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      <h1 class="title toc-ignore">FastRerandomize Package Tutorial</h1>
      <h4 class="date">2026-01-24</h4>

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      <!-- Introduction -->
      <div id="introduction" class="section level2">
        <h2>Introduction</h2>
        <p>
          This vignette demonstrates how to use the fastrerandomize package for
          generating and testing experimental designs in an agricultural study
          setting based on:
        </p>
        <ul>
          <li>
            Burchardi, K.B., Gulesci, S., Lerva, B. and Sulaiman, M., 2019.
            Moral hazard: Experimental Evidence from Tenancy Contracts. The
            Quarterly Journal of Economics, 134(1), pp.281-347.
          </li>
        </ul>
        <p>We will walk through:</p>
        <ul>
          <li>Generating synthetic covariate data</li>
          <li>Creating candidate randomizations</li>
          <li>
            Exploring the S3 methods for printing, summarizing, and plotting
            randomization objects
          </li>
          <li>Conducting a randomization-based inference test</li>
        </ul>
        <div id="key-concepts" class="concept-box">
          <h3>Key Concepts</h3>
          <ul>
            <li><strong>Balance statistic M:</strong> Mahalanobis distance between treatment and control covariate mean vectors. Smaller M = better balance.</li>
            <li><strong>Acceptance rule:</strong> keep an assignment if <code>M ≤ a</code> for a chosen cutoff a.</li>
            <li><strong>Acceptance probability q:</strong> q = P(M ≤ a) under complete randomization. Smaller q = stricter balance.</li>
            <li><strong>Accepted set:</strong> assignments that pass the rule; inference resamples from this set.</li>
            <li><strong>Shrinkage factor:</strong> how much rerandomization reduces covariate imbalance variance; larger reduction yields more precision when covariates predict outcomes.</li>
          </ul>
        </div>

        <p>
          Stringency refers to how selective the acceptance rule is, typically expressed by the acceptance
          probability q (lower q means more stringent). The shrinkage factor is the scalar reduction in covariate
          imbalance variance when conditioning on acceptance. In the paper's diagnostics, acceptance at q = 1%
          yields a shrinkage factor of about 0.45 at d = 30 but about 0.88 at d = 1000, so stringency has
          diminishing returns in very high-dimensional settings. In the
          same illustrative setup (n<sub>T</sub> = n<sub>C</sub> = 500, R² = 0.5), that corresponds to about 15% RMSE reduction at
          d = 30 versus about 3% at d = 1000.
        </p>
        <p>
          Note: This tutorial assumes you have installed the fastrerandomize
          package, either from source or via GitHub:
        </p>
        <div class="sourceCode" id="cb1">
          <pre class="sourceCode r"><code class="sourceCode r">
# If you haven't installed or set up the package:
# devtools::install_github("cjerzak/fastrerandomize-software", subdir = "fastrerandomize")

# (Done once) Optionally build the JAX backend if needed
# fastrerandomize::build_backend()

# note that this vignette can be found like so:
# vignette(package = "fastrerandomize")
          </code></pre>
        </div>
      </div>

      <!-- Covariate Data -->
      <div id="obtain-pre-treatment-covariate-data" class="section level2">
        <h2>1. Obtain Pre-treatment Covariate Data</h2>

        <div class="checklist">
          <h4>Covariate prep checklist</h4>
          <ul>
            <li>Use pre-treatment covariates only—never condition on post-treatment variables</li>
            <li>Standardize/scale when appropriate (especially if covariates have very different units)</li>
            <li>Avoid severe collinearity (Mahalanobis distance relies on the covariance matrix being invertible)</li>
          </ul>
        </div>

        <p>
          For illustration, we use several pre-treatment covariates from the
          original experiment.
        </p>
        <div class="sourceCode" id="cb2">
          <pre class="sourceCode r"><code class="sourceCode r">
library(fastrerandomize)

# Obtain pre-treatment covariates
data(QJEData, package = "fastrerandomize")
myCovariates <- c("children","married","hh_size","hh_sexrat")
QJEData <- QJEData[!is.na(rowSums(QJEData[,myCovariates])),]
X <- QJEData[,myCovariates]

# Analysis parameters
n_units   <- nrow(X)
n_treated <- round(nrow(X)/2)
          </code></pre>
        </div>
      </div>

      <!-- Generate Randomizations -->
      <div id="generate-randomizations" class="section level2">
        <h2>2. Generate Randomizations</h2>
        <p>
          We now create our candidate randomizations using the function
          <code>generate_randomizations()</code> from fastrerandomize. Depending
          on the scale and complexity of your experiment, you may choose either
          exact enumeration ("exact") or Monte Carlo ("monte_carlo")
          randomization.
        </p>

