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<!DOCTYPE html>
<html lang="en">
<head>
<title>Bootstrap in R: boot Package, CI & Hypothesis Tests Without Assumptions</title>
<meta charset="utf-8">
<meta name="Description" content="Bootstrap in R with the boot package: confidence intervals and hypothesis tests without distributional assumptions. Runnable BCa, percentile, basic examples.">
<meta name="Keywords" content="bootstrap in R, boot package R, bootstrap confidence interval, BCa interval, percentile bootstrap, bootstrap hypothesis test, nonparametric inference, resampling R">
<meta name="Distribution" content="Global">
<meta name="Author" content="Selva Prabhakaran">
<meta name="Robots" content="index, follow">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<meta name="referrer" content="strict-origin-when-cross-origin">
<link rel="icon" href="/screenshots/iconb-64.png" type="image/x-icon" />
<link rel="canonical" href="https://r-statistics.co/Bootstrap-in-R.html">
<link rel="alternate" type="application/atom+xml" title="r-statistics.co" href="https://r-statistics.co/feed.xml">
<link rel="preconnect" href="https://cdn.jsdelivr.net">
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<link rel="preload" href="/www/fonts/ibm-plex/ibm-plex-sans-latin.woff2" as="font" type="font/woff2" crossorigin>
<!-- Preload heading font (IBM Plex Serif 700) to prevent H1 font swap -->
<link rel="preload" href="/www/fonts/ibm-plex/ibm-plex-serif-700.woff2" as="font" type="font/woff2" crossorigin>
<!-- Critical CSS inlined for fast first paint -->
<style>
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@font-face{font-family:'IBM Plex Sans';font-style:normal;font-weight:500;font-display:optional;src:url('/www/fonts/ibm-plex/ibm-plex-sans-latin.woff2') format('woff2')}
@font-face{font-family:'IBM Plex Sans';font-style:normal;font-weight:600;font-display:optional;src:url('/www/fonts/ibm-plex/ibm-plex-sans-latin.woff2') format('woff2')}
@font-face{font-family:'IBM Plex Sans';font-style:normal;font-weight:700;font-display:optional;src:url('/www/fonts/ibm-plex/ibm-plex-sans-latin.woff2') format('woff2')}
@font-face{font-family:'IBM Plex Serif';font-style:normal;font-weight:600;font-display:optional;src:url('/www/fonts/ibm-plex/ibm-plex-serif-600.woff2') format('woff2')}
@font-face{font-family:'IBM Plex Serif';font-style:normal;font-weight:700;font-display:optional;src:url('/www/fonts/ibm-plex/ibm-plex-serif-700.woff2') format('woff2')}
@font-face{font-family:'IBM Plex Mono';font-style:normal;font-weight:400;font-display:optional;src:url('/www/fonts/ibm-plex/ibm-plex-mono-400.woff2') format('woff2')}
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body{margin:0;font-family:'IBM Plex Sans',-apple-system,BlinkMacSystemFont,sans-serif;font-size:18px;line-height:1.7;color:#0d1117;background:#fafbfc}
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p{margin:0 0 15px}
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a{color:#1d3158;text-decoration:none}a:hover{text-decoration:underline}
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#content h1{font-weight:700;font-size:2.1em;line-height:1.2;margin:0 0 0.4em 0;letter-spacing:-0.01em;color:#0d1117}
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html.dark .site-masthead{background:rgba(12,13,16,0.85)}
html.dark .site-masthead.scrolled{border-bottom-color:#262a31}
html.dark .masthead-wordmark{color:#e8eaee}
html.dark .masthead-tld{color:#6e7382}
html.dark .masthead-mark{background:#92a4d8}
html.dark .masthead-nav-link{color:#b0b5bf}
html.dark .masthead-nav-link:hover,html.dark .masthead-nav-link.active{background:#1a1d22;color:#e8eaee}
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.nav-cat{position:relative;display:flex;align-items:center;justify-content:space-between;padding:7px 18px;font-family:'IBM Plex Sans',sans-serif;font-size:13px;font-weight:500;color:#3a3f4a;cursor:pointer;transition:background 0.1s}
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html.dark .nav-dropdown-panel,html.dark .nav-sub{background:#161a1f;border-color:#2a2e35;box-shadow:0 8px 24px rgba(0,0,0,0.45)}
html.dark .nav-cat{color:#c5cad3}
html.dark .nav-cat:hover{background:#1a1d22;color:#fff}
html.dark .nav-sub a{color:#c5cad3}
html.dark .nav-sub a:hover{color:#fff}
html.dark .nav-sub h6{color:#7a808c}
@media (max-width: 768px){.nav-dropdown-panel{min-width:220px;left:auto;right:0}.nav-sub{position:static;border:none;box-shadow:none;padding:6px 0 6px 16px}.nav-cat::after{content:''}}
html.dark .masthead-search input{background:#14161a;border-color:#262a31;color:#e8eaee}
html.dark .masthead-search input:focus{border-color:#92a4d8;box-shadow:0 0 0 3px rgba(146,164,216,0.15)}
html.dark .masthead-icon-btn{color:#b0b5bf}
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<div id="sidebar-nav"><div class="continue-chip" data-continue-chip><span class="chip-label">Continue reading</span><a href="#" data-continue-link></a></div><div class="sidebar-tabs" role="tablist"><button class="sidebar-tab active" data-tab="posts" type="button" role="tab" onclick="var n=this.dataset.tab;document.querySelectorAll('.sidebar-tab').forEach(function(x){x.classList.toggle('active',x.dataset.tab===n)});document.querySelectorAll('.sidebar-panel').forEach(function(p){p.classList.toggle('active',p.dataset.panel===n)});try{localStorage.setItem('rstat_sidebar_tab',n)}catch(e){}">Posts</button><button class="sidebar-tab" data-tab="tools" type="button" role="tab" onclick="var n=this.dataset.tab;document.querySelectorAll('.sidebar-tab').forEach(function(x){x.classList.toggle('active',x.dataset.tab===n)});document.querySelectorAll('.sidebar-panel').forEach(function(p){p.classList.toggle('active',p.dataset.panel===n)});try{localStorage.setItem('rstat_sidebar_tab',n)}catch(e){}">Tools</button></div><div class="sidebar-panel active" data-panel="posts"><ul class="sidebar-menu list-unstyled"><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span> Learn R<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec0sub1" data-collapsed="false"><span class="subsec-chevron">▼</span> Getting Started</li><li data-subkey="sec0sub1"><a href="/Is-R-Worth-Learning-in-2026.html"><span class="progress-dot"></span>Is R Worth Learning?</a></li><li data-subkey="sec0sub1"><a href="/Install-R-and-RStudio-2026.html"><span class="progress-dot"></span>Install R & RStudio</a></li><li data-subkey="sec0sub1"><a href="/RStudio-IDE-Tour.html"><span class="progress-dot"></span>RStudio IDE Tour</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec0sub2" data-collapsed="false"><span class="subsec-chevron">▼</span> R Fundamentals</li><li data-subkey="sec0sub2"><a href="/R-Syntax-101.html"><span class="progress-dot"></span>R Syntax 101</a></li><li data-subkey="sec0sub2"><a href="/R-Data-Types.html"><span class="progress-dot"></span>R Data Types</a></li><li data-subkey="sec0sub2"><a href="/R-Vectors.html"><span class="progress-dot"></span>R Vectors</a></li><li data-subkey="sec0sub2"><a href="/R-Matrices.html"><span class="progress-dot"></span>R Matrices</a></li><li data-subkey="sec0sub2"><a href="/R-Factors.html"><span class="progress-dot"></span>R Factors</a></li><li data-subkey="sec0sub2"><a href="/R-Data-Frames.html"><span class="progress-dot"></span>R Data Frames</a></li><li data-subkey="sec0sub2"><a href="/R-Lists.html"><span class="progress-dot"></span>R Lists</a></li><li data-subkey="sec0sub2"><a href="/R-Control-Flow.html"><span class="progress-dot"></span>R Control Flow</a></li><li data-subkey="sec0sub2"><a href="/R-Special-Values.html"><span class="progress-dot"></span>R Special Values</a></li><li data-subkey="sec0sub2"><a href="/R-Type-Coercion.html"><span class="progress-dot"></span>R Type Coercion</a></li><li data-subkey="sec0sub2"><a href="/R-Functions.html"><span class="progress-dot"></span>Writing R Functions</a></li><li data-subkey="sec0sub2"><a href="/R-Beginner-Exercises-quiz.html"><span class="progress-dot"></span>R Fundamentals Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec0sub3" data-collapsed="false"><span class="subsec-chevron">▼</span> Working Effectively</li><li data-subkey="sec0sub3"><a href="/R-Subsetting.html"><span class="progress-dot"></span>R Subsetting</a></li><li data-subkey="sec0sub3"><a href="/Getting-Help-in-R.html"><span class="progress-dot"></span>Getting Help in R</a></li><li data-subkey="sec0sub3"><a href="/R-Project-Structure.html"><span class="progress-dot"></span>R Project Structure</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec0sub4" data-collapsed="false"><span class="subsec-chevron">▼</span> R Career & Resources</li><li data-subkey="sec0sub4"><a href="/R-vs-Python.html"><span class="progress-dot"></span>R vs Python</a></li><li data-subkey="sec0sub4"><a href="/How-to-Learn-R.html"><span class="progress-dot"></span>How to Learn R</a></li><li data-subkey="sec0sub4"><a href="/R-for-Excel-Users.html"><span class="progress-dot"></span>R for Excel Users</a></li><li data-subkey="sec0sub4"><a href="/R-Interview-Questions.html"><span class="progress-dot"></span>R Interview Questions</a></li><li data-subkey="sec0sub4"><a href="/R-Interview-Questions-quiz.html"><span class="progress-dot"></span>R Interview Readiness Quiz</a></li><li data-subkey="sec0sub4"><a href="/R-Cheat-Sheet.html"><span class="progress-dot"></span>R Cheat Sheet</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec0sub5" data-collapsed="false"><span class="subsec-chevron">▼</span> Professional R</li><li data-subkey="sec0sub5"><a href="/Data-Ethics-in-R.html"><span class="progress-dot"></span>Data Ethics</a></li><li data-subkey="sec0sub5"><a href="/Bias-in-Data-and-Models.html"><span class="progress-dot"></span>Bias in Data & Models</a></li><li data-subkey="sec0sub5"><a href="/Reproducibility-Crisis.html"><span class="progress-dot"></span>Reproducibility</a></li><li data-subkey="sec0sub5"><a href="/Data-Privacy-in-R.html"><span class="progress-dot"></span>Data Privacy</a></li><li data-subkey="sec0sub5"><a