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data-collapsed="false"><span class="subsec-chevron">▼</span> Join & Reshape</li><li data-subkey="sec1sub3"><a href="/R-Joins.html" title="R Joins"><span class="progress-dot"></span>R Joins</a></li><li data-subkey="sec1sub3"><a href="/pivot_longer-pivot_wider-Reshape-Data-in-R.html" title="pivot_longer & pivot_wider"><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" title="separate() & unite()"><span class="progress-dot"></span>separate() & unite()</a></li><li data-subkey="sec1sub3" class="is-quiz"><a href="/tidyr-Exercises-in-R-quiz.html" title="tidyr Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>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" title="Missing Values (NA)"><span class="progress-dot"></span>Missing Values (NA)</a></li><li data-subkey="sec1sub4"><a href="/Data-Quality-Checking-in-R.html" title="Data Quality Checking"><span class="progress-dot"></span>Data Quality Checking</a></li><li data-subkey="sec1sub4"><a href="/janitor-Package-in-R.html" title="janitor Package"><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" title="stringr"><span class="progress-dot"></span>stringr</a></li><li data-subkey="sec1sub5"><a href="/R-Regex-stringr-Pattern-Matching.html" title="Regex Patterns"><span class="progress-dot"></span>Regex Patterns</a></li><li data-subkey="sec1sub5"><a href="/lubridate-in-R.html" title="lubridate"><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" title="DBI & Databases"><span class="progress-dot"></span>DBI & Databases</a></li><li data-subkey="sec1sub6"><a href="/DuckDB-in-R.html" title="DuckDB & duckplyr"><span class="progress-dot"></span>DuckDB & duckplyr</a></li><li data-subkey="sec1sub6"><a href="/Web-Scraping-in-R-with-rvest.html" title="Web Scraping (rvest)"><span class="progress-dot"></span>Web Scraping (rvest)</a></li><li data-subkey="sec1sub6"><a href="/REST-APIs-in-R-with-httr2.html" title="REST APIs (httr2)"><span class="progress-dot"></span>REST APIs (httr2)</a></li><li data-subkey="sec1sub6"><a href="/Data-Wrangling-dplyr-Course.html" title="Data Wrangling with dplyr (Course)"><span class="progress-dot"></span>Data Wrangling with dplyr (Course)</a></li><li data-subkey="sec1sub6"><a href="/Join-Reshape-Course.html" title="Join & Reshape (Course)"><span class="progress-dot"></span>Join & Reshape (Course)</a></li><li data-subkey="sec1sub6"><a href="/data-table-Course.html" title="data.table (Course)"><span class="progress-dot"></span>data.table (Course)</a></li><li data-subkey="sec1sub6"><a href="/Report-Tables-Course.html" title="Report-Ready Tables (Course)"><span class="progress-dot"></span>Report-Ready Tables (Course)</a></li><li data-subkey="sec1sub6"><a href="/Communicate-Automate-Course.html" title="Communicate & Automate (Course)"><span class="progress-dot"></span>Communicate & Automate (Course)</a></li></ul></li><li class="sidebar-section expanded"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span><span class="sec-num">3.</span> <span class="sec-title-t">Statistics</span><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> EDA & Data Quality</li><li data-subkey="sec2sub1"><a href="/Automated-EDA-in-R.html" title="Automated EDA"><span class="progress-dot"></span>Automated EDA</a></li><li data-subkey="sec2sub1"><a href="/Missing-Data-Visualization-in-R-naniar.html" title="Missing Data Viz (naniar)"><span class="progress-dot"></span>Missing Data Viz (naniar)</a></li><li data-subkey="sec2sub1"><a href="/Outlier-Detection-in-R.html" title="Outlier Detection"><span class="progress-dot"></span>Outlier Detection</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub2" data-collapsed="false"><span class="subsec-chevron">▼</span> Probability</li><li data-subkey="sec2sub2"><a href="/Sample-Spaces-Events-and-Probability-Axioms-in-R-With-Monte-Carlo-Proof.html" title="Probability Axioms"><span class="progress-dot"></span>Probability Axioms</a></li><li data-subkey="sec2sub2"><a href="/Conditional-Probability-in-R.html" title="Conditional Probability"><span class="progress-dot"></span>Conditional Probability</a></li><li data-subkey="sec2sub2"><a href="/Random-Variables-in-R.html" title="Random Variables"><span class="progress-dot"></span>Random Variables</a></li><li data-subkey="sec2sub2"><a href="/Binomial-and-Poisson-Distributions-in-R.html" title="Binomial vs Poisson"><span class="progress-dot"></span>Binomial vs Poisson</a></li><li data-subkey="sec2sub2"><a href="/Normal-t-F-and-Chi-Squared-Distributions-in-R.html" title="Normal, t, F, Chi-Squared"><span class="progress-dot"></span>Normal, t, F, Chi-Squared</a></li><li data-subkey="sec2sub2"><a href="/Central-Limit-Theorem-in-R.html" title="Central Limit Theorem"><span class="progress-dot"></span>Central Limit Theorem</a></li><li data-subkey="sec2sub2"><a href="/Sampling-Distributions-in-R.html" title="Sampling Distributions"><span class="progress-dot"></span>Sampling Distributions</a></li><li data-subkey="sec2sub2"><a href="/Law-of-Large-Numbers-vs-CLT-in-R.html" title="LLN vs CLT"><span class="progress-dot"></span>LLN vs CLT</a></li><li data-subkey="sec2sub2"><a href="/What-Is-Probability-Simulation-First-Intuition-in-R-Before-the-Formulas.html" title="Probability (Simulation-First)"><span class="progress-dot"></span>Probability (Simulation-First)</a></li><li data-subkey="sec2sub2"><a href="/Expected-Value-and-Variance-in-R.html" title="Expected Value and Variance"><span class="progress-dot"></span>Expected Value and Variance</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub3" data-collapsed="false"><span class="subsec-chevron">▼</span> Inference & Estimation</li><li data-subkey="sec2sub3"><a href="/Maximum-Likelihood-Estimation-in-R.html" title="Maximum Likelihood Estimation"><span class="progress-dot"></span>Maximum Likelihood Estimation</a></li><li data-subkey="sec2sub3"><a href="/Hypothesis-Testing-in-R.html" title="Hypothesis Testing"><span class="progress-dot"></span>Hypothesis Testing</a></li><li data-subkey="sec2sub3"><a href="/Sample-Size-Planning-in-R.html" title="Sample Size Planning"><span class="progress-dot"></span>Sample Size Planning</a></li><li data-subkey="sec2sub3"><a href="/Which-Statistical-Test-in-R.html" title="Choosing the Right Test"><span class="progress-dot"></span>Choosing the Right Test</a></li><li data-subkey="sec2sub3"><a href="/Statistical-Tests-in-R.html" title="Statistical Tests"><span class="progress-dot"></span>Statistical Tests</a></li><li data-subkey="sec2sub3"><a href="/Measures-of-Association-in-R.html" title="Measures of Association"><span class="progress-dot"></span>Measures of Association</a></li><li data-subkey="sec2sub3"><a href="/Point-Estimation-in-R.html" title="Point Estimation"><span class="progress-dot"></span>Point Estimation</a></li><li data-subkey="sec2sub3"><a href="/Confidence-Intervals-in-R.html" title="Confidence Intervals"><span class="progress-dot"></span>Confidence Intervals</a></li><li data-subkey="sec2sub3"><a href="/Type-I-and-Type-II-Errors-in-R.html" title="Type I and II Errors"><span class="progress-dot"></span>Type I and II Errors</a></li><li data-subkey="sec2sub3"><a href="/Statistical-Power-Analysis-in-R.html" title="Power Analysis"><span class="progress-dot"></span>Power Analysis</a></li><li data-subkey="sec2sub3"><a href="/Effect-Size-in-R.html" title="Effect Size"><span class="progress-dot"></span>Effect Size</a></li><li data-subkey="sec2sub3"><a href="/t-Tests-in-R.html" title="t-Tests"><span class="progress-dot"></span>t-Tests</a></li><li data-subkey="sec2sub3"><a href="/Proportion-Tests-in-R.html" title="Proportion Tests"><span class="progress-dot"></span>Proportion Tests</a></li><li data-subkey="sec2sub3"><a href="/Normality-and-Variance-Tests-in-R.html" title="Normality & Variance Tests"><span class="progress-dot"></span>Normality & Variance Tests</a></li><li data-subkey="sec2sub3"><a href="/Chi-Square-Tests-in-R.html" title="Chi-Square Tests"><span class="progress-dot"></span>Chi-Square Tests</a></li><li data-subkey="sec2sub3"><a href="/Wilcoxon-Mann-Whitney-and-Kruskal-Wallis-in-R.html" title="Wilcoxon, Mann-Whitney & Kruskal-Wallis"><span class="progress-dot"></span>Wilcoxon, Mann-Whitney & Kruskal-Wallis</a></li><li data-subkey="sec2sub3"><a href="/Multiple-Comparisons-in-R.html" title="Multiple Testing Correction"><span class="progress-dot"></span>Multiple Testing Correction</a></li><li data-subkey="sec2sub3" class="is-quiz"><a href="/Hypothesis-Testing-Exercises-in-R-quiz.html" title="Hypothesis Testing Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub4" data-collapsed="false"><span class="subsec-chevron">▼</span> Regression</li><li data-subkey="sec2sub4"><a href="/Linear-Regression.html" title="Linear Regression"><span class="progress-dot"></span>Linear Regression</a></li><li data-subkey="sec2sub4"><a href="/Logistic-Regression-With-R.html" title="Logistic Regression"><span class="progress-dot"></span>Logistic Regression</a></li><li data-subkey="sec2sub4"><a href="/Variable-Selection-and-Importance-With-R.html" title="Feature Selection"><span class="progress-dot"></span>Feature Selection</a></li><li data-subkey="sec2sub4"><a href="/Model-Selection-in-R.html" title="Model Selection"><span class="progress-dot"></span>Model Selection</a></li><li data-subkey="sec2sub4"><a href="/Missing-Value-Treatment-With-R.html" title="Missing Value Treatment"><span class="progress-dot"></span>Missing Value Treatment</a></li><li data-subkey="sec2sub4"><a href="/Outlier-Treatment-With-R.html" title="Outlier Analysis"><span class="progress-dot"></span>Outlier Analysis</a></li><li data-subkey="sec2sub4"><a href="/adv-regression-models.html" title="Advanced Regression Models"><span class="progress-dot"></span>Advanced Regression Models</a></li><li data-subkey="sec2sub4" class="is-quiz"><a href="/Linear-Regression-Exercises-in-R-quiz.html" title="Linear Regression Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub5" data-collapsed="false"><span class="subsec-chevron">▼</span> Reporting & Communication</li><li data-subkey="sec2sub5"><a href="/Statistical-Consulting-in-R.html" title="Statistical Consulting"><span class="progress-dot"></span>Statistical Consulting</a></li><li data-subkey="sec2sub5"><a href="/Statistical-Report-Writing-in-R.html" title="Statistical Report Writing"><span class="progress-dot"></span>Statistical Report Writing</a></li><li data-subkey="sec2sub5"><a href="/Bootstrap-Confidence-Intervals-in-R.html" title="Bootstrap Confidence Intervals"><span class="progress-dot"></span>Bootstrap Confidence Intervals</a></li><li data-subkey="sec2sub5"><a href="/Reporting-Statistics-in-R.html" title="Reporting Statistics"><span class="progress-dot"></span>Reporting Statistics</a></li><li data-subkey="sec2sub5"><a href="/Regression-Tables-in-R.html" title="Regression Tables (3 packages)"><span class="progress-dot"></span>Regression Tables (3 packages)</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub6" data-collapsed="false"><span class="subsec-chevron">▼</span> Regression in Practice</li><li data-subkey="sec2sub6"><a href="/Simple-Linear-Regression-in-R.html" title="Simple Linear Regression"><span class="progress-dot"></span>Simple Linear Regression</a></li><li data-subkey="sec2sub6"><a href="/Multiple-Regression-in-R.html" title="Multiple Regression"><span class="progress-dot"></span>Multiple Regression</a></li><li data-subkey="sec2sub6"><a href="/Correlation-in-R.html" title="Correlation (Pearson, Spearman, Kendall)"><span class="progress-dot"></span>Correlation (Pearson, Spearman, Kendall)</a></li><li data-subkey="sec2sub6"><a href="/Linear-Regression-Assumptions-in-R.html" title="Linear Regression Assumptions"><span class="progress-dot"></span>Linear Regression Assumptions</a></li><li data-subkey="sec2sub6"><a href="/Dummy-Variables-in-R.html" title="Dummy Variables in R"><span class="progress-dot"></span>Dummy Variables in R</a></li><li data-subkey="sec2sub6"><a href="/Interaction-Effects-in-R.html" title="Interaction Effects"><span class="progress-dot"></span>Interaction Effects</a></li><li data-subkey="sec2sub6"><a href="/Regression-Diagnostics-in-R.html" title="Regression Diagnostics"><span class="progress-dot"></span>Regression Diagnostics</a></li><li data-subkey="sec2sub6"><a href="/Variable-Selection-in-R.html" title="Variable Selection"><span class="progress-dot"></span>Variable Selection</a></li><li data-subkey="sec2sub6"><a href="/Polynomial-and-Spline-Regression-in-R.html" title="Polynomial & Splines"><span class="progress-dot"></span>Polynomial & Splines</a></li><li data-subkey="sec2sub6"><a href="/Ridge-and-Lasso-Regression-in-R.html" title="Ridge & Lasso Regression"><span class="progress-dot"></span>Ridge & Lasso Regression</a></li><li data-subkey="sec2sub6"><a href="/Robust-Regression-in-R.html" title="Robust Regression (rlm)"><span class="progress-dot"></span>Robust Regression (rlm)</a></li><li data-subkey="sec2sub6"><a href="/Quantile-Regression-in-R-2.html" title="Quantile Regression"><span class="progress-dot"></span>Quantile Regression</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub7" data-collapsed="false"><span class="subsec-chevron">▼</span> ANOVA & Experiments</li><li data-subkey="sec2sub7"><a href="/One-Way-ANOVA-in-R.html" title="One-Way ANOVA"><span class="progress-dot"></span>One-Way ANOVA</a></li><li data-subkey="sec2sub7"><a href="/Post-Hoc-Tests-After-ANOVA.html" title="Post-Hoc Tests After ANOVA"><span class="progress-dot"></span>Post-Hoc Tests After ANOVA</a></li><li data-subkey="sec2sub7"><a href="/Two-Way-ANOVA-in-R.html" title="Two-Way ANOVA"><span class="progress-dot"></span>Two-Way ANOVA</a></li><li data-subkey="sec2sub7"><a href="/Repeated-Measures-ANOVA-in-R.html" title="Repeated Measures ANOVA"><span class="progress-dot"></span>Repeated Measures ANOVA</a></li><li data-subkey="sec2sub7"><a href="/ANCOVA-in-R.html" title="ANCOVA"><span class="progress-dot"></span>ANCOVA</a></li><li data-subkey="sec2sub7"><a href="/Experimental-Design-Principles-in-R.html" title="Experimental Design in R"><span class="progress-dot"></span>Experimental Design in R</a></li><li data-subkey="sec2sub7"><a href="/Factorial-Experiments-in-R.html" title="Factorial Designs (2^k)"><span class="progress-dot"></span>Factorial Designs (2^k)</a></li><li data-subkey="sec2sub7"><a href="/AB-Testing-in-R.html" title="A/B Testing"><span class="progress-dot"></span>A/B Testing</a></li><li data-subkey="sec2sub7"><a href="/MANOVA-in-R.html" title="MANOVA"><span class="progress-dot"></span>MANOVA</a></li><li data-subkey="sec2sub7"><a href="/Mixed-ANOVA-in-R.html" title="Mixed ANOVA"><span class="progress-dot"></span>Mixed ANOVA</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub8" data-collapsed="false"><span class="subsec-chevron">▼</span> GLMs & Categorical Data</li><li data-subkey="sec2sub8"><a href="/Categorical-Data-in-R.html" title="Categorical Data (Tables & Mosaic)"><span class="progress-dot"></span>Categorical Data (Tables & Mosaic)</a></li><li data-subkey="sec2sub8"><a href="/Chi-Square-Test-of-Independence-in-R.html" title="Chi-Square Test of Independence"><span class="progress-dot"></span>Chi-Square Test of Independence</a></li><li data-subkey="sec2sub8"><a href="/Chi-Square-Goodness-of-Fit-Test-in-R.html" title="Chi-Square Goodness-of-Fit"><span class="progress-dot"></span>Chi-Square Goodness-of-Fit</a></li><li data-subkey="sec2sub8"><a href="/Fishers-Exact-Test-in-R.html" title="Fisher's Exact Test"><span class="progress-dot"></span>Fisher's Exact Test</a></li><li