        <div class="guidance-box">
          <h4>Choosing stringency (q) in practice</h4>
          <ul>
            <li>Start with a target acceptance probability (common: <code>q = 0.01</code> for moderate, <code>q = 0.001</code> for high stringency).</li>
            <li>Use <code>diagnose_rerandomization()</code> to check implied balance improvement and computational implications.</li>
            <li>Plan candidate draws for a sufficiently large accepted pool (aim for thousands of accepted assignments for stable inference).</li>
            <li>Remember: smaller q → better balance, but more draws needed and diminishing returns in high dimensions.</li>
          </ul>
        </div>

        <p>
          In this example, we'll use the Monte Carlo approach. Note that
          <code>randomization_accept_prob</code> is set very low (0.0001) purely
          for demonstration.
        </p>
        <p>
          In high-dimensional planning, diagnostics can recommend extremely small q values. For example, in the
          paper's illustrative case (d = 1000, n<sub>T</sub> = n<sub>C</sub> = 500, R² = 0.4, σ = 1, |τ| = 0.2), the diagnostic
          inversion implies q* ≈ 0.000012 (about 1.2e-5), or roughly 81,226 draws per accepted assignment. Holding
          q fixed, more draws mainly increase the accepted pool (lowering the minimum attainable p-value) without
          changing expected balance.
        </p>
        <div class="sourceCode" id="cb3">
          <pre class="sourceCode r"><code class="sourceCode r">
RunMainAnalysis <- (!is.null(check_jax_availability()))
# if FALSE, consider calling build_backend()

if(RunMainAnalysis){
  CandRandomizations <- generate_randomizations(
    n_units = n_units,
    n_treated = n_treated,
    X = X,
    randomization_type = "monte_carlo",
    max_draws = 100000L,
    batch_size = 1000L,
    randomization_accept_prob = 0.0001
  )
}
          </code></pre>
        </div>

        <pre><code>## Using monte carlo randomization</code></pre>
        <pre><code>## Starting batch processing with 100 batches.</code></pre>
        <pre><code>## On batch 100 of 100</code></pre>
        <pre><code>## MC Loop Time (s): 1.2409</code></pre>
      </div>

      <!-- S3 Methods -->
      <div id="s3-methods-print-summary-and-plot" class="section level2">
        <h2>4. S3 Methods: Print, Summary, and Plot</h2>
        <p>
          <code>generate_randomizations()</code> returns an S3 object of class
          <code>fastrerandomize_randomizations</code>. We can use its associated
          print, summary, and plot methods to inspect the candidate
          randomizations.
        </p>
        <div class="sourceCode" id="cb107">
          <pre class="sourceCode r"><code class="sourceCode r">
if(RunMainAnalysis){
  # 4a. Print the object
  print(CandRandomizations)

  # 4b. Summary
  summary(CandRandomizations)

  # 4c. Plot the balance distribution
  plot(CandRandomizations)
}
          </code></pre>
        </div>
        <pre><code>## Object of class 'fastrerandomize_randomizations'
##
## Call:
##    generate_randomizations(n_units = n_units, n_treated = n_treated,
##       X = X, randomization_accept_prob = 1e-04, max_draws = 100000L,
##       batch_size = 1000L, randomization_type = "monte_carlo")
##
## Number of candidate randomizations: 10
## Number of units (columns): 302
## Balance statistics available for each randomization.
## Summary of 'fastrerandomize_randomizations' object:
##
## Balance statistics (summary):
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max.
## 0.003574 0.008231 0.008746 0.009835 0.013152 0.017181</code></pre>
      </div>

      <!-- Randomization Test -->
      <div id="randomization-test" class="section level2">
        <h2>3. Randomization Test</h2>
        <p>
          Next, we showcase a randomization-based inference procedure using
          fastrerandomize.
        </p>

        <div class="note-box">
          <h4>Why the randomization test conditions on acceptance</h4>
          <p>Under rerandomization, treatment was assigned from the subset of assignments that passed the balance rule. To keep p-values calibrated, the randomization test must resample only from assignments that would also pass the same rule. This is what "design-respecting inference" means.</p>
        </div>

        <p>
          Randomization tests are conditioned on the accepted assignments: the observed statistic is compared to
          its distribution over the accepted pool. As a result, p-values are lower-bounded by 1 / M_accept, and
          optional fiducial intervals are available by inverting the test.
        </p>

        <div id="a.-setup-simulated-outcomes" class="section level3">
          <h3>3a. Setup Simulated Outcomes</h3>
          <p>
            We simulate an outcome <code>Y</code> that depends on our covariates
            <code>X</code>, the chosen randomization, and some true treatment
            effect τ. We pick the first acceptable randomization from our set as
            our “observed” assignment <code>Wobs</code>.
          </p>
          <div class="sourceCode" id="cb109">
            <pre class="sourceCode r"><code class="sourceCode r">
if(RunMainAnalysis){
  CoefY <- rnorm(ncol(X))          # Coefficients for outcome model
  Wobs  <- CandRandomizations$randomizations[1, ]  # 1st acceptable randomization
  tau_true <- 1                    # true treatment effect