href="/Communicating-Uncertainty.html"><span class="progress-dot"></span>Communicating Uncertainty</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span> Data Wrangling<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub1" data-collapsed="false"><span class="subsec-chevron">▼</span> Import & Setup</li><li data-subkey="sec1sub1"><a href="/Importing-Data-in-R.html"><span class="progress-dot"></span>Importing Data</a></li><li data-subkey="sec1sub1"><a href="/R-Pipe-Operator.html"><span class="progress-dot"></span>Pipe Operator</a></li><li data-subkey="sec1sub1"><a href="/Tidy-Data-in-R.html"><span class="progress-dot"></span>Tidy Data</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub2" data-collapsed="false"><span class="subsec-chevron">▼</span> dplyr Essentials</li><li data-subkey="sec1sub2"><a href="/dplyr-filter-select.html"><span class="progress-dot"></span>dplyr filter & select</a></li><li data-subkey="sec1sub2"><a href="/dplyr-mutate-rename.html"><span class="progress-dot"></span>dplyr mutate & rename</a></li><li data-subkey="sec1sub2"><a href="/dplyr-group-by-summarise.html"><span class="progress-dot"></span>dplyr group_by & summarise</a></li><li data-subkey="sec1sub2"><a href="/dplyr-arrange-slice.html"><span class="progress-dot"></span>dplyr arrange & slice</a></li><li data-subkey="sec1sub2"><a href="/dplyr-across.html"><span class="progress-dot"></span>dplyr across()</a></li><li data-subkey="sec1sub2"><a href="/dplyr-case-when.html"><span class="progress-dot"></span>dplyr case_when()</a></li><li data-subkey="sec1sub2"><a href="/dplyr-Exercises-in-R-quiz.html"><span class="progress-dot"></span>dplyr Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub3" data-collapsed="false"><span class="subsec-chevron">▼</span> Join & Reshape</li><li data-subkey="sec1sub3"><a href="/R-Joins.html"><span class="progress-dot"></span>R Joins</a></li><li data-subkey="sec1sub3"><a href="/pivot_longer-pivot_wider-Reshape-Data-in-R.html"><span class="progress-dot"></span>pivot_longer & pivot_wider</a></li><li data-subkey="sec1sub3"><a href="/tidyr-separate-unite-Split-Combine-Columns-in-R.html"><span class="progress-dot"></span>separate() & unite()</a></li><li data-subkey="sec1sub3"><a href="/tidyr-Exercises-in-R-quiz.html"><span class="progress-dot"></span>tidyr Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub4" data-collapsed="false"><span class="subsec-chevron">▼</span> Clean & Quality</li><li data-subkey="sec1sub4"><a href="/Missing-Values-in-R-Detect-Count-Remove-Impute-NA.html"><span class="progress-dot"></span>Missing Values (NA)</a></li><li data-subkey="sec1sub4"><a href="/Data-Quality-Checking-in-R.html"><span class="progress-dot"></span>Data Quality Checking</a></li><li data-subkey="sec1sub4"><a href="/janitor-Package-in-R.html"><span class="progress-dot"></span>janitor Package</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub5" data-collapsed="false"><span class="subsec-chevron">▼</span> Strings & Dates</li><li data-subkey="sec1sub5"><a href="/stringr-in-R.html"><span class="progress-dot"></span>stringr</a></li><li data-subkey="sec1sub5"><a href="/R-Regex-stringr-Pattern-Matching.html"><span class="progress-dot"></span>Regex Patterns</a></li><li data-subkey="sec1sub5"><a href="/lubridate-in-R.html"><span class="progress-dot"></span>lubridate</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub6" data-collapsed="false"><span class="subsec-chevron">▼</span> Scale & Connect</li><li data-subkey="sec1sub6"><a href="/DBI-in-R.html"><span class="progress-dot"></span>DBI & Databases</a></li><li data-subkey="sec1sub6"><a href="/DuckDB-in-R.html"><span class="progress-dot"></span>DuckDB & duckplyr</a></li><li data-subkey="sec1sub6"><a href="/Web-Scraping-in-R-with-rvest.html"><span class="progress-dot"></span>Web Scraping (rvest)</a></li><li data-subkey="sec1sub6"><a href="/REST-APIs-in-R-with-httr2.html"><span class="progress-dot"></span>REST APIs (httr2)</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span> Visualization<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub1" data-collapsed="false"><span class="subsec-chevron">▼</span> ggplot2 Foundations</li><li data-subkey="sec2sub1"><a href="/ggplot2-Grammar-of-Graphics.html"><span class="progress-dot"></span>Grammar of Graphics</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Getting-Started.html"><span class="progress-dot"></span>ggplot2 Getting Started</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Aesthetics-aes-Map-Data.html"><span class="progress-dot"></span>ggplot2 Aesthetics (aes)</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Colours.html"><span class="progress-dot"></span>ggplot2 Colours</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Scales.html"><span class="progress-dot"></span>ggplot2 Scales</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Themes-in-R.html"><span class="progress-dot"></span>ggplot2 Themes</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Labels-and-Annotations.html"><span class="progress-dot"></span>Labels & Annotations</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Facets.html"><span class="progress-dot"></span>ggplot2 Facets</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Exercises-in-R-quiz.html"><span class="progress-dot"></span>ggplot2 Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub2" data-collapsed="false"><span class="subsec-chevron">▼</span> Core Charts</li><li data-subkey="sec2sub2"><a href="/ggplot2-Scatter-Plots.html"><span class="progress-dot"></span>Scatter Plots</a></li><li data-subkey="sec2sub2"><a href="/ggplot2-Line-Charts.html"><span class="progress-dot"></span>Line Charts</a></li><li data-subkey="sec2sub2"><a href="/ggplot2-Bar-Charts.html"><span class="progress-dot"></span>Bar Charts</a></li><li data-subkey="sec2sub2"><a href="/ggplot2-Distribution-Charts.html"><span class="progress-dot"></span>Distribution Charts</a></li><li data-subkey="sec2sub2"><a href="/Error-Bars-in-R.html"><span class="progress-dot"></span>Error Bars</a></li><li data-subkey="sec2sub2"><a href="/geom_smooth-in-R.html"><span class="progress-dot"></span>geom_smooth()</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub3" data-collapsed="false"><span class="subsec-chevron">▼</span> Distributions & Groups</li><li data-subkey="sec2sub3"><a href="/Violin-Plot-in-R.html"><span class="progress-dot"></span>Violin Plot</a></li><li data-subkey="sec2sub3"><a href="/Ridgeline-Plot-in-R.html"><span class="progress-dot"></span>Ridgeline Plot</a></li><li data-subkey="sec2sub3"><a href="/Lollipop-Chart-in-R.html"><span class="progress-dot"></span>Lollipop Chart</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub4" data-collapsed="false"><span class="subsec-chevron">▼</span> Relationships</li><li data-subkey="sec2sub4"><a href="/Bubble-Chart-in-R.html"><span class="progress-dot"></span>Bubble Chart</a></li><li data-subkey="sec2sub4"><a href="/Heatmap-in-R.html"><span class="progress-dot"></span>Heatmap in R</a></li><li data-subkey="sec2sub4"><a href="/Correlation-Matrix-Plot-in-R.html"><span class="progress-dot"></span>Correlation Matrix</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub5" data-collapsed="false"><span class="subsec-chevron">▼</span> Advanced Charts</li><li data-subkey="sec2sub5"><a href="/Pie-Donut-Chart-in-R.html"><span class="progress-dot"></span>Pie & Donut Chart</a></li><li data-subkey="sec2sub5"><a href="/Treemap-in-R.html"><span class="progress-dot"></span>Treemap</a></li><li data-subkey="sec2sub5"><a href="/Waffle-Chart-in-R.html"><span class="progress-dot"></span>Waffle Chart</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub6" data-collapsed="false"><span class="subsec-chevron">▼</span> Exploratory Analysis</li><li data-subkey="sec2sub6"><a href="/Exploratory-Data-Analysis-in-R.html"><span class="progress-dot"></span>EDA (7-Step Framework)</a></li><li data-subkey="sec2sub6"><a href="/Univariate-EDA-in-R.html"><span class="progress-dot"></span>Univariate EDA</a></li><li data-subkey="sec2sub6"><a href="/Bivariate-EDA-in-R.html"><span class="progress-dot"></span>Bivariate EDA</a></li><li data-subkey="sec2sub6"><a href="/Descriptive-Statistics-in-R.html"><span class="progress-dot"></span>Descriptive Statistics</a></li><li data-subkey="sec2sub6"><a href="/Correlation-Analysis-in-R.html"><span class="progress-dot"></span>Correlation Analysis</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub7" data-collapsed="false"><span class="subsec-chevron">▼</span> Interactive & Maps</li><li data-subkey="sec2sub7"><a href="/Combining-ggplot2-with-plotly.html"><span class="progress-dot"></span>ggplot2 + plotly Interactive</a></li><li data-subkey="sec2sub7"><a href="/Interactive-Maps-in-R-with-leaflet.html"><span class="progress-dot"></span>Leaflet Interactive Maps</a></li><li data-subkey="sec2sub7"><a href="/Spatial-Data-in-R-with-sf.html"><span class="progress-dot"></span>Spatial Data (sf)</a></li><li data-subkey="sec2sub7"><a href="/Choropleth-Maps-in-R.html"><span class="progress-dot"></span>Choropleth Maps (sf)</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub8" data-collapsed="false"><span class="subsec-chevron">▼</span> Customization & Reference</li><li data-subkey="sec2sub8"><a href="/ggplot2-Legends-in-R.html"><span class="progress-dot"></span>ggplot2 Legends</a></li><li data-subkey="sec2sub8"><a href="/ggplot2-Secondary-Axis.html"><span class="progress-dot"></span>Secondary Axis</a></li><li data-subkey="sec2sub8"><a href="/ggplot2-Log-Scale.html"><span class="progress-dot"></span>Log Scale</a></li><li data-subkey="sec2sub8"><a href="/patchwork-Package.html"><span class="progress-dot"></span>patchwork (Combine Plots)</a></li><li data-subkey="sec2sub8"><a href="/Publication-Quality-Figures-in-R.html"><span class="progress-dot"></span>Publication-Ready Figures</a></li><li data-subkey="sec2sub8"><a href="/ggplot2-cheatsheet.html"><span class="progress-dot"></span>ggplot2 Quickref</a></li></ul></li><li class="sidebar-section