data-subkey="sec2sub8"><a href="/Odds-Ratios-and-Relative-Risk-in-R.html" title="Odds Ratios & Relative Risk"><span class="progress-dot"></span>Odds Ratios & Relative Risk</a></li><li data-subkey="sec2sub8"><a href="/Logistic-Regression-in-R.html" title="Logistic Regression (glm + ROC)"><span class="progress-dot"></span>Logistic Regression (glm + ROC)</a></li><li data-subkey="sec2sub8"><a href="/Logistic-Regression-in-R-2.html" title="Logistic Regression (Diagnostics)"><span class="progress-dot"></span>Logistic Regression (Diagnostics)</a></li><li data-subkey="sec2sub8"><a href="/Poisson-Regression-in-R.html" title="Poisson Regression"><span class="progress-dot"></span>Poisson Regression</a></li><li data-subkey="sec2sub8"><a href="/Poisson-and-Negative-Binomial-Regression.html" title="Poisson & Negative Binomial Regression"><span class="progress-dot"></span>Poisson & Negative Binomial Regression</a></li><li data-subkey="sec2sub8"><a href="/Multinomial-and-Ordinal-Logistic-Regression-in-R.html" title="Multinomial & Ordinal Logistic Regression"><span class="progress-dot"></span>Multinomial & Ordinal Logistic Regression</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub9" data-collapsed="false"><span class="subsec-chevron">▼</span> Multivariate Methods</li><li data-subkey="sec2sub9"><a href="/Multivariate-Statistics-in-R.html" title="Multivariate Distances & Hotelling's T²"><span class="progress-dot"></span>Multivariate Distances & Hotelling's T²</a></li><li data-subkey="sec2sub9"><a href="/PCA-in-R.html" title="PCA with prcomp()"><span class="progress-dot"></span>PCA with prcomp()</a></li><li data-subkey="sec2sub9"><a href="/Interpreting-PCA-Results-in-R.html" title="Interpreting PCA Output"><span class="progress-dot"></span>Interpreting PCA Output</a></li><li data-subkey="sec2sub9"><a href="/factoextra-and-FactoMineR.html" title="factoextra (PCA + Clusters)"><span class="progress-dot"></span>factoextra (PCA + Clusters)</a></li><li data-subkey="sec2sub9"><a href="/Exploratory-Factor-Analysis-in-R.html" title="Exploratory Factor Analysis"><span class="progress-dot"></span>Exploratory Factor Analysis</a></li><li data-subkey="sec2sub9"><a href="/CFA-and-Structural-Equation-Modeling-in-R.html" title="SEM and CFA (lavaan)"><span class="progress-dot"></span>SEM and CFA (lavaan)</a></li><li data-subkey="sec2sub9"><a href="/Linear-Discriminant-Analysis-in-R.html" title="LDA (Linear Discriminant Analysis)"><span class="progress-dot"></span>LDA (Linear Discriminant Analysis)</a></li><li data-subkey="sec2sub9"><a href="/Cluster-Analysis-in-R.html" title="Clustering (k-Means / HC / DBSCAN)"><span class="progress-dot"></span>Clustering (k-Means / HC / DBSCAN)</a></li><li data-subkey="sec2sub9"><a href="/Correspondence-Analysis-in-R.html" title="Correspondence Analysis"><span class="progress-dot"></span>Correspondence Analysis</a></li><li data-subkey="sec2sub9"><a href="/t-SNE-and-UMAP-in-R.html" title="t-SNE and UMAP"><span class="progress-dot"></span>t-SNE and UMAP</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub10" data-collapsed="false"><span class="subsec-chevron">▼</span> Nonparametric & Resampling</li><li data-subkey="sec2sub10"><a href="/When-to-Use-Nonparametric-Tests-in-R.html" title="When to Use Nonparametric Tests"><span class="progress-dot"></span>When to Use Nonparametric Tests</a></li><li data-subkey="sec2sub10"><a href="/Wilcoxon-Signed-Rank-Test-in-R.html" title="Wilcoxon Signed-Rank Test"><span class="progress-dot"></span>Wilcoxon Signed-Rank Test</a></li><li data-subkey="sec2sub10"><a href="/Mann-Whitney-U-Test-in-R.html" title="Mann-Whitney U Test"><span class="progress-dot"></span>Mann-Whitney U Test</a></li><li data-subkey="sec2sub10"><a href="/Kruskal-Wallis-Test-in-R-2.html" title="Kruskal-Wallis Test"><span class="progress-dot"></span>Kruskal-Wallis Test</a></li><li data-subkey="sec2sub10"><a href="/Friedman-Test-in-R.html" title="Friedman Test"><span class="progress-dot"></span>Friedman Test</a></li><li data-subkey="sec2sub10"><a href="/Spearman-and-Kendall-Correlation-in-R.html" title="Spearman & Kendall Correlation"><span class="progress-dot"></span>Spearman & Kendall Correlation</a></li><li data-subkey="sec2sub10"><a href="/Bootstrap-in-R.html" title="Bootstrap (boot package)"><span class="progress-dot"></span>Bootstrap (boot package)</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub11" data-collapsed="false"><span class="subsec-chevron">▼</span> Linear Algebra for Statistics</li><li data-subkey="sec2sub11"><a href="/Matrix-Operations-in-R.html" title="Matrix Operations in R"><span class="progress-dot"></span>Matrix Operations in R</a></li><li data-subkey="sec2sub11"><a href="/Solving-Linear-Systems-in-R.html" title="Solving Linear Systems in R"><span class="progress-dot"></span>Solving Linear Systems in R</a></li><li data-subkey="sec2sub11"><a href="/Eigenvalues-and-Eigenvectors-in-R.html" title="Eigenvalues & Eigenvectors in R"><span class="progress-dot"></span>Eigenvalues & Eigenvectors in R</a></li><li data-subkey="sec2sub11"><a href="/Singular-Value-Decomposition-in-R.html" title="Singular Value Decomposition in R"><span class="progress-dot"></span>Singular Value Decomposition in R</a></li><li data-subkey="sec2sub11"><a href="/Projections-and-the-Hat-Matrix-in-R.html" title="Projections & the Hat Matrix"><span class="progress-dot"></span>Projections & the Hat Matrix</a></li><li data-subkey="sec2sub11"><a href="/QR-Decomposition-in-R.html" title="QR Decomposition in R"><span class="progress-dot"></span>QR Decomposition in R</a></li><li data-subkey="sec2sub11"><a href="/Quadratic-Forms-in-R.html" title="Quadratic Forms"><span class="progress-dot"></span>Quadratic Forms</a></li><li data-subkey="sec2sub11"><a href="/Matrix-Derivatives-and-the-Hessian-in-R.html" title="Matrix Derivatives & Hessian"><span class="progress-dot"></span>Matrix Derivatives & Hessian</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub12" data-collapsed="false"><span class="subsec-chevron">▼</span> Statistical Theory</li><li data-subkey="sec2sub12"><a href="/Exponential-Family-Distributions-in-R.html" title="Exponential Family Distributions"><span class="progress-dot"></span>Exponential Family Distributions</a></li><li data-subkey="sec2sub12"><a href="/Sufficient-Statistics-in-R.html" title="Sufficient Statistics"><span class="progress-dot"></span>Sufficient Statistics</a></li><li data-subkey="sec2sub12"><a href="/Complete-and-Ancillary-Statistics-in-R.html" title="Complete & Ancillary Statistics"><span class="progress-dot"></span>Complete & Ancillary Statistics</a></li><li data-subkey="sec2sub12"><a href="/UMVUE-in-R-2.html" title="UMVUE (Rao-Blackwell & Lehmann-Scheffé)"><span class="progress-dot"></span>UMVUE (Rao-Blackwell & Lehmann-Scheffé)</a></li><li data-subkey="sec2sub12"><a href="/Cramer-Rao-Lower-Bound-in-R-2.html" title="Cramér-Rao Lower Bound"><span class="progress-dot"></span>Cramér-Rao Lower Bound</a></li><li data-subkey="sec2sub12"><a href="/Asymptotic-Theory-in-R-2.html" title="Asymptotic Theory"><span class="progress-dot"></span>Asymptotic Theory</a></li><li data-subkey="sec2sub12"><a href="/Neyman-Pearson-Lemma-in-R-2.html" title="Neyman-Pearson Lemma"><span class="progress-dot"></span>Neyman-Pearson Lemma</a></li><li data-subkey="sec2sub12"><a href="/Likelihood-Ratio-Tests-and-Pivotal-Methods.html" title="Likelihood Ratio & Pivotal Methods"><span class="progress-dot"></span>Likelihood Ratio & Pivotal Methods</a></li><li data-subkey="sec2sub12"><a href="/Decision-Theory-in-R.html" title="Decision Theory"><span class="progress-dot"></span>Decision Theory</a></li><li data-subkey="sec2sub12"><a href="/Asymptotic-Relative-Efficiency-in-R.html" title="Asymptotic Relative Efficiency"><span