 100)
)

Simulation Loop

We will simulate 50 trials. For each trial, we perform the interim and final analyses.

set.seed(2024)
n_sims <- 50
results <- data.frame(
  sim_id = integer(n_sims),
  interim_date = numeric(n_sims),
  interim_z = numeric(n_sims),
  interim_n = integer(n_sims),
  interim_info = numeric(n_sims),
  final_date = numeric(n_sims),
  final_z = numeric(n_sims),
  final_n = integer(n_sims),
  final_info = numeric(n_sims),
  info_frac = numeric(n_sims)
)

# Target completers for interim (40%)
target_completers <- 0.4 * n_total

for (i in 1:n_sims) {
  # 1. Simulate Trial Data
  sim_data <- nb_sim(
    enroll_rate = enroll_rate,
    fail_rate = fail_rate,
    dropout_rate = dropout_rate,
    max_followup = max_followup,
    n = n_total
  )
  
  # 2. Interim Analysis
  # Find date when target_completers is reached
  interim_date <- cut_date_for_completers(sim_data, target_completers)
  
  # Cut data for completers at this date
  data_interim <- cut_completers(sim_data, interim_date)
  
  # Analyze (Mütze Test)
  res_interim <- mutze_test(data_interim)
  # Extract Z-statistic
  z_interim <- res_interim$z
  # Extract Information
  info_interim <- 1 / res_interim$se^2
  
  # 3. Final Analysis (All Data)
  # Date is when last patient completes (or max follow-up reached)
  # For final analysis, we use all data collected up to the end of the study.
  # The end of the study is when the last patient reaches max_followup.
  date_final <- max(sim_data$calendar_time)
  
  # Cut data at final date (includes partial follow-up for dropouts, full for completers)
  data_final <- cut_data_by_date(sim_data, date_final)
  
  res_final <- mutze_test(data_final)
  z_final <- res_final$z
  # Extract Information
  info_final <- 1 / res_final$se^2
  
  # Store results
  results$sim_id[i] <- i
  results$interim_date[i] <- interim_date
  results$interim_z[i] <- z_interim
  results$interim_n[i] <- nrow(data_interim)
  results$interim_info[i] <- info_interim
  results$final_date[i] <- date_final
  results$final_z[i] <- z_final
  results$final_n[i] <- nrow(data_final)
  results$final_info[i] <- info_final
  results$info_frac[i] <- info_interim / info_final
}

Results Summary

We summarize the distribution of the test statistics (Z-scores) at the interim and final analyses.

summary(results[, c("interim_date", "interim_z", "interim_info", "final_date", "final_z", "final_info", "info_frac")])
#>   interim_date     interim_z        interim_info     final_date   
#>  Min.   :1.329   Min.   :-4.0441   Min.   :10.67   Min.   :1.883  
#>  1st Qu.:1.390   1st Qu.:-2.3709   1st Qu.:13.73   1st Qu.:1.970  
#>  Median :1.411   Median :-1.5500   Median :15.42   Median :2.015  
#>  Mean   :1.420   Mean   :-1.5144   Mean   :15.59   Mean   :2.014  
#>  3rd Qu.:1.455   3rd Qu.:-0.7135   3rd Qu.:17.59   3rd Qu.:2.061  
#>  Max.   :1.536   Max.   : 1.1014   Max.   :20.24   Max.   :2.220  
#>     final_z          final_info      info_frac     
#>  Min.   :-4.0901   Min.   :14.58   Min.   :0.6962  
#>  1st Qu.:-2.2880   1st Qu.:17.78   1st Qu.:0.7631  
#>  Median :-1.3878   Median :18.75   Median :0.8239  
#>  Mean   :-1.5391   Mean   :19.11   Mean   :0.8145  
#>  3rd Qu.:-0.9025   3rd Qu.:20.99   3rd Qu.:0.8474  
#>  Max.   : 1.0082   Max.   :23.67   Max.   :0.9303

Visualization

Comparison of Z-scores at Interim vs Final Analysis.


# Correlation between interim and final Z-scores
cor_z <- cor(results$interim_z, results$final_z)
cat("Correlation between interim and final Z-scores:", round(cor_z, 3), "\n")
#> Correlation between interim and final Z-scores: 0.905

ggplot(results, aes(x = interim_z, y = final_z)) +
  geom_point(alpha = 0.7) +
  geom_abline(intercept = 0, slope = 1, linetype = "dashed", color = "gray") +
  labs(
    title = paste0("Z-Scores: Interim vs Final Analysis (Cor = ", round(cor_z, 3), ")"),
    x = "Interim Z-Score (Completers Only)",
    y = "Final Z-Score (Full Data)"
  ) +
  theme_minimal()

The plot shows the correlation between the interim statistic (based on 40% completers) and the final statistic.