expanded"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span> Statistics<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub1" data-collapsed="false"><span class="subsec-chevron">▼</span> EDA & Data Quality</li><li data-subkey="sec3sub1"><a href="/Automated-EDA-in-R.html"><span class="progress-dot"></span>Automated EDA</a></li><li data-subkey="sec3sub1"><a href="/Missing-Data-Visualization-in-R-naniar.html"><span class="progress-dot"></span>Missing Data Viz (naniar)</a></li><li data-subkey="sec3sub1"><a href="/Outlier-Detection-in-R.html"><span class="progress-dot"></span>Outlier Detection</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub2" data-collapsed="false"><span class="subsec-chevron">▼</span> Probability</li><li data-subkey="sec3sub2"><a href="/Sample-Spaces-Events-and-Probability-Axioms-in-R-With-Monte-Carlo-Proof.html"><span class="progress-dot"></span>Probability Axioms</a></li><li data-subkey="sec3sub2"><a href="/Conditional-Probability-in-R.html"><span class="progress-dot"></span>Conditional Probability</a></li><li data-subkey="sec3sub2"><a href="/Random-Variables-in-R.html"><span class="progress-dot"></span>Random Variables</a></li><li data-subkey="sec3sub2"><a href="/Binomial-and-Poisson-Distributions-in-R.html"><span class="progress-dot"></span>Binomial vs Poisson</a></li><li data-subkey="sec3sub2"><a href="/Normal-t-F-and-Chi-Squared-Distributions-in-R.html"><span class="progress-dot"></span>Normal, t, F, Chi-Squared</a></li><li data-subkey="sec3sub2"><a href="/Central-Limit-Theorem-in-R.html"><span class="progress-dot"></span>Central Limit Theorem</a></li><li data-subkey="sec3sub2"><a href="/Sampling-Distributions-in-R.html"><span class="progress-dot"></span>Sampling Distributions</a></li><li data-subkey="sec3sub2"><a href="/Law-of-Large-Numbers-vs-CLT-in-R.html"><span class="progress-dot"></span>LLN vs CLT</a></li><li data-subkey="sec3sub2"><a href="/What-Is-Probability-Simulation-First-Intuition-in-R-Before-the-Formulas.html"><span class="progress-dot"></span>Probability (Simulation-First)</a></li><li data-subkey="sec3sub2"><a href="/Expected-Value-and-Variance-in-R.html"><span class="progress-dot"></span>Expected Value and Variance</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub3" data-collapsed="false"><span class="subsec-chevron">▼</span> Inference & Estimation</li><li data-subkey="sec3sub3"><a href="/Maximum-Likelihood-Estimation-in-R.html"><span class="progress-dot"></span>Maximum Likelihood Estimation</a></li><li data-subkey="sec3sub3"><a href="/Hypothesis-Testing-in-R.html"><span class="progress-dot"></span>Hypothesis Testing</a></li><li data-subkey="sec3sub3"><a href="/Sample-Size-Planning-in-R.html"><span class="progress-dot"></span>Sample Size Planning</a></li><li data-subkey="sec3sub3"><a href="/Which-Statistical-Test-in-R.html"><span class="progress-dot"></span>Choosing the Right Test</a></li><li data-subkey="sec3sub3"><a href="/Statistical-Tests-in-R.html"><span class="progress-dot"></span>Statistical Tests</a></li><li data-subkey="sec3sub3"><a href="/Measures-of-Association-in-R.html"><span class="progress-dot"></span>Measures of Association</a></li><li data-subkey="sec3sub3"><a href="/Point-Estimation-in-R.html"><span class="progress-dot"></span>Point Estimation</a></li><li data-subkey="sec3sub3"><a href="/Confidence-Intervals-in-R.html"><span class="progress-dot"></span>Confidence Intervals</a></li><li data-subkey="sec3sub3"><a href="/Type-I-and-Type-II-Errors-in-R.html"><span class="progress-dot"></span>Type I and II Errors</a></li><li data-subkey="sec3sub3"><a href="/Statistical-Power-Analysis-in-R.html"><span class="progress-dot"></span>Power Analysis</a></li><li data-subkey="sec3sub3"><a href="/Effect-Size-in-R.html"><span class="progress-dot"></span>Effect Size</a></li><li data-subkey="sec3sub3"><a href="/t-Tests-in-R.html"><span class="progress-dot"></span>t-Tests</a></li><li data-subkey="sec3sub3"><a href="/Proportion-Tests-in-R.html"><span class="progress-dot"></span>Proportion Tests</a></li><li data-subkey="sec3sub3"><a href="/Normality-and-Variance-Tests-in-R.html"><span class="progress-dot"></span>Normality & Variance Tests</a></li><li data-subkey="sec3sub3"><a href="/Chi-Square-Tests-in-R.html"><span class="progress-dot"></span>Chi-Square Tests</a></li><li data-subkey="sec3sub3"><a href="/Wilcoxon-Mann-Whitney-and-Kruskal-Wallis-in-R.html"><span class="progress-dot"></span>Wilcoxon, Mann-Whitney & Kruskal-Wallis</a></li><li data-subkey="sec3sub3"><a href="/Multiple-Comparisons-in-R.html"><span class="progress-dot"></span>Multiple Testing Correction</a></li><li data-subkey="sec3sub3"><a href="/Hypothesis-Testing-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Hypothesis Testing Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub4" data-collapsed="false"><span class="subsec-chevron">▼</span> Regression</li><li data-subkey="sec3sub4"><a href="/Linear-Regression.html"><span class="progress-dot"></span>Linear Regression</a></li><li data-subkey="sec3sub4"><a href="/Logistic-Regression-With-R.html"><span class="progress-dot"></span>Logistic Regression</a></li><li data-subkey="sec3sub4"><a href="/Variable-Selection-and-Importance-With-R.html"><span class="progress-dot"></span>Feature Selection</a></li><li data-subkey="sec3sub4"><a href="/Model-Selection-in-R.html"><span class="progress-dot"></span>Model Selection</a></li><li data-subkey="sec3sub4"><a href="/Missing-Value-Treatment-With-R.html"><span class="progress-dot"></span>Missing Value Treatment</a></li><li data-subkey="sec3sub4"><a href="/Outlier-Treatment-With-R.html"><span class="progress-dot"></span>Outlier Analysis</a></li><li data-subkey="sec3sub4"><a href="/adv-regression-models.html"><span class="progress-dot"></span>Advanced Regression Models</a></li><li data-subkey="sec3sub4"><a href="/Linear-Regression-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Linear Regression Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub5" data-collapsed="false"><span class="subsec-chevron">▼</span> Reporting</li><li data-subkey="sec3sub5"><a href="/Statistical-Consulting-in-R.html"><span class="progress-dot"></span>Statistical Consulting</a></li><li data-subkey="sec3sub5"><a href="/Statistical-Report-Writing-in-R.html"><span class="progress-dot"></span>Statistical Report Writing</a></li><li data-subkey="sec3sub5"><a href="/Bootstrap-Confidence-Intervals-in-R.html"><span class="progress-dot"></span>Bootstrap Confidence Intervals</a></li><li data-subkey="sec3sub5"><a href="/Reporting-Statistics-in-R.html"><span class="progress-dot"></span>Reporting Statistics</a></li><li data-subkey="sec3sub5"><a href="/Correlation-in-R.html"><span class="progress-dot"></span>Correlation (Pearson, Spearman, Kendall)</a></li><li data-subkey="sec3sub5"><a href="/Linear-Regression-Assumptions-in-R.html"><span class="progress-dot"></span>Linear Regression Assumptions</a></li><li data-subkey="sec3sub5"><a href="/Dummy-Variables-in-R.html"><span class="progress-dot"></span>Dummy Variables in R</a></li><li data-subkey="sec3sub5"><a href="/Interaction-Effects-in-R.html"><span class="progress-dot"></span>Interaction Effects</a></li><li data-subkey="sec3sub5"><a href="/Regression-Diagnostics-in-R.html"><span class="progress-dot"></span>Regression Diagnostics</a></li><li data-subkey="sec3sub5"><a href="/Logistic-Regression-in-R.html"><span class="progress-dot"></span>Logistic Regression (glm + ROC)</a></li><li data-subkey="sec3sub5"><a href="/Variable-Selection-in-R.html"><span class="progress-dot"></span>Variable Selection</a></li><li data-subkey="sec3sub5"><a href="/Poisson-Regression-in-R.html"><span class="progress-dot"></span>Poisson Regression</a></li><li data-subkey="sec3sub5"><a href="/Ridge-and-Lasso-Regression-in-R.html"><span class="progress-dot"></span>Ridge & Lasso Regression</a></li><li data-subkey="sec3sub5"><a href="/Polynomial-and-Spline-Regression-in-R.html"><span class="progress-dot"></span>Polynomial & Splines</a></li><li data-subkey="sec3sub5"><a href="/Regression-Tables-in-R.html"><span class="progress-dot"></span>Regression Tables (3 packages)</a></li><li data-subkey="sec3sub5"><a href="/One-Way-ANOVA-in-R.html"><span class="progress-dot"></span>One-Way ANOVA</a></li><li data-subkey="sec3sub5"><a href="/Post-Hoc-Tests-After-ANOVA.html"><span class="progress-dot"></span>Post-Hoc Tests After ANOVA</a></li><li data-subkey="sec3sub5"><a href="/Two-Way-ANOVA-in-R.html"><span class="progress-dot"></span>Two-Way ANOVA</a></li><li data-subkey="sec3sub5"><a href="/Repeated-Measures-ANOVA-in-R.html"><span class="progress-dot"></span>Repeated Measures ANOVA</a></li><li data-subkey="sec3sub5"><a href="/ANCOVA-in-R.html"><span class="progress-dot"></span>ANCOVA</a></li><li data-subkey="sec3sub5"><a href="/Experimental-Design-Principles-in-R.html"><span class="progress-dot"></span>Experimental Design in R</a></li><li data-subkey="sec3sub5"><a href="/Factorial-Experiments-in-R.html"><span class="progress-dot"></span>Factorial Designs (2^k)</a></li><li data-subkey="sec3sub5"><a href="/AB-Testing-in-R.html"><span class="progress-dot"></span>A/B Testing</a></li><li data-subkey="sec3sub5"><a href="/MANOVA-in-R.html"><span class="progress-dot"></span>MANOVA</a></li><li data-subkey="sec3sub5"><a href="/Mixed-ANOVA-in-R.html"><span class="progress-dot"></span>Mixed ANOVA</a></li><li data-subkey="sec3sub5"><a href="/Multivariate-Statistics-in-R.html"><span class="progress-dot"></span>Multivariate Distances & Hotelling's T²</a></li><li data-subkey="sec3sub5"><a href="/PCA-in-R.html"><span class="progress-dot"></span>PCA with prcomp()</a></li><li data-subkey="sec3sub5"><a href="/Interpreting-PCA-Results-in-R.html"><span class="progress-dot"></span>Interpreting PCA Output</a></li><li data-subkey="sec3sub5"><a href="/Exploratory-Factor-Analysis-in-R.html"><span class="progress-dot"></span>Exploratory Factor Analysis</a></li><li 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href="/Robust-Regression-in-R.html"><span