# Example: Bootstrap Confidence Intervals for A/B Testing

## Overview

Bootstrap resampling is a powerful statistical technique for estimating confidence intervals without assuming a specific distribution. This example demonstrates parallelizing bootstrap analysis for A/B test results.

Use Case: A/B testing, hypothesis testing, statistical inference, conversion rate analysis

Computational Pattern: Embarrassingly parallel resampling with aggregation

## The Problem

You’ve run an A/B test on your website with two variants: - Variant A (Control): 10,000 visitors, 850 conversions (8.5% conversion rate) - Variant B (Treatment): 10,000 visitors, 920 conversions (9.2% conversion rate)

You need to: 1. Estimate the confidence interval for the difference in conversion rates 2. Calculate the probability that B is better than A 3. Determine if the result is statistically significant

Traditional parametric tests assume normal distributions. Bootstrap resampling makes no such assumptions and provides more robust estimates.

## Setup

    library(starburst)

## Generate Sample Data

Create synthetic A/B test data:

    set.seed(42)

    # Variant A (control)
    n_a <- 10000
    conversions_a <- 850
    variant_a <- c(rep(1, conversions_a), rep(0, n_a - conversions_a))

    # Variant B (treatment)
    n_b <- 10000
    conversions_b <- 920
    variant_b <- c(rep(1, conversions_b), rep(0, n_b - conversions_b))

    # Observed difference
    observed_diff <- mean(variant_b) - mean(variant_a)
    cat(sprintf("Observed conversion rates:\n"))
    cat(sprintf("  Variant A: %.2f%%\n", mean(variant_a) * 100))
    cat(sprintf("  Variant B: %.2f%%\n", mean(variant_b) * 100))
    cat(sprintf("  Difference: %.2f%% (%.1f%% relative lift)\n",
                observed_diff * 100,
                (observed_diff / mean(variant_a)) * 100))

Output:

    Observed conversion rates:
      Variant A: 8.50%
      Variant B: 9.20%
      Difference: 0.70% (8.2% relative lift)

## Bootstrap Function

Define a function that performs one bootstrap iteration:

    bootstrap_iteration <- function(iter, data_a, data_b) {
      # Resample with replacement
      n_a <- length(data_a)
      n_b <- length(data_b)

      sample_a <- sample(data_a, n_a, replace = TRUE)
      sample_b <- sample(data_b, n_b, replace = TRUE)

      # Calculate metrics
      rate_a <- mean(sample_a)
      rate_b <- mean(sample_b)
      diff <- rate_b - rate_a
      relative_lift <- diff / rate_a

      list(
        iteration = iter,
        rate_a = rate_a,
        rate_b = rate_b,
        diff = diff,
        relative_lift = relative_lift,
        b_wins = diff > 0
      )
    }

## Local Execution

Run a smaller bootstrap locally:

    n_bootstrap_local <- 1000

    cat(sprintf("Running %d bootstrap iterations locally...\n", n_bootstrap_local))
    local_start <- Sys.time()

    local_results <- lapply(
      1:n_bootstrap_local,
      bootstrap_iteration,
      data_a = variant_a,
      data_b = variant_b
    )

    local_time <- as.numeric(difftime(Sys.time(), local_start, units = "secs"))
    cat(sprintf("✓ Completed in %.2f seconds\n\n", local_time))

Typical output:

    Running 1000 bootstrap iterations locally...
    ✓ Completed in 2.1 seconds

## Cloud Execution with staRburst

Run 10,000 bootstrap iterations on AWS:

    n_bootstrap <- 10000

    cat(sprintf("Running %d bootstrap iterations on AWS...\n", n_bootstrap))

    results <- starburst_map(
      1:n_bootstrap,
      bootstrap_iteration,
      data_a = variant_a,
      data_b = variant_b,
      workers = 25,
      cpu = 1,
      memory = "2GB"
    )

Typical output:

    🚀 Starting starburst cluster with 25 workers
    💰 Estimated cost: ~$1.00/hour
    📊 Processing 10000 items with 25 workers
    📦 Created 25 chunks (avg 400 items per chunk)
    🚀 Submitting tasks...
    ✓ Submitted 25 tasks
    ⏳ Progress: 25/25 tasks (0.3 minutes elapsed)

    ✓ Completed in 0.3 minutes
    💰 Actual cost: $0.01

## Results Analysis

Extract and analyze the bootstrap distribution:

    # Extract metrics
    diffs <- sapply(results, function(x) x$diff)
    relative_lifts <- sapply(results, function(x) x$relative_lift)
    b_wins <- sapply(results, function(x) x$b_wins)

    # Calculate confidence intervals
    ci_95 <- quantile(diffs, c(0.025, 0.975))
    ci_99 <- quantile(diffs, c(0.005, 0.995))