class="progress-dot"></span>Asymptotic Relative Efficiency</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub13" data-collapsed="false"><span class="subsec-chevron">▼</span> Bayesian Foundations</li><li data-subkey="sec2sub13"><a href="/Bayes-Theorem-in-R.html" title="Bayes' Theorem"><span class="progress-dot"></span>Bayes' Theorem</a></li><li data-subkey="sec2sub13"><a href="/Bayesian-Statistics-in-R.html" title="Bayesian Statistics"><span class="progress-dot"></span>Bayesian Statistics</a></li><li data-subkey="sec2sub13"><a href="/Conjugate-Priors-in-R.html" title="Conjugate Priors"><span class="progress-dot"></span>Conjugate Priors</a></li><li data-subkey="sec2sub13"><a href="/Grid-Approximation-in-R.html" title="Grid Approximation"><span class="progress-dot"></span>Grid Approximation</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub14" data-collapsed="false"><span class="subsec-chevron">▼</span> MCMC & Stan</li><li data-subkey="sec2sub14"><a href="/MCMC-in-R.html" title="MCMC in R"><span class="progress-dot"></span>MCMC in R</a></li><li data-subkey="sec2sub14"><a href="/Gibbs-Sampling-in-R.html" title="Gibbs Sampling"><span class="progress-dot"></span>Gibbs Sampling</a></li><li data-subkey="sec2sub14"><a href="/Hamiltonian-Monte-Carlo-in-R.html" title="Hamiltonian Monte Carlo"><span class="progress-dot"></span>Hamiltonian Monte Carlo</a></li><li data-subkey="sec2sub14"><a href="/Stan-in-R.html" title="Stan"><span class="progress-dot"></span>Stan</a></li><li data-subkey="sec2sub14"><a href="/brms-in-R.html" title="brms"><span class="progress-dot"></span>brms</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub15" data-collapsed="false"><span class="subsec-chevron">▼</span> Bayesian Modeling</li><li data-subkey="sec2sub15"><a href="/Choosing-Priors-in-R.html" title="Choosing Priors"><span class="progress-dot"></span>Choosing Priors</a></li><li data-subkey="sec2sub15"><a href="/Prior-Predictive-Checks-in-R.html" title="Prior Predictive Checks"><span class="progress-dot"></span>Prior Predictive Checks</a></li><li data-subkey="sec2sub15"><a href="/Compare-Bayesian-Models-in-R.html" title="Compare Bayesian Models"><span class="progress-dot"></span>Compare Bayesian Models</a></li><li data-subkey="sec2sub15"><a href="/Posterior-Predictive-Checks-in-R.html" title="Posterior Predictive Checks"><span class="progress-dot"></span>Posterior Predictive Checks</a></li><li data-subkey="sec2sub15"><a href="/Bayesian-Linear-Regression-in-R.html" title="Bayesian Linear Regression"><span class="progress-dot"></span>Bayesian Linear Regression</a></li><li data-subkey="sec2sub15"><a href="/Bayesian-Logistic-Regression-in-R.html" title="Bayesian Logistic Regression"><span class="progress-dot"></span>Bayesian Logistic Regression</a></li><li data-subkey="sec2sub15"><a href="/Bayesian-Hierarchical-Models-in-R.html" title="Bayesian Hierarchical Models"><span class="progress-dot"></span>Bayesian Hierarchical Models</a></li><li data-subkey="sec2sub15"><a href="/Multilevel-Models-in-R.html" title="Multilevel Models"><span class="progress-dot"></span>Multilevel Models</a></li><li data-subkey="sec2sub15"><a href="/Bayesian-ANOVA-in-R.html" title="Bayesian ANOVA"><span class="progress-dot"></span>Bayesian ANOVA</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub16" data-collapsed="false"><span class="subsec-chevron">▼</span> Machine Learning</li><li data-subkey="sec2sub16"><a href="/Random-Forest-Course.html" title="Random Forests (Course)"><span class="progress-dot"></span>Random Forests (Course)</a></li><li data-subkey="sec2sub16"><a href="/R-Gradient-Boosting-Course.html" title="Gradient Boosting (Course)"><span class="progress-dot"></span>Gradient Boosting (Course)</a></li><li data-subkey="sec2sub16"><a href="/R-tidymodels-Course.html" title="tidymodels (Course)"><span class="progress-dot"></span>tidymodels (Course)</a></li><li data-subkey="sec2sub16" class="is-quiz"><a href="/Machine-Learning-Exercises-in-R-quiz.html" title="Machine Learning Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec2sub16"><a href="/T-Test-Course.html" title="The t-test (Lesson)"><span class="progress-dot"></span>The t-test (Lesson)</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span><span class="sec-num">4.</span> <span class="sec-title-t">Visualization</span><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> ggplot2 Foundations</li><li data-subkey="sec3sub1"><a href="/ggplot2-Grammar-of-Graphics.html" title="Grammar of Graphics"><span class="progress-dot"></span>Grammar of Graphics</a></li><li data-subkey="sec3sub1"><a href="/ggplot2-Getting-Started.html" title="ggplot2 Getting Started"><span class="progress-dot"></span>ggplot2 Getting Started</a></li><li data-subkey="sec3sub1"><a href="/ggplot2-Aesthetics-aes-Map-Data.html" title="ggplot2 Aesthetics (aes)"><span class="progress-dot"></span>ggplot2 Aesthetics (aes)</a></li><li data-subkey="sec3sub1"><a href="/ggplot2-Colours.html" title="ggplot2 Colours"><span class="progress-dot"></span>ggplot2 Colours</a></li><li data-subkey="sec3sub1"><a href="/ggplot2-Scales.html" title="ggplot2 Scales"><span class="progress-dot"></span>ggplot2 Scales</a></li><li data-subkey="sec3sub1"><a href="/ggplot2-Themes-in-R.html" title="ggplot2 Themes"><span class="progress-dot"></span>ggplot2 Themes</a></li><li data-subkey="sec3sub1"><a href="/ggplot2-Labels-and-Annotations.html" title="Labels & Annotations"><span class="progress-dot"></span>Labels & Annotations</a></li><li data-subkey="sec3sub1"><a href="/ggplot2-Facets.html" title="ggplot2 Facets"><span class="progress-dot"></span>ggplot2 Facets</a></li><li data-subkey="sec3sub1" class="is-quiz"><a href="/ggplot2-Exercises-in-R-quiz.html" title="ggplot2 Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub2" data-collapsed="false"><span class="subsec-chevron">▼</span> Core Charts</li><li data-subkey="sec3sub2"><a href="/ggplot2-Scatter-Plots.html" title="Scatter Plots"><span class="progress-dot"></span>Scatter Plots</a></li><li data-subkey="sec3sub2"><a href="/ggplot2-Line-Charts.html" title="Line Charts"><span class="progress-dot"></span>Line Charts</a></li><li data-subkey="sec3sub2"><a href="/ggplot2-Bar-Charts.html" title="Bar Charts"><span class="progress-dot"></span>Bar Charts</a></li><li data-subkey="sec3sub2"><a href="/ggplot2-Distribution-Charts.html" title="Distribution Charts"><span class="progress-dot"></span>Distribution Charts</a></li><li data-subkey="sec3sub2"><a href="/Error-Bars-in-R.html" title="Error Bars"><span class="progress-dot"></span>Error Bars</a></li><li data-subkey="sec3sub2"><a href="/geom_smooth-in-R.html" title="geom_smooth()"><span class="progress-dot"></span>geom_smooth()</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub3" data-collapsed="false"><span class="subsec-chevron">▼</span> Distributions & Groups</li><li data-subkey="sec3sub3"><a href="/Violin-Plot-in-R.html" title="Violin Plot"><span class="progress-dot"></span>Violin Plot</a></li><li data-subkey="sec3sub3"><a href="/Ridgeline-Plot-in-R.html" title="Ridgeline Plot"><span class="progress-dot"></span>Ridgeline Plot</a></li><li data-subkey="sec3sub3"><a href="/Lollipop-Chart-in-R.html" title="Lollipop Chart"><span class="progress-dot"></span>Lollipop Chart</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub4" data-collapsed="false"><span class="subsec-chevron">▼</span> Relationships</li><li data-subkey="sec3sub4"><a href="/Bubble-Chart-in-R.html" title="Bubble Chart"><span class="progress-dot"></span>Bubble Chart</a></li><li data-subkey="sec3sub4"><a href="/Heatmap-in-R.html" title="Heatmap in R"><span class="progress-dot"></span>Heatmap in R</a></li><li data-subkey="sec3sub4"><a href="/Correlation-Matrix-Plot-in-R.html" title="Correlation Matrix"><span