class="progress-dot"></span>Robust Regression (rlm)</a></li><li data-subkey="sec3sub5"><a href="/factoextra-and-FactoMineR.html"><span class="progress-dot"></span>factoextra (PCA + Clusters)</a></li><li data-subkey="sec3sub5"><a href="/Categorical-Data-in-R.html"><span class="progress-dot"></span>Categorical Data (Tables & Mosaic)</a></li><li data-subkey="sec3sub5"><a href="/Chi-Square-Test-of-Independence-in-R.html"><span class="progress-dot"></span>Chi-Square Test of Independence</a></li><li data-subkey="sec3sub5"><a href="/Chi-Square-Goodness-of-Fit-Test-in-R.html"><span class="progress-dot"></span>Chi-Square Goodness-of-Fit</a></li><li data-subkey="sec3sub5"><a href="/Fishers-Exact-Test-in-R.html"><span class="progress-dot"></span>Fisher's Exact Test</a></li><li data-subkey="sec3sub5"><a href="/Odds-Ratios-and-Relative-Risk-in-R.html"><span class="progress-dot"></span>Odds Ratios & Relative Risk</a></li><li data-subkey="sec3sub5"><a href="/Logistic-Regression-in-R-2.html"><span class="progress-dot"></span>Logistic Regression (Diagnostics)</a></li><li data-subkey="sec3sub5"><a href="/Poisson-and-Negative-Binomial-Regression.html"><span class="progress-dot"></span>Poisson & Negative Binomial Regression</a></li><li data-subkey="sec3sub5"><a href="/Multinomial-and-Ordinal-Logistic-Regression-in-R.html"><span class="progress-dot"></span>Multinomial & Ordinal Logistic Regression</a></li><li data-subkey="sec3sub5"><a href="/When-to-Use-Nonparametric-Tests-in-R.html"><span class="progress-dot"></span>When to Use Nonparametric Tests</a></li><li data-subkey="sec3sub5"><a href="/Wilcoxon-Signed-Rank-Test-in-R.html"><span class="progress-dot"></span>Wilcoxon Signed-Rank Test</a></li><li data-subkey="sec3sub5"><a href="/Mann-Whitney-U-Test-in-R.html"><span class="progress-dot"></span>Mann-Whitney U Test</a></li><li data-subkey="sec3sub5"><a href="/Kruskal-Wallis-Test-in-R-2.html"><span class="progress-dot"></span>Kruskal-Wallis Test</a></li><li data-subkey="sec3sub5"><a href="/Friedman-Test-in-R.html"><span class="progress-dot"></span>Friedman Test</a></li><li data-subkey="sec3sub5"><a href="/Spearman-and-Kendall-Correlation-in-R.html"><span class="progress-dot"></span>Spearman & Kendall Correlation</a></li><li data-subkey="sec3sub5"><a href="/Bootstrap-in-R.html" class="active"><span class="progress-dot"></span>Bootstrap (boot package)</a></li><li data-subkey="sec3sub5"><a href="/Quantile-Regression-in-R-2.html"><span class="progress-dot"></span>Quantile Regression</a></li><li data-subkey="sec3sub5"><a href="/Matrix-Operations-in-R.html"><span class="progress-dot"></span>Matrix Operations in R</a></li><li data-subkey="sec3sub5"><a href="/Solving-Linear-Systems-in-R.html"><span class="progress-dot"></span>Solving Linear Systems in R</a></li><li data-subkey="sec3sub5"><a href="/Eigenvalues-and-Eigenvectors-in-R.html"><span class="progress-dot"></span>Eigenvalues & Eigenvectors in R</a></li><li data-subkey="sec3sub5"><a href="/Singular-Value-Decomposition-in-R.html"><span class="progress-dot"></span>Singular Value Decomposition in R</a></li><li data-subkey="sec3sub5"><a href="/Projections-and-the-Hat-Matrix-in-R.html"><span class="progress-dot"></span>Projections & the Hat Matrix</a></li><li data-subkey="sec3sub5"><a href="/QR-Decomposition-in-R.html"><span class="progress-dot"></span>QR Decomposition in R</a></li><li data-subkey="sec3sub5"><a href="/Quadratic-Forms-in-R.html"><span class="progress-dot"></span>Quadratic Forms</a></li><li data-subkey="sec3sub5"><a href="/Matrix-Derivatives-and-the-Hessian-in-R.html"><span class="progress-dot"></span>Matrix Derivatives & Hessian</a></li><li data-subkey="sec3sub5"><a href="/Exponential-Family-Distributions-in-R.html"><span class="progress-dot"></span>Exponential Family Distributions</a></li><li data-subkey="sec3sub5"><a href="/Sufficient-Statistics-in-R.html"><span class="progress-dot"></span>Sufficient Statistics</a></li><li data-subkey="sec3sub5"><a href="/Complete-and-Ancillary-Statistics-in-R.html"><span class="progress-dot"></span>Complete & Ancillary Statistics</a></li><li data-subkey="sec3sub5"><a href="/UMVUE-in-R-2.html"><span class="progress-dot"></span>UMVUE (Rao-Blackwell & Lehmann-Scheffé)</a></li><li data-subkey="sec3sub5"><a href="/Cramer-Rao-Lower-Bound-in-R-2.html"><span class="progress-dot"></span>Cramér-Rao Lower Bound</a></li><li data-subkey="sec3sub5"><a href="/Asymptotic-Theory-in-R-2.html"><span class="progress-dot"></span>Asymptotic Theory</a></li><li data-subkey="sec3sub5"><a href="/Neyman-Pearson-Lemma-in-R-2.html"><span class="progress-dot"></span>Neyman-Pearson Lemma</a></li><li data-subkey="sec3sub5"><a href="/Likelihood-Ratio-Tests-and-Pivotal-Methods.html"><span class="progress-dot"></span>Likelihood Ratio & Pivotal Methods</a></li><li data-subkey="sec3sub5"><a href="/Decision-Theory-in-R.html"><span class="progress-dot"></span>Decision Theory</a></li><li data-subkey="sec3sub5"><a href="/Asymptotic-Relative-Efficiency-in-R.html"><span class="progress-dot"></span>Asymptotic Relative Efficiency</a></li><li data-subkey="sec3sub5"><a href="/Bayes-Theorem-in-R.html"><span class="progress-dot"></span>Bayes' Theorem</a></li><li data-subkey="sec3sub5"><a href="/Bayesian-Statistics-in-R.html"><span class="progress-dot"></span>Bayesian Statistics</a></li><li data-subkey="sec3sub5"><a href="/Conjugate-Priors-in-R.html"><span class="progress-dot"></span>Conjugate Priors</a></li><li data-subkey="sec3sub5"><a href="/Grid-Approximation-in-R.html"><span class="progress-dot"></span>Grid Approximation</a></li><li data-subkey="sec3sub5"><a href="/MCMC-in-R.html"><span class="progress-dot"></span>MCMC in R</a></li><li data-subkey="sec3sub5"><a href="/Gibbs-Sampling-in-R.html"><span class="progress-dot"></span>Gibbs Sampling</a></li><li data-subkey="sec3sub5"><a href="/Hamiltonian-Monte-Carlo-in-R.html"><span class="progress-dot"></span>Hamiltonian Monte Carlo</a></li><li data-subkey="sec3sub5"><a href="/Stan-in-R.html"><span class="progress-dot"></span>Stan</a></li><li data-subkey="sec3sub5"><a href="/brms-in-R.html"><span class="progress-dot"></span>brms</a></li><li data-subkey="sec3sub5"><a href="/Choosing-Priors-in-R.html"><span class="progress-dot"></span>Choosing Priors</a></li><li data-subkey="sec3sub5"><a href="/Prior-Predictive-Checks-in-R.html"><span class="progress-dot"></span>Prior Predictive Checks</a></li><li data-subkey="sec3sub5"><a href="/Compare-Bayesian-Models-in-R.html"><span class="progress-dot"></span>Compare Bayesian Models</a></li><li data-subkey="sec3sub5"><a href="/Posterior-Predictive-Checks-in-R.html"><span class="progress-dot"></span>Posterior Predictive Checks</a></li><li data-subkey="sec3sub5"><a href="/Bayesian-Linear-Regression-in-R.html"><span class="progress-dot"></span>Bayesian Linear Regression</a></li><li data-subkey="sec3sub5"><a href="/Bayesian-Logistic-Regression-in-R.html"><span class="progress-dot"></span>Bayesian Logistic Regression</a></li><li data-subkey="sec3sub5"><a href="/Bayesian-Hierarchical-Models-in-R.html"><span class="progress-dot"></span>Bayesian Hierarchical Models</a></li><li data-subkey="sec3sub5"><a href="/Multilevel-Models-in-R.html"><span class="progress-dot"></span>Multilevel Models</a></li><li data-subkey="sec3sub5"><a href="/Bayesian-ANOVA-in-R.html"><span class="progress-dot"></span>Bayesian ANOVA</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub6" data-collapsed="false"><span class="subsec-chevron">▼</span> Machine Learning</li><li data-subkey="sec3sub6"><a href="/Machine-Learning-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Machine Learning Quiz</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span> Time Series<span class="section-meta" 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class="subsec-chevron">▼</span> Functional Programming</li><li data-subkey="sec5sub1"><a href="/Functional-Programming-in-R.html"><span class="progress-dot"></span>Functional Programming</a></li><li data-subkey="sec5sub1"><a href="/R-Functional-Programming-Exercises-quiz.html"><span class="progress-dot"></span>Functional Programming Quiz</a></li><li data-subkey="sec5sub1"><a href="/purrr-map-Variants.html"><span class="progress-dot"></span>purrr map() Variants</a></li><li data-subkey="sec5sub1"><a href="/R-Anonymous-Functions.html"><span class="progress-dot"></span>R Anonymous Functions</a></li><li data-subkey="sec5sub1"><a href="/R-Function-Factories.html"><span class="progress-dot"></span>R Function Factories</a></li><li data-subkey="sec5sub1"><a href="/R-Function-Operators.html"><span class="progress-dot"></span>R Function Operators</a></li><li data-subkey="sec5sub1"><a href="/Reduce-Filter-Map-in-R.html"><span class="progress-dot"></span>Reduce, Filter, Map</a></li><li data-subkey="sec5sub1"><a href="/Memoization-in-R.html"><span class="progress-dot"></span>Memoization in R</a></li><li data-subkey="sec5sub1"><a href="/Writing-Composable-R-Code.html"><span class="progress-dot"></span>Composable R Code</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec5sub2" data-collapsed="false"><span class="subsec-chevron">▼</span> OOP in R</li><li data-subkey="sec5sub2"><a href="/OOP-in-R.html"><span class="progress-dot"></span>OOP in R: S3/S4/R6</a></li><li data-subkey="sec5sub2"><a href="/S3-Classes-in-R.html"><span class="progress-dot"></span>S3 Classes</a></li><li data-subkey="sec5sub2"><a href="/S3-Method-Dispatch-in-R.html"><span class="progress-dot"></span>S3 Method Dispatch</a></li><li data-subkey="sec5sub2"><a href="/S4-Classes-in-R.html"><span class="progress-dot"></span>S4 Classes</a></li><li data-subkey="sec5sub2"><a href="/S4-Methods-in-R.html"><span class="progress-dot"></span>S4 Methods & Dispatch</a></li><li data-subkey="sec5sub2"><a href="/R6-Classes-in-R.html"><span class="progress-dot"></span>R6 Classes</a></li><li data-subkey="sec5sub2"><a href="/R6-Advanced.html"><span class="progress-dot"></span>R6 Advanced</a></li><li data-subkey="sec5sub2"><a href="/Operator-Overloading-in-R.html"><span