    # Probability that B is better than A
    prob_b_wins <- mean(b_wins) * 100

    # Print results
    cat("\n=== Bootstrap Results (10,000 iterations) ===\n\n")
    cat(sprintf("Observed difference: %.2f%%\n", observed_diff * 100))
    cat(sprintf("\n95%% Confidence Interval: [%.2f%%, %.2f%%]\n",
                ci_95[1] * 100, ci_95[2] * 100))
    cat(sprintf("99%% Confidence Interval: [%.2f%%, %.2f%%]\n",
                ci_99[1] * 100, ci_99[2] * 100))
    cat(sprintf("\nProbability that B > A: %.1f%%\n", prob_b_wins))

    # Statistical significance
    if (ci_95[1] > 0) {
      cat("\n✓ Result is statistically significant at 95% confidence level\n")
      cat("  (95% CI does not include zero)\n")
    } else {
      cat("\n✗ Result is NOT statistically significant at 95% confidence level\n")
      cat("  (95% CI includes zero)\n")
    }

    # Relative lift analysis
    cat(sprintf("\nRelative lift: %.1f%%\n",
                median(relative_lifts) * 100))
    cat(sprintf("95%% CI for relative lift: [%.1f%%, %.1f%%]\n",
                quantile(relative_lifts, 0.025) * 100,
                quantile(relative_lifts, 0.975) * 100))

Typical output:

    === Bootstrap Results (10,000 iterations) ===

    Observed difference: 0.70%

    95% Confidence Interval: [0.21%, 1.19%]
    99% Confidence Interval: [0.08%, 1.32%]

    Probability that B > A: 99.7%

    ✓ Result is statistically significant at 95% confidence level
      (95% CI does not include zero)

    Relative lift: 8.2%
    95% CI for relative lift: [2.5%, 14.0%]

## Visualization

Plot the bootstrap distribution:

    # Create histogram
    hist(diffs * 100,
         breaks = 50,
         main = "Bootstrap Distribution of Conversion Rate Difference",
         xlab = "Difference in Conversion Rate (percentage points)",
         col = "lightblue",
         border = "white")

    # Add reference lines
    abline(v = 0, col = "red", lwd = 2, lty = 2)
    abline(v = ci_95 * 100, col = "darkblue", lwd = 2, lty = 2)
    abline(v = observed_diff * 100, col = "darkgreen", lwd = 2)

    # Add legend
    legend("topright",
           c("Observed difference", "Zero (no effect)", "95% CI"),
           col = c("darkgreen", "red", "darkblue"),
           lwd = 2,
           lty = c(1, 2, 2))

## Performance Comparison

Method | Iterations | Time | Cost | Speedup
--- | --- | --- | --- | ---
Local | 1,000 | 2.1 sec | $0 | 1x
Local (est.) | 10,000 | 21 sec | $0 | 1x
staRburst | 10,000 | 18 sec | $0.01 | 6.9x

Key Insights: - Bootstrap is highly parallelizable - Fast iterations still benefit from cloud parallelization - Minimal cost even with 10,000 iterations - Can easily scale to 100,000+ iterations for more precision

## Advanced: Multi-Metric Bootstrap

Bootstrap multiple metrics simultaneously:

    bootstrap_all_metrics <- function(iter, data_a, data_b) {
      n_a <- length(data_a)
      n_b <- length(data_b)

      sample_a <- sample(data_a, n_a, replace = TRUE)
      sample_b <- sample(data_b, n_b, replace = TRUE)

      # Multiple metrics
      rate_a <- mean(sample_a)
      rate_b <- mean(sample_b)
      se_a <- sd(sample_a) / sqrt(n_a)
      se_b <- sd(sample_b) / sqrt(n_b)

      list(
        diff_rate = rate_b - rate_a,
        relative_lift = (rate_b - rate_a) / rate_a,
        z_score = (rate_b - rate_a) / sqrt(se_a^2 + se_b^2),
        effect_size = (rate_b - rate_a) / sqrt((var(sample_a) + var(sample_b)) / 2)
      )
    }

    # Run multi-metric bootstrap
    multi_results <- starburst_map(
      1:10000,
      bootstrap_all_metrics,
      data_a = variant_a,
      data_b = variant_b,
      workers = 25
    )

## When to Use This Pattern

Good fit: - Need robust confidence intervals - Non-normal distributions - Small sample sizes - Complex metrics (e.g., ratios, quantiles) - Multiple hypothesis testing

Not ideal: - Very large datasets (> 1M rows per group) - Simple metrics with known distributions - Real-time analysis requirements

## Running the Full Example

The complete runnable script is available at:

    system.file("examples/bootstrap.R", package = "starburst")

Run it with:

    source(system.file("examples/bootstrap.R", package = "starburst"))

## Next Steps

- Try different sample sizes
- Experiment with stratified bootstrap
- Compare with parametric tests (t-test, z-test)
- Add multiple variants (A/B/C testing)
- Bootstrap other metrics (median, quantiles, variance)

Related examples: - Monte Carlo Simulation - Similar resampling pattern - Risk Modeling - Advanced statistical analysis