  # Generate Y = X * Coef + W*tau + noise
  Yobs <- as.numeric(as.matrix(X) %*% as.matrix(CoefY) +
                     Wobs * tau_true + rnorm(n_units, sd = 0.1))
}
            </code></pre>
          </div>
        </div>

        <div id="b.-run-the-randomization-test" class="section level3">
          <h3>3b. Run the Randomization Test</h3>
          <p>
            We pass our observed treatment assignment (<code>obsW</code>),
            observed outcomes (<code>obsY</code>), and the full matrix of
            candidate randomizations
            (<code>candidate_randomizations</code>) to
            <code>randomization_test()</code>. This test:
          </p>
          <ul>
            <li>Computes the observed difference in means</li>
            <li>Permutes the treatment assignment across our candidate
            randomizations</li>
            <li>
              Estimates the p-value by comparing the observed statistic to its
              permutation distribution
            </li>
          </ul>
          <div class="sourceCode" id="cb110">
            <pre class="sourceCode r"><code class="sourceCode r">
if(RunMainAnalysis){
  randomization_test_results <- randomization_test(
    obsW = Wobs,
    obsY = Yobs,
    candidate_randomizations = CandRandomizations$randomizations,
    findFI = TRUE
  )

  # Inspect results
  print(randomization_test_results)
  summary(randomization_test_results)
  plot(randomization_test_results)
}
            </code></pre>
          </div>
          <pre><code>## Object of class 'fastrerandomize_test'
##
## Call:
##    randomization_test(obsW = Wobs, obsY = Yobs, candidate_randomizations = CandRandomizations$randomizations,
##       findFI = TRUE)
##
## P-value:  0.1
## Observed effect (tau_obs):  0.9888765
## Fiducial interval: [-547.2352, 498.731]
## Summary of 'fastrerandomize_test' object:
##
## Call:
##    randomization_test(obsW = Wobs, obsY = Yobs, candidate_randomizations = CandRandomizations$randomizations,
##       findFI = TRUE)
##
## P-value:
##    0.1
##
## Observed effect (tau_obs):
##    0.9888765
##
## Fiducial Interval:
##   [-547.2352, 498.731]</code></pre>
          <p>
            The <code>findFI = TRUE</code> flag attempts to locate a fiducial
            interval (FI) for the treatment effect.
          </p>
        </div>
      </div>

      <!-- What to Report -->
      <div id="what-to-report" class="section level2">
        <h2>5. What to Report</h2>
        <div class="reporting-guide">
          <h4>Recommended reporting checklist</h4>
          <ul>
            <li><strong>Covariates used</strong> in the balance check and balance statistic (e.g., Mahalanobis M)</li>
            <li><strong>Acceptance rule</strong> (M ≤ a) and either a or the implied acceptance probability q</li>
            <li><strong>Number of candidate draws</strong> and the resulting accepted pool size</li>
            <li><strong>That inference was conducted conditional on the acceptance rule</strong> (design-respecting inference)</li>
          </ul>
        </div>
        <p>
          Transparent reporting helps readers understand how the design affects inference and enables replication.
        </p>
      </div>

      <!-- Conclusion -->
      <div id="conclusion" class="section level2">
        <h2>Conclusion</h2>
        <p>
          This tutorial demonstrated a basic workflow with fastrerandomize:
        </p>
        <ul>
          <li>Generating synthetic or real covariate data</li>
          <li>
            Producing a set of acceptable treatment assignments via
            randomization
          </li>
          <li>Visualizing and summarizing these assignments</li>
          <li>
            Testing for treatment effects with a randomization-based approach
          </li>
        </ul>
        <p>
          We encourage you to consult the package documentation for more
          advanced functionalities, including GPU-accelerated computations via
          JAX and flexible definitions of balance criteria.
        </p>
      </div>

      <div class="section-card animate-fade-in">
        <p class="eyebrow">Citation</p>
        <p>
          Connor T. Jerzak, Rebecca Goldstein, Aniket Kamat, and Fucheng Warren Zhu.
          FastRerandomize: An R Package for Fast Rerandomization Using Accelerated Computing.
          SoftwareX, 2026.
        </p>
        <div class="citation-card">
          <pre><code>@article{jerzak2025fastrerandomize,
  title={FastRerandomize: An R Package for Fast Rerandomization Using Accelerated Computing},
  author={Jerzak, Connor T. and Rebecca Goldstein and Aniket Kamat and Fucheng Warren Zhu},
  journal={SoftwareX},
  year={2026}
}</code></pre>
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