class="progress-dot"></span>Operator Overloading</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec5sub3" data-collapsed="false"><span class="subsec-chevron">▼</span> How R Works</li><li data-subkey="sec5sub3"><a href="/R-Names-and-Values.html"><span class="progress-dot"></span>R Names & Values</a></li><li data-subkey="sec5sub3"><a href="/R-Assignment-Deep-Dive.html"><span class="progress-dot"></span>R Assignment Deep Dive</a></li><li data-subkey="sec5sub3"><a href="/R-Memory-lobstr.html"><span class="progress-dot"></span>R Memory & lobstr</a></li><li data-subkey="sec5sub3"><a href="/R-Environments.html"><span class="progress-dot"></span>R Environments</a></li><li data-subkey="sec5sub3"><a href="/R-Lexical-Scoping.html"><span class="progress-dot"></span>Lexical Scoping</a></li><li data-subkey="sec5sub3"><a href="/R-Closures.html"><span class="progress-dot"></span>R Closures</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec5sub4" data-collapsed="false"><span class="subsec-chevron">▼</span> Debugging & Performance</li><li data-subkey="sec5sub4"><a href="/R-Conditions-System.html"><span class="progress-dot"></span>Conditions System</a></li><li data-subkey="sec5sub4"><a href="/R-Debugging.html"><span class="progress-dot"></span>Debugging R Code</a></li><li data-subkey="sec5sub4"><a href="/R-Common-Errors.html"><span class="progress-dot"></span>50 Common R Errors</a></li><li data-subkey="sec5sub4"><a href="/Parallel-Computing-With-R.html"><span class="progress-dot"></span>Parallel Computing</a></li><li data-subkey="sec5sub4"><a href="/Strategies-To-Improve-And-Speedup-R-Code.html"><span class="progress-dot"></span>Speedup R Code</a></li><li data-subkey="sec5sub4"><a href="/Shiny-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Shiny Quiz</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span> Classic Tutorials<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li data-subkey="sec6sub0"><a href="/R-Tutorial.html"><span class="progress-dot"></span>R Tutorial (Classic)</a></li><li data-subkey="sec6sub0"><a href="/ggplot2-Tutorial-With-R.html"><span class="progress-dot"></span>ggplot2 Short Tutorial</a></li><li data-subkey="sec6sub0"><a href="/Complete-Ggplot2-Tutorial-Part1-With-R-Code.html"><span class="progress-dot"></span>ggplot2 Tutorial 1 - Intro</a></li><li data-subkey="sec6sub0"><a href="/Complete-Ggplot2-Tutorial-Part2-Customizing-Theme-With-R-Code.html"><span class="progress-dot"></span>ggplot2 Tutorial 2 - Theme</a></li><li data-subkey="sec6sub0"><a href="/Top50-Ggplot2-Visualizations-MasterList-R-Code.html"><span class="progress-dot"></span>ggplot2 Tutorial 3 - Masterlist</a></li><li data-subkey="sec6sub0"><a href="/Association-Mining-With-R.html"><span class="progress-dot"></span>Association Mining</a></li><li data-subkey="sec6sub0"><a href="/Multi-Dimensional-Scaling-With-R.html"><span class="progress-dot"></span>Multi Dimensional Scaling</a></li><li data-subkey="sec6sub0"><a href="/Optimization-With-R.html"><span class="progress-dot"></span>Optimization</a></li><li data-subkey="sec6sub0"><a href="/Information-Value-With-R.html"><span class="progress-dot"></span>InformationValue Package</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span> Practice Exercises<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec7sub1" data-collapsed="false"><span class="subsec-chevron">▼</span> Mastery Quizzes (Certificate)</li><li data-subkey="sec7sub1"><a href="/R-Beginner-Exercises-quiz.html"><span class="progress-dot"></span>R Fundamentals Quiz</a></li><li data-subkey="sec7sub1"><a href="/dplyr-Exercises-in-R-quiz.html"><span class="progress-dot"></span>dplyr Quiz</a></li><li data-subkey="sec7sub1"><a href="/ggplot2-Exercises-in-R-quiz.html"><span class="progress-dot"></span>ggplot2 Quiz</a></li><li data-subkey="sec7sub1"><a href="/Hypothesis-Testing-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Hypothesis Testing Quiz</a></li><li data-subkey="sec7sub1"><a href="/Linear-Regression-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Linear Regression Quiz</a></li><li data-subkey="sec7sub1"><a href="/Machine-Learning-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Machine Learning Quiz</a></li><li data-subkey="sec7sub1"><a href="/tidyr-Exercises-in-R-quiz.html"><span 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<h1>Bootstrap in R: boot Package, CI & Hypothesis Tests Without Assumptions</h1>
<p class="lead">The bootstrap resamples your data with replacement to estimate confidence intervals and run hypothesis tests without assuming a <a class="auto-link" href="Normal-t-F-and-Chi-Squared-Distributions-in-R.html" title="Normal, t, F, and Chi-Squared in R: Understand Each Distribution and When It Arises">normal distribution</a>. The <code>boot</code> package in R turns the workflow into two calls: <code>boot()</code> to generate replicates and <code>boot.ci()</code> to extract BCa, percentile, basic, or normal intervals.</p>
<div class="post-byline" style="color:#6b7280;font-size:14px;margin:2px 0 18px 0;line-height:1.5;">By <strong>Selva Prabhakaran</strong> · Published May 10, 2026 · Last updated May 10, 2026</div>
<div class="engagement-header" data-difficulty="Intermediate" data-time="30" data-exercises="8" data-xp="120"></div>
<h2>How do you bootstrap in R with the boot package?</h2>
<p>The fastest way to see the bootstrap in action is to compute a 95% CI for a statistic that has no clean textbook formula. The median of a small sample fits: there is no closed-form CI, and its sampling distribution is rarely normal. Here is the entire bootstrap pipeline for the median MPG in <code>mtcars</code>.</p>
<p>The pattern is always the same: define a statistic function that takes data and an index vector, hand it to <code>boot()</code>, then pass the result to <code>boot.ci()</code>.</p>
<div class="webr-container" data-block-title="First bootstrap CI for median MPG">
<div class="webr-code-block">
<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">First bootstrap CI for median MPG</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl"><span class="nf">library</span>(boot)</span>
<span class="cl"><span class="nf">set.seed</span>(<span class="m">42</span>)</span>
<span class="cl"></span>
<span class="cl">median_stat <span class="o"><-</span> <span class="kr">function</span>(d, i) <span class="nf">median</span>(d[i])</span>
<span class="cl">boot_out <span class="o"><-</span> <span class="nf">boot</span>(data <span class="o">=</span> mtcars<span class="o">$</span>mpg, statistic <span class="o">=</span> median_stat, R <span class="o">=</span> <span class="m">2000</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="nf">boot.ci</span>(boot_out, type <span class="o">=</span> <span class="nf">c</span>(<span class="s">"perc"</span>, <span class="s">"bca"</span>))</span>
<span class="cl"><span class="c1">#> BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS</span></span>
<span class="cl"><span class="c1">#> Based on 2000 bootstrap replicates</span></span>
<span class="cl"><span class="c1">#></span></span>
<span class="cl"><span class="c1">#> Intervals :</span></span>
<span class="cl"><span class="c1">#> Level Percentile BCa</span></span>
<span class="cl"><span class="c1">#> 95% (16.40, 21.40) (16.40, 22.05)</span></span></div>
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<p>Two thousand resamples produced two 95% intervals for the median. The percentile CI runs from 16.4 to 21.4 MPG. The BCa CI runs slightly wider on the upper end (22.05) because BCa adjusts for <a class="auto-link" href="Descriptive-Statistics-in-R.html" title="Descriptive Statistics in R: The 8 Numbers That Tell You What Your Data Is Doing">skewness</a> in the <a class="auto-link" href="Bootstrap-Confidence-Intervals-in-R.html" title="Bootstrap CIs in R: Distribution-Free Confidence Intervals for Any Statistic">bootstrap distribution</a>. Both agree the true population median plausibly sits in this range, and you got there without a single distributional assumption.</p>
<div class="callout callout-tip"><div class="callout-label">Tip</div><div class="callout-body"><strong>Always set a seed before <code>boot()</code>.</strong> Two <code>boot()</code> calls with the same seed produce identical replicates, which means your CIs are reproducible across reviewers and across re-runs. Pick a fresh integer for each major example so you do not silently reuse the same random pattern.</div></div>
<p>The <code>statistic</code> function has a strict signature: first argument is the data, second is an integer index vector. Inside the function, <code>d[i]</code> (for vectors) or <code>d[i, ]</code> (for data frames) builds the bootstrap sample. <code>boot()</code> calls your function <code>R</code> times with random indices and stores every replicate in <code>boot_out$t</code>.</p>
<section class="tryit-block">
<p><strong>Try it:</strong> Write a function <code>ex_mean_stat()</code> that returns the mean instead of the median, then bootstrap a 95% percentile CI for the mean MPG. Use 1000 replicates and seed <code>123</code>.</p>
<div class="webr-container" data-block-title="Your turn: bootstrap a CI for the mean">
<div class="webr-code-block">
<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Your turn: bootstrap a CI for the mean</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl"><span class="c1"># Try it: bootstrap a CI for the mean MPG</span></span>
<span class="cl">ex_mean_stat <span class="o"><-</span> <span class="kr">function</span>(d, i) {</span>
<span class="cl"> <span class="c1"># your code here</span></span>
<span class="cl">}</span>
<span class="cl"></span>
<span class="cl"><span class="nf">set.seed</span>(<span class="m">123</span>)</span>
<span class="cl">ex_boot <span class="o"><-</span> <span class="nf">boot</span>(mtcars<span class="o">$</span>mpg, ex_mean_stat, R <span class="o">=</span> <span class="m">1000</span>)</span>
