Author: Gosuke Homma
Date: 2025-06-18

# Introduction to bbssr: Blinded Sample Size Re-estimation for Binary Endpoints

#### Gosuke Homma

#### 2025-06-18

## Introduction

The `bbssr` package provides comprehensive tools for implementing blinded sample size re-estimation (BSSR) in two-arm clinical trials with binary endpoints. This vignette introduces the key concepts and demonstrates how to use the main functions in the package.

## What is Blinded Sample Size Re-estimation?

Traditional clinical trials use fixed sample sizes determined during the planning phase. However, these calculations often rely on assumptions about treatment effects and response rates that may prove incorrect. BSSR addresses this limitation by allowing adaptive sample size adjustments during the trial while maintaining:

- Statistical validity through exact Type I error control
- Study blinding by using only pooled response data
- Regulatory acceptance through well-established methodology

## Key Advantages of BSSR

1. Maintains Blinding: Only pooled response rates are used, preserving treatment allocation concealment
2. Exact Statistical Control: Uses exact tests rather than asymptotic approximations
3. Flexible Designs: Supports different adaptation rules (restricted/unrestricted/weighted)
4. Regulatory Compliance: Based on established statistical methodology accepted by regulatory agencies

## Getting Started

    library(bbssr)
    library(dplyr)
    library(ggplot2)

## Basic Usage

### Power Calculation for Traditional Design

Let’s start with a basic power calculation for a traditional fixed-sample design:

    # Calculate power for a traditional design
    power_result <- BinaryPower(
      p1 = 0.5,          # Response rate in treatment group
      p2 = 0.2,          # Response rate in control group
      N1 = 40,           # Sample size in treatment group
      N2 = 40,           # Sample size in control group
      alpha = 0.025,     # One-sided significance level
      Test = 'Fisher'    # Fisher exact test
    )

    print(paste("Power:", round(power_result, 3)))
    #> [1] "Power: 0.749"

### Sample Size Calculation

Now let’s calculate the required sample size for a desired power:

    # Calculate required sample size
    sample_size_result <- BinarySampleSize(
      p1 = 0.5,           # Expected response rate in treatment group
      p2 = 0.2,           # Expected response rate in control group
      r = 1,              # Allocation ratio (1:1)
      alpha = 0.025,      # One-sided significance level
      tar.power = 0.8,    # Target power
      Test = 'Boschloo'   # Boschloo exact test (most powerful)
    )

    print(sample_size_result)
    #>    p1  p2 r alpha tar.power     Test     Power N1 N2  N
    #> 1 0.5 0.2 1 0.025       0.8 Boschloo 0.8096508 40 40 80

## Blinded Sample Size Re-estimation (BSSR)

### Basic BSSR Example

Here’s how to perform BSSR analysis:

    # Basic BSSR calculation
    bssr_result <- BinaryPowerBSSR(
      asmd.p1 = 0.45,      # Assumed response rate in treatment group
      asmd.p2 = 0.09,      # Assumed response rate in control group
      p = seq(0.1, 0.9, by = 0.1), # Range of pooled response rates
      Delta.A = 0.36,      # Assumed treatment effect (risk difference)
      Delta.T = 0.36,      # True treatment effect
      N1 = 24,             # Initial sample size in treatment group
      N2 = 24,             # Initial sample size in control group
      omega = 0.5,         # Fraction of data for interim analysis
      r = 1,               # Allocation ratio
      alpha = 0.025,       # Significance level
      tar.power = 0.8,     # Target power
      Test = 'Z-pool',     # Statistical test
      restricted = FALSE,  # Unrestricted design
      weighted = FALSE     # Non-weighted approach
    )

    # Display first few rows
    head(bssr_result)
    #>     p1   p2   p power.BSSR power.TRAD
    #> 1 0.38 0.02 0.2  0.8728849  0.9310364
    #> 2 0.48 0.12 0.3  0.7714587  0.7880436
    #> 3 0.58 0.22 0.4  0.7816618  0.7168506
    #> 4 0.68 0.32 0.5  0.7851850  0.6677197
    #> 5 0.78 0.42 0.6  0.7816618  0.7168506
    #> 6 0.88 0.52 0.7  0.7714587  0.7880436

### Comparing BSSR Design Rules

One of the key features of `bbssr` is the ability to compare different BSSR design rules. Here’s the comprehensive comparison as requested:

    # Calculate power of the BSSR with binary endpoint
    power.BSSR <- tibble(
      Rule = factor(
        c('Restricted', 'Unrestricted', 'Weighted'),
        levels = c('Restricted', 'Unrestricted', 'Weighted')
      ),
      restricted = c(TRUE, FALSE, FALSE),
      weighted = c(FALSE, FALSE, TRUE)
    ) %>%
      group_by_all() %>%
      reframe(r = c(1, 2), N1 = c(24, 36), N2 = c(24, 18)) %>%
      group_by_all() %>%
      reframe(
        BinaryPowerBSSR(
          asmd.p1 = 0.45, asmd.p2 = 0.09, p = seq(0, 1, by = 0.01),
          Delta.A = 0.36, Delta.T = 0.36, N1, N2, omega = 0.5, r,
          alpha = 0.025, tar.power = 0.8, Test = 'Z-pool',
          restricted, weighted
        )
      ) %>%
      mutate(
        Rule = factor(Rule, levels = c('Restricted', 'Unrestricted', 'Weighted')),
        r_label = paste0('Allocation ratio = ', r, ':1')
      ) %>%
      rename(Theta = p, Power = power.BSSR)