class="progress-dot"></span>Correlation Matrix</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub5" data-collapsed="false"><span class="subsec-chevron">▼</span> Advanced Charts</li><li data-subkey="sec3sub5"><a href="/Pie-Donut-Chart-in-R.html" title="Pie & Donut Chart"><span class="progress-dot"></span>Pie & Donut Chart</a></li><li data-subkey="sec3sub5"><a href="/Treemap-in-R.html" title="Treemap"><span class="progress-dot"></span>Treemap</a></li><li data-subkey="sec3sub5"><a href="/Waffle-Chart-in-R.html" title="Waffle Chart"><span class="progress-dot"></span>Waffle Chart</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub6" data-collapsed="false"><span class="subsec-chevron">▼</span> <a class="auto-link" href="Exploratory-Data-Analysis-in-R.html" title="EDA in R: A 7-Step Framework That Works on Every Dataset You'll Encounter">Exploratory Analysis</a></li><li data-subkey="sec3sub6"><a href="/Exploratory-Data-Analysis-in-R.html" title="EDA (7-Step Framework)"><span class="progress-dot"></span>EDA (7-Step Framework)</a></li><li data-subkey="sec3sub6"><a href="/Univariate-EDA-in-R.html" title="Univariate EDA"><span class="progress-dot"></span>Univariate EDA</a></li><li data-subkey="sec3sub6"><a href="/Bivariate-EDA-in-R.html" title="Bivariate EDA"><span class="progress-dot"></span>Bivariate EDA</a></li><li data-subkey="sec3sub6"><a href="/Descriptive-Statistics-in-R.html" title="Descriptive Statistics"><span class="progress-dot"></span>Descriptive Statistics</a></li><li data-subkey="sec3sub6"><a href="/Correlation-Analysis-in-R.html" title="Correlation Analysis"><span class="progress-dot"></span>Correlation Analysis</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub7" data-collapsed="false"><span class="subsec-chevron">▼</span> Interactive & Maps</li><li data-subkey="sec3sub7"><a href="/Combining-ggplot2-with-plotly.html" title="ggplot2 + plotly Interactive"><span class="progress-dot"></span>ggplot2 + plotly Interactive</a></li><li data-subkey="sec3sub7"><a href="/Interactive-Maps-in-R-with-leaflet.html" title="Leaflet Interactive Maps"><span class="progress-dot"></span>Leaflet Interactive Maps</a></li><li data-subkey="sec3sub7"><a href="/Spatial-Data-in-R-with-sf.html" title="Spatial Data (sf)"><span class="progress-dot"></span>Spatial Data (sf)</a></li><li data-subkey="sec3sub7"><a href="/Choropleth-Maps-in-R.html" title="Choropleth Maps (sf)"><span class="progress-dot"></span>Choropleth Maps (sf)</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub8" data-collapsed="false"><span class="subsec-chevron">▼</span> Customization & Reference</li><li data-subkey="sec3sub8"><a href="/ggplot2-Legends-in-R.html" title="ggplot2 Legends"><span class="progress-dot"></span>ggplot2 Legends</a></li><li data-subkey="sec3sub8"><a href="/ggplot2-Secondary-Axis.html" title="Secondary Axis"><span class="progress-dot"></span>Secondary Axis</a></li><li data-subkey="sec3sub8"><a href="/ggplot2-Log-Scale.html" title="Log Scale"><span class="progress-dot"></span>Log Scale</a></li><li data-subkey="sec3sub8"><a href="/patchwork-Package.html" title="patchwork (Combine Plots)"><span class="progress-dot"></span>patchwork (Combine Plots)</a></li><li data-subkey="sec3sub8"><a href="/Publication-Quality-Figures-in-R.html" title="Publication-Ready Figures"><span class="progress-dot"></span>Publication-Ready Figures</a></li><li data-subkey="sec3sub8"><a href="/ggplot2-cheatsheet.html" title="ggplot2 Quickref"><span class="progress-dot"></span>ggplot2 Quickref</a></li><li data-subkey="sec3sub8"><a href="/Advanced-ggplot2-Course.html" title="Advanced ggplot2 (Course)"><span class="progress-dot"></span>Advanced ggplot2 (Course)</a></li><li data-subkey="sec3sub8"><a href="/ggplot2-Course.html" title="ggplot2 (Course)"><span class="progress-dot"></span>ggplot2 (Course)</a></li><li data-subkey="sec3sub8"><a href="/Dashboards-Course.html" title="Interactive Dashboards (Course)"><span class="progress-dot"></span>Interactive Dashboards (Course)</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span><span class="sec-num">5.</span> <span class="sec-title-t">Time Series</span><span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li data-subkey="sec4sub0"><a href="/Time-Series-Analysis-With-R.html" title="Time Series Analysis"><span class="progress-dot"></span>Time Series Analysis</a></li><li data-subkey="sec4sub0"><a href="/Time-Series-Forecasting-With-R.html" title="Time Series Forecasting"><span class="progress-dot"></span>Time Series Forecasting</a></li><li data-subkey="sec4sub0"><a href="/Time-Series-Forecasting-With-R-part2.html" title="More Time Series Forecasting"><span class="progress-dot"></span>More Time Series Forecasting</a></li><li data-subkey="sec4sub0" class="is-quiz"><a href="/Time-Series-Exercises-in-R-quiz.html" title="Time Series Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span><span class="sec-num">6.</span> <span class="sec-title-t">Advanced R</span><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="sec5sub1" data-collapsed="false"><span class="subsec-chevron">▼</span> Functional Programming</li><li data-subkey="sec5sub1"><a href="/Functional-Programming-in-R.html" title="Functional Programming"><span class="progress-dot"></span>Functional Programming</a></li><li data-subkey="sec5sub1" class="is-quiz"><a href="/R-Functional-Programming-Exercises-quiz.html" title="Functional Programming Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec5sub1"><a href="/purrr-map-Variants.html" title="purrr map() Variants"><span class="progress-dot"></span>purrr map() Variants</a></li><li data-subkey="sec5sub1"><a href="/R-Anonymous-Functions.html" title="R Anonymous Functions"><span class="progress-dot"></span>R Anonymous Functions</a></li><li data-subkey="sec5sub1"><a href="/R-Function-Factories.html" title="R Function Factories"><span class="progress-dot"></span>R Function Factories</a></li><li data-subkey="sec5sub1"><a href="/R-Function-Operators.html" title="R Function Operators"><span class="progress-dot"></span>R Function Operators</a></li><li data-subkey="sec5sub1"><a href="/Reduce-Filter-Map-in-R.html" title="Reduce, Filter, Map"><span class="progress-dot"></span>Reduce, Filter, Map</a></li><li data-subkey="sec5sub1"><a href="/Memoization-in-R.html" title="Memoization in R"><span class="progress-dot"></span>Memoization in R</a></li><li data-subkey="sec5sub1"><a href="/Writing-Composable-R-Code.html" title="Composable R Code"><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" title="OOP in R: S3/S4/R6"><span class="progress-dot"></span>OOP in R: S3/S4/R6</a></li><li data-subkey="sec5sub2"><a href="/S3-Classes-in-R.html" title="S3 Classes"><span class="progress-dot"></span>S3 Classes</a></li><li data-subkey="sec5sub2"><a href="/S3-Method-Dispatch-in-R.html" title="S3 Method Dispatch"><span class="progress-dot"></span>S3 Method Dispatch</a></li><li data-subkey="sec5sub2"><a href="/S4-Classes-in-R.html" title="S4 Classes"><span class="progress-dot"></span>S4 Classes</a></li><li data-subkey="sec5sub2"><a href="/S4-Methods-in-R.html" title="S4 Methods & Dispatch"><span class="progress-dot"></span>S4 Methods & Dispatch</a></li><li data-subkey="sec5sub2"><a href="/R6-Classes-in-R.html" title="R6 Classes"><span class="progress-dot"></span>R6 Classes</a></li><li data-subkey="sec5sub2"><a href="/R6-Advanced.html" title="R6 Advanced"><span class="progress-dot"></span>R6 Advanced</a></li><li data-subkey="sec5sub2"><a href="/Operator-Overloading-in-R.html" title="Operator Overloading"><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" title="R Names & Values"><span class="progress-dot"></span>R Names & Values</a></li><li data-subkey="sec5sub3"><a href="/R-Assignment-Deep-Dive.html" title="R Assignment Deep Dive"><span class="progress-dot"></span>R