<span class="cl"><span class="nf">boot.ci</span>(ex_boot, type <span class="o">=</span> <span class="s">"perc"</span>)</span>
<span class="cl"><span class="c1">#> Expected: a 95% CI roughly (18.0, 22.3)</span></span></div>
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<details>
<summary>Click to reveal solution</summary>
<div class="webr-container" data-block-title="Mean MPG bootstrap solution">
<div class="webr-code-block">
<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Mean MPG bootstrap solution</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl">ex_mean_stat <span class="o"><-</span> <span class="kr">function</span>(d, i) <span class="nf">mean</span>(d[i])</span>
<span class="cl"></span>
<span class="cl"><span class="nf">set.seed</span>(<span class="m">123</span>)</span>
<span class="cl">ex_boot <span class="o"><-</span> <span class="nf">boot</span>(mtcars<span class="o">$</span>mpg, ex_mean_stat, R <span class="o">=</span> <span class="m">1000</span>)</span>
<span class="cl"><span class="nf">boot.ci</span>(ex_boot, type <span class="o">=</span> <span class="s">"perc"</span>)</span>
<span class="cl"><span class="c1">#> Level Percentile</span></span>
<span class="cl"><span class="c1">#> 95% (18.04, 22.27)</span></span></div>
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<p><strong>Explanation:</strong> Replacing <code>median</code> with <code>mean</code> is the only change. Every other line stays the same, which is the whole point of the boot package: the workflow is statistic-agnostic.</p>
</details>
</section>
<h2>Why does bootstrap work without distributional assumptions?</h2>
<p>Classical confidence intervals lean on a formula like $\bar{x} \pm 1.96 \cdot s/\sqrt{n}$. That formula is correct only if the <a class="auto-link" href="Central-Limit-Theorem-in-R.html" title="Central Limit Theorem in R: Simulate It From Skewed, Bimodal, and Uniform Distributions">sampling distribution of the mean</a> is approximately normal, which itself requires assumptions about the population or a sufficiently large sample. The bootstrap sidesteps that machinery by replacing "what would the population look like?" with "the sample is the best guess we have for the population."</p>
<p>Mathematically, the bootstrap uses the <strong>empirical distribution function</strong>, the discrete distribution that puts probability $1/n$ on each observed data point. Sampling with replacement from your data is sampling from that empirical distribution. As $n$ grows, it converges to the true population, so resampled statistics converge to true sampling statistics.</p>
<p>$$\hat{F}_n(x) = \frac{1}{n} \sum_{i=1}^{n} \mathbb{1}\{x_i \leq x\}$$</p>
<p>Where:</p>
<ul>
<li>$\hat{F}_n(x)$ = the empirical distribution function evaluated at $x$</li>
<li>$n$ = sample size</li>
<li>$\mathbb{1}\{x_i \leq x\}$ = an indicator that is 1 if $x_i \leq x$, else 0</li>
</ul>
<p>If you are not interested in the math, skip to the next paragraph. The practical takeaway is unchanged: every replicate is a "what could my sample have looked like?" scenario, and the spread of replicates is the spread of the statistic.</p>
<p><img src="screenshots/Bootstrap-in-R-resampling-flow.webp" alt="How the bootstrap turns one sample into an inference" class="img-responsive img-zoomable" loading="lazy" width="2248" height="638" /></p>
<p><em>Figure 1: The bootstrap resamples one observed sample many times to build a distribution of any statistic.</em></p>
<p>The bootstrap distribution is just the histogram of <code>boot_out$t</code>. Plotting it tells you the shape of the sampling distribution, the centre (your point estimate), and the spread (the standard error).</p>
<div class="webr-container" data-block-title="Visualise the bootstrap distribution">
<div class="webr-code-block">
<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Visualise the bootstrap distribution</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl"><span class="nf">hist</span>(boot_out<span class="o">$</span>t,</span>
<span class="cl"> breaks <span class="o">=</span> <span class="m">30</span>,</span>
<span class="cl"> col <span class="o">=</span> <span class="s">"steelblue"</span>,</span>
<span class="cl"> border <span class="o">=</span> <span class="s">"white"</span>,</span>
<span class="cl"> main <span class="o">=</span> <span class="s">"Bootstrap distribution of median(mpg)"</span>,</span>
<span class="cl"> xlab <span class="o">=</span> <span class="s">"Bootstrap median MPG"</span>)</span>
<span class="cl"><span class="nf">abline</span>(v <span class="o">=</span> <span class="nf">median</span>(mtcars<span class="o">$</span>mpg), col <span class="o">=</span> <span class="s">"red"</span>, lwd <span class="o">=</span> <span class="m">2</span>)</span>
<span class="cl"><span class="c1">#> Histogram with a peak near 19, the red vertical at the observed median</span></span></div>
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<p>The peak of the histogram lines up with the observed median (red line), confirming the bootstrap is unbiased on average. The width of the histogram is the bootstrap standard error: roughly the standard deviation of the replicates. If you saw a wildly skewed shape or two distinct modes, you would know that a normal-based CI would be wrong here, even before computing the interval.</p>
<div class="callout callout-insight"><div class="callout-label">Key Insight</div><div class="callout-body"><strong>The bootstrap replaces "trust the formula" with "trust the data."</strong> Classical CIs require the sampling distribution to be approximately normal; the bootstrap simulates the sampling distribution from the data itself. The trade-off is computation time, which is cheap, against assumptions, which are expensive when wrong.</div></div>
<section class="tryit-block">
<p><strong>Try it:</strong> The bootstrap standard error is just the standard deviation of replicates. Compute it two ways and check they agree: with <code>sd(boot_out$t)</code> and by reading the <code>std. error</code> printed by <code>boot_out</code>.</p>
<div class="webr-container" data-block-title="Your turn: two views of the bootstrap SE">
<div class="webr-code-block">
<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Your turn: two views of the bootstrap SE</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl"><span class="c1"># Try it: SE from sd() of replicates</span></span>
<span class="cl">ex_se_manual <span class="o"><-</span> <span class="c1"># your code here</span></span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Compare to:</span></span>
<span class="cl">boot_out</span>
<span class="cl"><span class="c1">#> Expected: ex_se_manual matches the std. error printed by boot_out</span></span></div>
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<div class="webr-container" data-block-title="Bootstrap SE solution">
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<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Bootstrap SE solution</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl">ex_se_manual <span class="o"><-</span> <span class="nf">sd</span>(boot_out<span class="o">$</span>t)</span>
<span class="cl">ex_se_manual</span>
<span class="cl"><span class="c1">#> [1] 1.412</span></span>
<span class="cl"></span>
<span class="cl"><span class="nf">print</span>(boot_out)</span>
<span class="cl"><span class="c1">#> Bootstrap Statistics :</span></span>
<span class="cl"><span class="c1">#> original bias std. error</span></span>
<span class="cl"><span class="c1">#> t1* 19.2 0.0723 1.412</span></span></div>
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<p><strong>Explanation:</strong> <code>boot_out$t</code> is the matrix of replicates. Its column standard deviation is the bootstrap SE that <code>boot_out</code> prints. Two views, same number.</p>
</details>
</section>
<h2>Which bootstrap confidence interval should you use?</h2>
<p>The boot package offers five interval types: percentile, basic, normal, BCa, and studentized. The first four are one-line calls; studentized needs an extra variance estimate per replicate. For most applied work, the choice narrows to BCa or percentile.</p>
<p><img src="screenshots/Bootstrap-in-R-ci-decision.webp" alt="Which bootstrap CI should you pick?" class="img-responsive img-zoomable" loading="lazy" width="1796" height="806" /></p>
<p><em>Figure 2: A quick decision rule for picking the right <code>boot.ci()</code> type.</em></p>
<p>Asking for all four at once is a quick way to compare them on the same data. A large gap between them is a warning that one or more is misbehaving.</p>
<div class="webr-container" data-block-title="Compare four CI types side by side">
<div class="webr-code-block">
<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Compare four CI types side by side</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl">ci_all <span class="o"><-</span> <span class="nf">boot.ci</span>(boot_out, type <span class="o">=</span> <span class="nf">c</span>(<span class="s">"norm"</span>, <span class="s">"basic"</span>, <span class="s">"perc"</span>, <span class="s">"bca"</span>))</span>
<span class="cl">ci_all</span>
<span class="cl"><span class="c1">#> Intervals :</span></span>
<span class="cl"><span class="c1">#> Level Normal Basic</span></span>
<span class="cl"><span class="c1">#> 95% (16.36, 21.91 ) (16.40, 21.40 )</span></span>
<span class="cl"><span class="c1">#></span></span>
<span class="cl"><span class="c1">#> Level Percentile BCa</span></span>
<span class="cl"><span class="c1">#> 95% (16.40, 21.40 ) (16.40, 22.05 )</span></span></div>
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<p>Three of the four CIs share their lower bound (16.4) because the bootstrap distribution of the median has a hard floor near that value. Their upper bounds differ: BCa stretches to 22.05 while percentile and basic stop at 21.40. BCa is doing what it was designed for, correcting for the asymmetric tail of the distribution. The Normal CI extends below 16.40, which is impossible since the percentile bound is 16.40, signalling that a Gaussian approximation does not fit this distribution.</p>