    # Create the improved figure
    power.BSSR %>%
      ggplot(aes(x = Theta, y = Power)) +
      geom_line(aes(color = Rule), linewidth = 1.0) +
      theme_bw() +
      facet_grid(. ~ r_label) +
      geom_hline(yintercept = 0.8, color = 'gray', linetype = 'longdash', linewidth = 1.0) +
      scale_x_continuous(
        limits = c(0.1, 0.9),
        breaks = seq(0.2, 0.8, by = 0.2),  # Fewer x-axis breaks
        labels = c("0.2", "0.4", "0.6", "0.8")
      ) +
      scale_y_continuous(
        limits = c(0.7, 1.0),
        breaks = seq(0.7, 1.0, by = 0.1),
        labels = c("0.7", "0.8", "0.9", "1.0")
      ) +
      scale_color_manual(
        values = c("Restricted" = "#E31A1C", "Unrestricted" = "#1F78B4", "Weighted" = "#33A02C")
      ) +
      labs(
        x = expression(theta),
        y = "Power",
        title = "Power Comparison: BSSR Design Rules",
        subtitle = "Dashed line indicates target power = 0.8"
      ) +
      theme(
        text = element_text(size = 11),
        plot.title = element_text(size = 13, hjust = 0.5, margin = margin(b = 5)),
        plot.subtitle = element_text(size = 10, hjust = 0.5, margin = margin(b = 10)),
        panel.spacing = unit(0.8, 'lines'),
        legend.position = 'bottom',
        legend.key.width = unit(1.2, 'cm'),
        legend.text = element_text(size = 10),
        legend.title = element_blank(),
        legend.margin = margin(t = 8, r = 0, b = 0, l = 0),
        axis.title.x = element_text(size = 11, margin = margin(t = 8)),
        axis.title.y = element_text(size = 11, margin = margin(r = 8)),
        axis.text = element_text(size = 9),
        strip.text = element_text(size = 10),
        plot.margin = margin(t = 10, r = 10, b = 5, l = 5)
      )

## Understanding the Results

### Design Rule Comparison

The plot above shows three different BSSR approaches:

- Restricted Design: Conservative approach where final sample size ≥ initial sample size
- Unrestricted Design: Flexible approach allowing both increases and decreases
- Weighted Design: Advanced approach using weighted averaging across scenarios

### Key Observations

1. All designs maintain target power: The horizontal dashed line shows the target power of 0.8
2. Weighted design shows robust performance: More consistent power across different pooled response rates
3. Allocation ratio matters: Different allocation ratios (r=1 vs r=2) show different patterns

## Statistical Tests Available

The package supports five different exact statistical tests:

Test | Code | Description | Characteristics
--- | --- | --- | ---
Pearson chi-squared | `'Chisq'` | One-sided exact test | Good for large samples
Fisher exact | `'Fisher'` | Classical conditional test | Conservative, widely accepted
Fisher mid-p | `'Fisher-midP'` | Less conservative Fisher | Better power than Fisher exact
Z-pooled | `'Z-pool'` | Unconditional exact test | Good compromise
Boschloo | `'Boschloo'` | Most powerful unconditional | Highest power, computationally intensive

### Test Comparison Example

    # Compare different statistical tests
    tests <- c('Chisq', 'Fisher', 'Fisher-midP', 'Z-pool', 'Boschloo')
    test_results <- sapply(tests, function(test) {
      BinaryPower(p1 = 0.5, p2 = 0.2, N1 = 30, N2 = 30, alpha = 0.025, Test = test)
    })

    test_comparison_df <- data.frame(
      Test = tests,
      Power = round(test_results, 3)
    )

    print(test_comparison_df)
    #>                    Test Power
    #> Chisq             Chisq 0.692
    #> Fisher           Fisher 0.596
    #> Fisher-midP Fisher-midP 0.679
    #> Z-pool           Z-pool 0.678
    #> Boschloo       Boschloo 0.671

## Practical Considerations

### When to Use BSSR

BSSR is particularly valuable when: - Initial assumptions about response rates are uncertain - Regulatory efficiency is important - Maintaining study blinding is critical - Exact statistical control is required

### Design Choice Guidelines

- Restricted: Use when regulatory/logistical constraints limit sample size increases
- Unrestricted: Use when flexibility is paramount and resources allow
- Weighted: Use when robust performance across scenarios is desired

## Next Steps

This vignette provided a basic introduction to `bbssr`. For more advanced topics, see:

- `vignette("bbssr-statistical-methods")` - Detailed statistical methodology
- Package documentation: `help(package = "bbssr")`