Assignment Deep Dive</a></li><li data-subkey="sec5sub3"><a href="/R-Memory-lobstr.html" title="R Memory & lobstr"><span class="progress-dot"></span>R Memory & lobstr</a></li><li data-subkey="sec5sub3"><a href="/R-Environments.html" title="R Environments"><span class="progress-dot"></span>R Environments</a></li><li data-subkey="sec5sub3"><a href="/R-Lexical-Scoping.html" title="Lexical Scoping"><span class="progress-dot"></span>Lexical Scoping</a></li><li data-subkey="sec5sub3"><a href="/R-Closures.html" title="R Closures"><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" title="Conditions System"><span class="progress-dot"></span>Conditions System</a></li><li data-subkey="sec5sub4"><a href="/R-Debugging.html" title="Debugging R Code"><span class="progress-dot"></span>Debugging R Code</a></li><li data-subkey="sec5sub4"><a href="/R-Common-Errors.html" title="50 Common R Errors"><span class="progress-dot"></span>50 Common R Errors</a></li><li data-subkey="sec5sub4"><a href="/Parallel-Computing-With-R.html" title="Parallel Computing"><span class="progress-dot"></span>Parallel Computing</a></li><li data-subkey="sec5sub4"><a href="/Strategies-To-Improve-And-Speedup-R-Code.html" title="Speedup R Code"><span class="progress-dot"></span>Speedup R Code</a></li><li data-subkey="sec5sub4" class="is-quiz"><a href="/Shiny-Exercises-in-R-quiz.html" title="Shiny Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li></ul></li><li class="sidebar-section expanded"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span><span class="sec-num">7.</span> <span class="sec-title-t">Classic Tutorials</span><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" title="R Tutorial (Classic)"><span class="progress-dot"></span>R Tutorial (Classic)</a></li><li data-subkey="sec6sub0"><a href="/ggplot2-Tutorial-With-R.html" title="ggplot2 Short Tutorial"><span class="progress-dot"></span>ggplot2 Short Tutorial</a></li><li data-subkey="sec6sub0"><a href="/Complete-Ggplot2-Tutorial-Part1-With-R-Code.html" title="ggplot2 Tutorial 1 - Intro"><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" title="ggplot2 Tutorial 2 - Theme"><span class="progress-dot"></span>ggplot2 Tutorial 2 - Theme</a></li><li data-subkey="sec6sub0"><a href="/Top50-Ggplot2-Visualizations-MasterList-R-Code.html" title="ggplot2 Tutorial 3 - Masterlist"><span class="progress-dot"></span>ggplot2 Tutorial 3 - Masterlist</a></li><li data-subkey="sec6sub0"><a href="/Association-Mining-With-R.html" class="active" title="Association Mining"><span class="progress-dot"></span>Association Mining</a></li><li data-subkey="sec6sub0"><a href="/Multi-Dimensional-Scaling-With-R.html" title="Multi Dimensional Scaling"><span class="progress-dot"></span>Multi Dimensional Scaling</a></li><li data-subkey="sec6sub0"><a href="/Optimization-With-R.html" title="Optimization"><span class="progress-dot"></span>Optimization</a></li><li data-subkey="sec6sub0"><a href="/Information-Value-With-R.html" title="InformationValue Package"><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><span class="sec-num">8.</span> <span class="sec-title-t">Practice Exercises</span><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" class="is-quiz"><a href="/R-Beginner-Exercises-quiz.html" title="R Fundamentals Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec7sub1" class="is-quiz"><a href="/dplyr-Exercises-in-R-quiz.html" title="dplyr Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec7sub1" class="is-quiz"><a href="/ggplot2-Exercises-in-R-quiz.html" title="ggplot2 Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec7sub1" class="is-quiz"><a href="/Hypothesis-Testing-Exercises-in-R-quiz.html" title="Hypothesis Testing Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec7sub1" class="is-quiz"><a href="/Linear-Regression-Exercises-in-R-quiz.html" title="Linear Regression Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec7sub1" class="is-quiz"><a href="/Machine-Learning-Exercises-in-R-quiz.html" title="Machine Learning Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec7sub1" class="is-quiz"><a href="/tidyr-Exercises-in-R-quiz.html" title="tidyr Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec7sub1" class="is-quiz"><a href="/Time-Series-Exercises-in-R-quiz.html" title="Time Series Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec7sub1" class="is-quiz"><a href="/Shiny-Exercises-in-R-quiz.html" title="Shiny Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec7sub1" class="is-quiz"><a href="/R-Interview-Questions-quiz.html" title="R Interview Readiness Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec7sub1" class="is-quiz"><a href="/R-Functional-Programming-Exercises-quiz.html" title="Functional Programming Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec7sub2" data-collapsed="false"><span class="subsec-chevron">▼</span> R Fundamentals</li><li data-subkey="sec7sub2"><a href="/R-Basics-Exercises.html" title="R Basics (15 problems)"><span class="progress-dot"></span>R Basics (15 problems)</a></li><li data-subkey="sec7sub2"><a href="/R-Vectors-Exercises.html" title="R Vectors (12 problems)"><span class="progress-dot"></span>R Vectors (12 problems)</a></li><li data-subkey="sec7sub2"><a href="/R-Data-Frames-Exercises.html" title="R Data Frames (15 problems)"><span class="progress-dot"></span>R Data Frames (15 problems)</a></li><li data-subkey="sec7sub2"><a href="/R-Lists-Exercises.html" title="R Lists (10 problems)"><span class="progress-dot"></span>R Lists (10 problems)</a></li><li data-subkey="sec7sub2"><a href="/R-Control-Flow-Exercises.html" title="R Control Flow (12 problems)"><span class="progress-dot"></span>R Control Flow (12 problems)</a></li><li data-subkey="sec7sub2"><a href="/R-Functions-Exercises.html" title="R Functions (10 problems)"><span class="progress-dot"></span>R Functions (10 problems)</a></li><li data-subkey="sec7sub2"><a href="/R-String-Exercises.html" title="R Strings (10 problems)"><span class="progress-dot"></span>R Strings (10 problems)</a></li><li data-subkey="sec7sub2"><a href="/R-Date-Time-Exercises.html" title="R Date & Time (10 problems)"><span class="progress-dot"></span>R Date & Time (10 problems)</a></li><li data-subkey="sec7sub2"><a href="/R-Apply-Exercises.html" title="R apply Family (12 problems)"><span class="progress-dot"></span>R apply Family (12 problems)</a></li><li data-subkey="sec7sub2"><a href="/R-Subsetting-Exercises.html" title="R Subsetting (10 problems)"><span class="progress-dot"></span>R Subsetting (10 problems)</a></li><li data-subkey="sec7sub2"><a href="/R-Functional-Programming-Exercises.html" title="Functional Programming (10 problems)"><span class="progress-dot"></span>Functional Programming (10 problems)</a></li><li data-subkey="sec7sub2"><a href="/R-OOP-Exercises.html" title="OOP in R (8 problems)"><span class="progress-dot"></span>OOP in R (8 problems)</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec7sub3" data-collapsed="false"><span class="subsec-chevron">▼</span> Data Wrangling</li><li data-subkey="sec7sub3"><a href="/R-Data-Import-Exercises.html" title="Data Import (10 problems)"><span class="progress-dot"></span>Data Import (10 problems)</a></li><li data-subkey="sec7sub3"><a href="/dplyr-Exercises.html" title="dplyr (15 problems)"><span class="progress-dot"></span>dplyr (15 problems)</a></li><li data-subkey="sec7sub3"><a href="/dplyr-filter-select-Exercises.html" title="dplyr filter() & select() (12 problems)"><span class="progress-dot"></span>dplyr filter() & select() (12 problems)</a></li><li data-subkey="sec7sub3"><a href="/dplyr-group-by-summarise-Exercises.html" title="dplyr group_by() & summarise() (10 problems)"><span class="progress-dot"></span>dplyr group_by() & summarise() (10 problems)</a></li><li