<div class="callout callout-tip"><div class="callout-label">Tip</div><div class="callout-body"><strong>Default to BCa unless you have a reason not to.</strong> BCa adjusts for both bias and skewness and is recommended by Davison & Hinkley as the all-purpose CI. Percentile is fine when the bootstrap distribution looks symmetric; Normal is fastest but assumes symmetry; Basic is rarely the best choice but works as a sanity check.</div></div>
<p>You can also pull out just the BCa interval as a numeric vector. The <code>bca</code> element holds the level, the bootstrap quantile indices, and the lower and upper bounds in its last two columns.</p>
<section class="tryit-block">
<p><strong>Try it:</strong> Extract the lower and upper BCa bounds from <code>ci_all</code> as a length-2 numeric vector named <code>ex_bca</code>.</p>
<div class="webr-container" data-block-title="Your turn: extract BCa bounds">
<div class="webr-code-block">
<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Your turn: extract BCa bounds</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl"><span class="c1"># Try it: pull out just the BCa bounds</span></span>
<span class="cl">ex_bca <span class="o"><-</span> <span class="c1"># your code here</span></span>
<span class="cl">ex_bca</span>
<span class="cl"><span class="c1">#> Expected: c(16.4, 22.05) (or similar)</span></span></div>
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<details>
<summary>Click to reveal solution</summary>
<div class="webr-container" data-block-title="Extract BCa bounds solution">
<div class="webr-code-block">
<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Extract BCa bounds solution</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl">ex_bca <span class="o"><-</span> ci_all<span class="o">$</span>bca[<span class="m">1</span>, <span class="m">4</span><span class="o">:</span><span class="m">5</span>]</span>
<span class="cl">ex_bca</span>
<span class="cl"><span class="c1">#> [1] 16.40 22.05</span></span></div>
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<p><strong>Explanation:</strong> <code>ci_all$bca</code> is a 1x5 matrix. Columns 4 and 5 are the lower and upper bounds. Other CI types live in <code>$normal</code>, <code>$basic</code>, and <code>$percent</code>.</p>
</details>
</section>
<h2>How do you run a bootstrap hypothesis test in R?</h2>
<p>There are two common routes. The CI route is the simplest: bootstrap a CI for the <a class="auto-link" href="Hypothesis-Testing-in-R.html" title="Hypothesis Testing in R: Understand the Framework, Not Just the p-Value">test statistic</a> and reject the null at level $\alpha$ if the null value falls outside the $1-\alpha$ CI. The shift route is more direct: simulate replicates that satisfy the null, then count how often they are at least as extreme as the observed statistic.</p>
<p>The CI route shines for "is the mean of group A different from group B?" questions. Bootstrap a CI for the difference in means; if zero is outside the CI, reject the null of equal means.</p>
<div class="webr-container" data-block-title="Bootstrap CI for the difference in means">
<div class="webr-code-block">
<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Bootstrap CI for the difference in means</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl">mpg_diff <span class="o"><-</span> <span class="kr">function</span>(d, i) {</span>
<span class="cl"> s <span class="o"><-</span> d[i, ]</span>
<span class="cl"> <span class="nf">mean</span>(s<span class="o">$</span>mpg[s<span class="o">$</span>cyl <span class="o">==</span> <span class="m">4</span>]) <span class="o">-</span> <span class="nf">mean</span>(s<span class="o">$</span>mpg[s<span class="o">$</span>cyl <span class="o">==</span> <span class="m">6</span>])</span>
<span class="cl">}</span>
<span class="cl"></span>
<span class="cl">data4_6 <span class="o"><-</span> <span class="nf">subset</span>(mtcars, cyl <span class="o">%in%</span> <span class="nf">c</span>(<span class="m">4</span>, <span class="m">6</span>))</span>
<span class="cl"><span class="nf">set.seed</span>(<span class="m">7</span>)</span>
<span class="cl">boot_diff <span class="o"><-</span> <span class="nf">boot</span>(data4_6, mpg_diff, R <span class="o">=</span> <span class="m">2000</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="nf">boot.ci</span>(boot_diff, type <span class="o">=</span> <span class="s">"bca"</span>)</span>
<span class="cl"><span class="c1">#> Level BCa</span></span>
<span class="cl"><span class="c1">#> 95% ( 4.74, 9.83 )</span></span></div>
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<p>The 95% BCa CI for the difference in mean MPG (4-cyl minus 6-cyl) runs from 4.74 to 9.83. Zero is far outside. We reject the null that the two cylinder groups have the same mean MPG, with the same conclusion you would get from a <a class="auto-link" href="https://r-statistics.co/How-to-do-One-Sample-t-Test-in-R.html" title="t-Test in R: One-Sample, Two-Sample and Paired (With Examples)">t-test</a>, but without the equal-variance or normality assumption.</p>
<p>The shift route gives you a proper bootstrap p-value. Centre the replicates so their mean is zero (the null), then count how many of them are at least as far from zero as the observed difference.</p>
<div class="webr-container" data-block-title="Bootstrap p-value via shifted replicates">
<div class="webr-code-block">
<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Bootstrap p-value via shifted replicates</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl">obs_diff <span class="o"><-</span> boot_diff<span class="o">$</span>t0</span>
<span class="cl">centered <span class="o"><-</span> boot_diff<span class="o">$</span>t <span class="o">-</span> <span class="nf">mean</span>(boot_diff<span class="o">$</span>t)</span>
<span class="cl">pval <span class="o"><-</span> <span class="nf">mean</span>(<span class="nf">abs</span>(centered) <span class="o">>=</span> <span class="nf">abs</span>(obs_diff))</span>
<span class="cl"></span>
<span class="cl">pval</span>
<span class="cl"><span class="c1">#> [1] 0</span></span></div>
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<p>The two-sided p-value is effectively zero, again rejecting the null. This is the same logic as a <a class="auto-link" href="Permutation-Tests-in-R.html" title="Permutation Tests in R: Exact p-Values via Randomization">permutation test</a>: build a distribution under H0 and ask how surprising the data are. The CI route asks the dual question, "is the null inside the plausible range?", and gives the same answer when the test is well-defined.</p>
<div class="callout callout-warning"><div class="callout-label">Warning</div><div class="callout-body"><strong>A bootstrap CI is not a permutation test.</strong> The CI route asks "could the true value be 0?". The shift route asks "how often would I see a value this extreme if the true value were 0?". Both lead to the same accept/reject decision in symmetric cases, but the shift route is the one to report when you need a p-value.</div></div>
<section class="tryit-block">
<p><strong>Try it:</strong> Repeat the analysis using the difference in <strong>medians</strong> instead of means. Define <code>ex_med_diff()</code>, bootstrap with <code>R = 2000</code> and seed <code>9</code>, then print the BCa CI.</p>
<div class="webr-container" data-block-title="Your turn: difference in medians">
<div class="webr-code-block">
<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Your turn: difference in medians</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl"><span class="c1"># Try it: replace mean with median</span></span>
<span class="cl">ex_med_diff <span class="o"><-</span> <span class="kr">function</span>(d, i) {</span>
<span class="cl"> s <span class="o"><-</span> d[i, ]</span>
<span class="cl"> <span class="c1"># your code here</span></span>
<span class="cl">}</span>
<span class="cl"></span>
<span class="cl"><span class="nf">set.seed</span>(<span class="m">9</span>)</span>
<span class="cl">ex_boot_med <span class="o"><-</span> <span class="nf">boot</span>(data4_6, ex_med_diff, R <span class="o">=</span> <span class="m">2000</span>)</span>
<span class="cl"><span class="nf">boot.ci</span>(ex_boot_med, type <span class="o">=</span> <span class="s">"bca"</span>)</span>
<span class="cl"><span class="c1">#> Expected: BCa CI well above zero, similar to the mean version</span></span></div>
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<details>
<summary>Click to reveal solution</summary>
<div class="webr-container" data-block-title="Difference in medians solution">
<div class="webr-code-block">
<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Difference in medians solution</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl">ex_med_diff <span class="o"><-</span> <span class="kr">function</span>(d, i) {</span>
<span class="cl"> s <span class="o"><-</span> d[i, ]</span>
<span class="cl"> <span class="nf">median</span>(s<span class="o">$</span>mpg[s<span class="o">$</span>cyl <span class="o">==</span> <span class="m">4</span>]) <span class="o">-</span> <span class="nf">median</span>(s<span class="o">$</span>mpg[s<span class="o">$</span>cyl <span class="o">==</span> <span class="m">6</span>])</span>
<span class="cl">}</span>
<span class="cl"></span>
<span class="cl"><span class="nf">set.seed</span>(<span class="m">9</span>)</span>
<span class="cl">ex_boot_med <span class="o"><-</span> <span class="nf">boot</span>(data4_6, ex_med_diff, R <span class="o">=</span> <span class="m">2000</span>)</span>
<span class="cl"><span class="nf">boot.ci</span>(ex_boot_med, type <span class="o">=</span> <span class="s">"bca"</span>)</span>
<span class="cl"><span class="c1">#> Level BCa</span></span>
<span class="cl"><span class="c1">#> 95% ( 4.10, 8.85 )</span></span></div>
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<p><strong>Explanation:</strong> Swapping <code>mean</code> for <code>median</code> gives a robust difference. The CI shifts slightly but the conclusion (reject H0) is unchanged.</p>
</details>
</section>
<h2>When does bootstrap fail and how do you spot it?</h2>
<p>The bootstrap is not magic. Three failure modes show up often enough that you should know what they look like.</p>
<p>The first is <strong>tiny samples</strong>. With $n < 10$, there are very few distinct resamples and the bootstrap distribution becomes blocky and unreliable. The CI you get back is overconfident.</p>
<p>The second is <strong>boundary statistics</strong>. The maximum, minimum, and other extremes have a sampling distribution that piles up against the observed value. Bootstrap mimics this pile-up and underestimates the true uncertainty.</p>
<div class="webr-container" data-block-title="Boundary stat failure: bootstrap of max">
<div class="webr-code-block">
<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Boundary stat failure: bootstrap of max</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl"><span class="nf">set.seed</span>(<span class="m">1</span>)</span>
<span class="cl">x_unif <span class="o"><-</span> <span class="nf">runif</span>(<span class="m">50</span>, <span class="m">0</span>, <span class="m">100</span>)</span>
<span class="cl">max_stat <span class="o"><-</span> <span class="kr">function</span>(d, i) <span class="nf">max</span>(d[i])</span>
<span class="cl"></span>
<span class="cl">boot_max <span class="o"><-</span> <span class="nf">boot</span>(x_unif, max_stat, R <span class="o">=</span> <span class="m">2000</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="nf">hist</span>(boot_max<span class="o">$</span>t,</span>
<span class="cl"> breaks <span class="o">=</span> <span class="m">30</span>,</span>
<span class="cl"> col <span class="o">=</span> <span class="s">"indianred"</span>,</span>
<span class="cl"> border <span class="o">=</span> <span class="s">"white"</span>,</span>
<span class="cl"> main <span class="o">=</span> <span class="s">"Bootstrap distribution of max"</span>,</span>
<span class="cl"> xlab <span class="o">=</span> <span class="s">"Bootstrap max"</span>)</span>
<span class="cl"><span class="nf">abline</span>(v <span class="o">=</span> <span class="nf">max</span>(x_unif), col <span class="o">=</span> <span class="s">"blue"</span>, lwd <span class="o">=</span> <span class="m">2</span>)</span>
<span class="cl"><span class="c1">#> Histogram with a tall spike at the observed max (blue line)</span></span>
<span class="cl"><span class="c1">#> and a thin tail to the left</span></span></div>
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<p>The histogram is a single spike at the observed maximum with a sparse tail to its left. The bootstrap can never produce a value larger than the observed maximum, because the maximum of a resample with replacement cannot exceed the original maximum. So the upper tail is zero by construction, and any CI would be misleading. For boundary statistics, look at parametric methods or order-statistic theory instead.</p>
<p>The third is <strong>dependent data</strong>. If your observations are a <a class="auto-link" href="Time-Series-Analysis-With-R.html" title="Time Series Analysis with R">time series</a>, ordinary bootstrap destroys the autocorrelation. Use <code>tsboot()</code> from the same package with the block bootstrap (<code>sim = "fixed"</code> or <code>"geom"</code>) to preserve local dependence.</p>
<div class="callout callout-note"><div class="callout-label">Note</div><div class="callout-body"><strong>Use <code>tsboot()</code> for time series.</strong> It resamples blocks of consecutive observations rather than single points, preserving short-range autocorrelation. Pick the block length with care; too short loses dependence, too long reduces effective replicates.</div></div>
<section class="tryit-block">
<p><strong>Try it:</strong> <a class="auto-link" href="Reduce-Filter-Map-in-R.html" title="Reduce(), Filter(), Map() in Base R: Functional Trifecta Explained">Reduce</a> <code>x_unif</code> to 10 observations and re-run the max bootstrap. Confirm visually that the distribution is now nearly degenerate.</p>
<div class="webr-container" data-block-title="Your turn: bootstrap max with tiny n">
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<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Your turn: bootstrap max with tiny n</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl"><span class="c1"># Try it: same code, smaller n</span></span>
<span class="cl"><span class="nf">set.seed</span>(<span class="m">2</span>)</span>
<span class="cl">ex_x <span class="o"><-</span> <span class="nf">runif</span>(<span class="m">10</span>, <span class="m">0</span>, <span class="m">100</span>)</span>
<span class="cl"></span>
<span class="cl">ex_boot_max <span class="o"><-</span> <span class="nf">boot</span>(ex_x, max_stat, R <span class="o">=</span> <span class="m">2000</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="nf">hist</span>(ex_boot_max<span class="o">$</span>t, breaks <span class="o">=</span> <span class="m">30</span>, col <span class="o">=</span> <span class="s">"indianred"</span>, border <span class="o">=</span> <span class="s">"white"</span>,</span>
<span class="cl"> main <span class="o">=</span> <span class="s">"Bootstrap max, n = 10"</span>,</span>
<span class="cl"> xlab <span class="o">=</span> <span class="s">"Bootstrap max"</span>)</span>
<span class="cl"><span class="c1">#> Expected: an even more concentrated spike with very few distinct values</span></span></div>
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<details>
<summary>Click to reveal solution</summary>
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<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Tiny-n bootstrap max solution</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl"><span class="nf">set.seed</span>(<span class="m">2</span>)</span>
<span class="cl">ex_x <span class="o"><-</span> <span class="nf">runif</span>(<span class="m">10</span>, <span class="m">0</span>, <span class="m">100</span>)</span>
<span class="cl"></span>
<span class="cl">ex_boot_max <span class="o"><-</span> <span class="nf">boot</span>(ex_x, max_stat, R <span class="o">=</span> <span class="m">2000</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="nf">length</span>(<span class="nf">unique</span>(ex_boot_max<span class="o">$</span>t))</span>
<span class="cl"><span class="c1">#> [1] 10</span></span>
<span class="cl"></span>
<span class="cl"><span class="nf">hist</span>(ex_boot_max<span class="o">$</span>t, breaks <span class="o">=</span> <span class="m">30</span>, col <span class="o">=</span> <span class="s">"indianred"</span>, border <span class="o">=</span> <span class="s">"white"</span>,</span>
<span class="cl"> main <span class="o">=</span> <span class="s">"Bootstrap max, n = 10"</span>,</span>
<span class="cl"> xlab <span class="o">=</span> <span class="s">"Bootstrap max"</span>)</span></div>
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<p><strong>Explanation:</strong> With only 10 distinct data points, the bootstrap max can take only 10 distinct values. The histogram is a barcode rather than a smooth distribution. No CI from this is trustworthy.</p>
</details>
</section>
<h2>How do you run a stratified bootstrap?</h2>
<p>When the data has groups and you want each replicate to preserve group sizes, pass a <code>strata</code> vector to <code>boot()</code>. This is essential when groups have very different sizes or behaviours and you do not want bootstrap replicates that accidentally drop one group.</p>
<p>The example below estimates the ratio of mean MPG between automatic (<code>am = 1</code>) and manual (<code>am = 0</code>) cars in <code>mtcars</code>, with <code>strata = mtcars$am</code> so each replicate keeps the original 13 automatic and 19 manual cars.</p>
<div class="webr-container" data-block-title="Stratified bootstrap of a ratio">
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<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Stratified bootstrap of a ratio</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl">ratio_stat <span class="o"><-</span> <span class="kr">function</span>(d, i) {</span>
<span class="cl"> s <span class="o"><-</span> d[i, ]</span>
<span class="cl"> <span class="nf">mean</span>(s<span class="o">$</span>mpg[s<span class="o">$</span>am <span class="o">==</span> <span class="m">1</span>]) <span class="o">/</span> <span class="nf">mean</span>(s<span class="o">$</span>mpg[s<span class="o">$</span>am <span class="o">==</span> <span class="m">0</span>])</span>
<span class="cl">}</span>
<span class="cl"></span>
<span class="cl"><span class="nf">set.seed</span>(<span class="m">11</span>)</span>
<span class="cl">boot_strat <span class="o"><-</span> <span class="nf">boot</span>(mtcars, ratio_stat, R <span class="o">=</span> <span class="m">2000</span>, strata <span class="o">=</span> mtcars<span class="o">$</span>am)</span>
<span class="cl"><span class="nf">boot.ci</span>(boot_strat, type <span class="o">=</span> <span class="s">"bca"</span>)</span>
<span class="cl"><span class="c1">#> Level BCa</span></span>
<span class="cl"><span class="c1">#> 95% ( 1.13, 1.62 )</span></span></div>
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<p>The 95% BCa CI for the ratio runs from 1.13 to 1.62. Because the entire CI is above 1, automatic-transmission cars get more MPG on average than manual-transmission ones in this dataset, by a factor of 13% to 62%. Without <code>strata</code>, some replicates would have very few automatic cars, inflating the variance and the CI width.</p>
<section class="tryit-block">
<p><strong>Try it:</strong> Re-run the same analysis without <code>strata</code> and compare the BCa CI width. The unstratified CI should be wider.</p>
<div class="webr-container" data-block-title="Your turn: unstratified version for comparison">