data-subkey="sec7sub3"><a href="/dplyr-Join-Exercises.html" title="dplyr Joins (10 problems)"><span class="progress-dot"></span>dplyr Joins (10 problems)</a></li><li data-subkey="sec7sub3"><a href="/data-table-Exercises.html" title="data.table (12 problems)"><span class="progress-dot"></span>data.table (12 problems)</a></li><li data-subkey="sec7sub3"><a href="/purrr-Exercises.html" title="purrr (10 problems)"><span class="progress-dot"></span>purrr (10 problems)</a></li><li data-subkey="sec7sub3"><a href="/tidyr-Reshaping-Exercises.html" title="tidyr Reshaping (10 problems)"><span class="progress-dot"></span>tidyr Reshaping (10 problems)</a></li><li data-subkey="sec7sub3"><a href="/Missing-Data-in-R-Exercises.html" title="Missing Data in R (10 problems)"><span class="progress-dot"></span>Missing Data in R (10 problems)</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec7sub4" data-collapsed="false"><span class="subsec-chevron">▼</span> Visualization</li><li data-subkey="sec7sub4"><a href="/ggplot2-Exercises.html" title="ggplot2 (15 problems)"><span class="progress-dot"></span>ggplot2 (15 problems)</a></li><li data-subkey="sec7sub4"><a href="/ggplot2-Geom-Exercises.html" title="ggplot2 Geoms (12 problems)"><span class="progress-dot"></span>ggplot2 Geoms (12 problems)</a></li><li data-subkey="sec7sub4"><a href="/ggplot2-Aesthetics-Exercises.html" title="ggplot2 Aesthetics (10 problems)"><span class="progress-dot"></span>ggplot2 Aesthetics (10 problems)</a></li><li data-subkey="sec7sub4"><a href="/ggplot2-Customization-Exercises.html" title="ggplot2 Customization (10 problems)"><span class="progress-dot"></span>ggplot2 Customization (10 problems)</a></li><li data-subkey="sec7sub4"><a href="/ggplot2-Facet-Exercises.html" title="ggplot2 Facets (8 problems)"><span class="progress-dot"></span>ggplot2 Facets (8 problems)</a></li><li data-subkey="sec7sub4"><a href="/R-Visualization-Project.html" title="R Visualization Project (5 charts)"><span class="progress-dot"></span>R Visualization Project (5 charts)</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec7sub5" data-collapsed="false"><span class="subsec-chevron">▼</span> Statistics</li><li data-subkey="sec7sub5"><a href="/Probability-in-R-Exercises.html" title="Probability in R Exercises"><span class="progress-dot"></span>Probability in R Exercises</a></li><li data-subkey="sec7sub5"><a href="/R-Probability-Distributions-Exercises.html" title="R Probability Distributions (12 problems)"><span class="progress-dot"></span>R Probability Distributions (12 problems)</a></li><li data-subkey="sec7sub5"><a href="/Binomial-Distribution-Exercises-in-R.html" title="Binomial Distribution Exercises"><span class="progress-dot"></span>Binomial Distribution Exercises</a></li><li data-subkey="sec7sub5"><a href="/Poisson-Distribution-Exercises-in-R.html" title="Poisson Distribution Exercises"><span class="progress-dot"></span>Poisson Distribution Exercises</a></li><li data-subkey="sec7sub5"><a href="/Central-Limit-Theorem-Exercises-in-R.html" title="Central Limit Theorem Exercises"><span class="progress-dot"></span>Central Limit Theorem Exercises</a></li><li data-subkey="sec7sub5"><a href="/Hypothesis-Testing-Exercises-in-R.html" title="Hypothesis Testing Exercises"><span class="progress-dot"></span>Hypothesis Testing Exercises</a></li><li data-subkey="sec7sub5"><a href="/t-Test-Exercises-in-R.html" title="t-Test Exercises (12 problems)"><span class="progress-dot"></span>t-Test Exercises (12 problems)</a></li><li data-subkey="sec7sub5"><a href="/Chi-Square-Test-Exercises-in-R.html" title="Chi-Square Exercises (10 problems)"><span class="progress-dot"></span>Chi-Square Exercises (10 problems)</a></li><li data-subkey="sec7sub5"><a href="/Confidence-Interval-Exercises-in-R.html" title="Confidence Interval (10 problems)"><span class="progress-dot"></span>Confidence Interval (10 problems)</a></li><li data-subkey="sec7sub5"><a href="/Power-Analysis-Exercises-in-R.html" title="Power Analysis Exercises (8 problems)"><span class="progress-dot"></span>Power Analysis Exercises (8 problems)</a></li><li data-subkey="sec7sub5"><a href="/Nonparametric-Tests-Exercises-in-R.html" title="Nonparametric Exercises (10 problems)"><span class="progress-dot"></span>Nonparametric Exercises (10 problems)</a></li><li data-subkey="sec7sub5"><a href="/Multiple-Testing-Exercises-in-R.html" title="Multiple Testing (8 problems)"><span class="progress-dot"></span>Multiple Testing (8 problems)</a></li><li data-subkey="sec7sub5"><a href="/Multiple-Regression-Exercises-in-R.html" title="Multiple Regression Exercises"><span class="progress-dot"></span>Multiple Regression Exercises</a></li><li data-subkey="sec7sub5"><a href="/Logistic-Regression-Exercises-in-R.html" title="Logistic Regression Exercises (10 problems)"><span class="progress-dot"></span>Logistic Regression Exercises (10 problems)</a></li><li data-subkey="sec7sub5"><a href="/Regression-Diagnostics-Exercises-in-R.html" title="Regression Diagnostics Exercises"><span class="progress-dot"></span>Regression Diagnostics Exercises</a></li><li data-subkey="sec7sub5"><a href="/Ridge-and-Lasso-Exercises-in-R.html" title="Ridge & Lasso Exercises"><span class="progress-dot"></span>Ridge & Lasso Exercises</a></li><li data-subkey="sec7sub5"><a href="/GLM-Exercises-in-R.html" title="GLM Exercises (10 problems)"><span class="progress-dot"></span>GLM Exercises (10 problems)</a></li><li data-subkey="sec7sub5"><a href="/ANOVA-Exercises-in-R.html" title="ANOVA Exercises (15 problems)"><span class="progress-dot"></span>ANOVA Exercises (15 problems)</a></li><li data-subkey="sec7sub5"><a href="/Post-Hoc-Tests-Exercises-in-R.html" title="Post-Hoc Tests Exercises (8 problems)"><span class="progress-dot"></span>Post-Hoc Tests Exercises (8 problems)</a></li><li data-subkey="sec7sub5"><a href="/Repeated-Measures-Exercises-in-R.html" title="Repeated Measures (8 problems)"><span class="progress-dot"></span>Repeated Measures (8 problems)</a></li><li data-subkey="sec7sub5"><a href="/Experimental-Design-Exercises-in-R.html" title="Experimental Design Exercises (8 problems)"><span class="progress-dot"></span>Experimental Design Exercises (8 problems)</a></li><li data-subkey="sec7sub5"><a href="/AB-Testing-Exercises-in-R.html" title="A/B Testing Exercises (8 problems)"><span class="progress-dot"></span>A/B Testing Exercises (8 problems)</a></li><li data-subkey="sec7sub5"><a href="/Linear-Regression-Exercises-in-R.html" title="Linear Regression (15 problems)"><span class="progress-dot"></span>Linear Regression (15 problems)</a></li><li data-subkey="sec7sub5"><a href="/PCA-Exercises-in-R.html" title="PCA Exercises (10 problems)"><span class="progress-dot"></span>PCA Exercises (10 problems)</a></li><li data-subkey="sec7sub5"><a href="/Cluster-Analysis-Exercises-in-R.html" title="Clustering Exercises (10 problems)"><span class="progress-dot"></span>Clustering Exercises (10 problems)</a></li><li data-subkey="sec7sub5"><a href="/SEM-Exercises-in-R.html" title="SEM Exercises (8 problems)"><span class="progress-dot"></span>SEM Exercises (8 problems)</a></li><li data-subkey="sec7sub5"><a href="/A-B-Testing-Exercises-in-R.html" title="A/B Testing Exercises"><span class="progress-dot"></span>A/B Testing Exercises</a></li><li data-subkey="sec7sub5"><a href="/API-Calls-Exercises-in-R.html" title="API Calls Exercises"><span class="progress-dot"></span>API Calls Exercises</a></li><li data-subkey="sec7sub5"><a href="/ARIMA-Exercises-in-R.html" title="ARIMA Exercises"><span class="progress-dot"></span>ARIMA Exercises</a></li><li data-subkey="sec7sub5"><a href="/Apply-Family-Exercises-in-R.html" title="Apply Family Exercises"><span class="progress-dot"></span>Apply Family Exercises</a></li><li data-subkey="sec7sub5"><a href="/Bayesian-Statistics-Exercises-in-R.html" title="Bayesian Statistics Exercises"><span class="progress-dot"></span>Bayesian Statistics Exercises</a></li><li data-subkey="sec7sub5"><a href="/Clustering-Exercises-in-R.html" title="Clustering Exercises"><span class="progress-dot"></span>Clustering Exercises</a></li><li data-subkey="sec7sub5"><a href="/Correlation-Exercises-in-R.html" title="Correlation Exercises"><span class="progress-dot"></span>Correlation Exercises</a></li><li data-subkey="sec7sub5"><a href="/Cross-Validation-Exercises-in-R.html" title="Cross Validation Exercises"><span class="progress-dot"></span>Cross Validation Exercises</a></li><li data-subkey="sec7sub5"><a href="/Data-Cleaning-Exercises-in-R.html" title="Data Cleaning Exercises"><span class="progress-dot"></span>Data Cleaning Exercises</a></li><li data-subkey="sec7sub5"><a href="/Data-Visualization-Exercises-in-R.html" title="Data Viz Exercises"><span class="progress-dot"></span>Data Viz Exercises</a></li><li data-subkey="sec7sub5"><a href="/Data-Wrangling-Exercises-in-R.html" title="Data Wrangling Exercises"><span class="progress-dot"></span>Data Wrangling Exercises</a></li><li data-subkey="sec7sub5"><a href="/Decision-Tree-Exercises-in-R.html" title="Decision Tree Exercises"><span class="progress-dot"></span>Decision Tree Exercises</a></li><li data-subkey="sec7sub5"><a href="/EDA-Exercises-in-R.html" title="EDA Exercises"><span class="progress-dot"></span>EDA Exercises</a></li><li data-subkey="sec7sub5"><a href="/GAM-Exercises-in-R.html" title="GAM Exercises"><span class="progress-dot"></span>GAM Exercises</a></li><li data-subkey="sec7sub5"><a href="/Machine-Learning-Exercises-in-R.html" title="Machine Learning Exercises"><span class="progress-dot"></span>Machine Learning Exercises</a></li><li data-subkey="sec7sub5"><a href="/Mixed-Effects-Models-Exercises-in-R.html" title="Mixed Effects Exercises"><span class="progress-dot"></span>Mixed Effects Exercises</a></li><li data-subkey="sec7sub5"><a href="/Network-Analysis-Exercises-in-R.html" title="Network Analysis Exercises"><span class="progress-dot"></span>Network Analysis Exercises</a></li><li data-subkey="sec7sub5"><a href="/Parallel-Computing-in-R-Exercises.html" title="Parallel Computing Exercises"><span class="progress-dot"></span>Parallel Computing Exercises</a></li><li data-subkey="sec7sub5"><a href="/Poisson-Regression-Exercises-in-R.html" title="Poisson Regression"><span class="progress-dot"></span>Poisson Regression</a></li><li data-subkey="sec7sub5"><a href="/Probability-Distributions-Exercises-in-R.html" title="Probability Distributions"><span class="progress-dot"></span>Probability Distributions</a></li><li data-subkey="sec7sub5"><a href="/R-Beginner-Exercises.html" title="R Beginner Exercises"><span class="progress-dot"></span>R Beginner Exercises</a></li><li data-subkey="sec7sub5"><a href="/R-Debugging-Exercises.html" title="R Debugging Exercises"><span class="progress-dot"></span>R 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<h1>Association Mining (Market Basket Analysis)</h1>
<blockquote>
<p>Association mining is commonly used to make product recommendations by identifying products that are frequently bought together. But, if you are not careful, the rules can give misleading results in certain cases.</p>
</blockquote>
<p>Association mining is usually done on transactions data from a retail market or from an online e-commerce store. Since most transactions data is large, the <code>apriori</code> algorithm makes it easier to find these patterns or <em>rules</em> quickly.</p>
<p>So, What is a <em>rule</em>?</p>
<p>A rule is a notation that represents which item/s is frequently bought with what item/s. It has an <em>LHS</em> and an <em>RHS</em> part and can be represented as follows:</p>
<p><strong>itemset A => itemset B</strong></p>
<p>This means, the item/s on the right were frequently purchased along with items on the left.</p>
<h2>How to measure the strength of a rule?</h2>
<p>The <code>apriori()</code> generates the most relevent set of rules from a given transaction data. It also shows the <em>support</em>, <em>confidence</em> and <em>lift</em> of those rules. These three measure can be used to decide the relative strength of the rules. So what do these terms mean?</p>
<p>Lets consider the rule <strong>A => B</strong> in order to compute these metrics.</p>
<p><br /><span class="math display">$$Support = \frac{Number\ of\ transactions\ with\ both\ A\ and\ B}{Total\ number\ of\ transactions} = P\left(A \cap B\right)$$</span><br /></p>
<p><br /><span class="math display">$$Confidence = \frac{Number\ of\ transactions\ with\ both\ A\ and\ B}{Total\ number\ of\ transactions\ with\ A} = \frac{P\left(A \cap B\right)}{P\left(A\right)}$$</span><br /></p>
<p><br /><span class="math display">$$Expected Confidence = \frac{Number\ of\ transactions\ with\ B}{Total\ number\ of\ transactions} = P\left(B\right)$$</span><br /></p>
<p><br /><span class="math display">$$Lift = \frac{Confidence}{Expected\ Confidence} = \frac{P\left(A \cap B\right)}{P\left(A\right).P\left(B\right)}$$</span><br /></p>
<p><em>Lift</em> is the factor by which, the co-occurence of A and B exceeds the expected probability of A and B co-occuring, had they been independent. So, higher the lift, higher the chance of A and B occurring together.</p>
<p>Lets see how to get the rules, confidence, lift etc using the <code>arules</code> package in R.</p>
<h2>Example</h2>
<h4>Transactions data</h4>
<p>Lets play with the <code>Groceries</code> data that comes with the <code>arules</code> pkg. Unlike dataframe, using <code>head(Groceries)</code> does not display the transaction items in the data. To view the transactions, use the <code>inspect()</code> function instead.</p>
<p>Since association mining deals with transactions, the data has to be converted to one of class <code>transactions</code>, made available in R through the <code>arules</code> pkg. This is a necessary step because the <code>apriori()</code> function accepts transactions data of class <code>transactions</code> only.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">library</span>(arules)
<span class="kw">class</span>(Groceries)
<span class="co">#> [1] "transactions"</span>
<span class="co">#> attr(,"package")</span>
<span class="co">#> [1] "arules"</span>
<span class="kw">inspect</span>(<span class="kw">head</span>(Groceries, <span class="dv">3</span>))
<span class="co">#> items </span>
<span class="co">#> 1 {citrus fruit, </span>
<span class="co">#> semi-finished bread, </span>
<span class="co">#> margarine, </span>
<span class="co">#> ready soups} </span>
<span class="co">#> 2 {tropical fruit, </span>
<span class="co">#> yogurt, </span>
<span class="co">#> coffee} </span>
<span class="co">#> 3 {whole milk} </span></code></pre></div>
<p>If you have to read data from a file as a transactions data, use <code>read.transactions()</code>.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">tdata <-<span class="st"> </span><span class="kw">read.transactions</span>(<span class="st">"transactions_data.txt"</span>, <span class="dt">sep=</span><span class="st">"</span><span class="ch">\t</span><span class="st">"</span>)</code></pre></div>
<p>If you already have your transactions stored as a dataframe, you could convert it to class <code>transactions</code> as follows,</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">tData <-<span class="st"> </span><span class="kw">as</span> (myDataFrame, <span class="st">"transactions"</span>) <span class="co"># convert to 'transactions' class</span></code></pre></div>
<p>Here are couple more utility functions that are good to know:</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">size</span>(<span class="kw">head</span>(Groceries)) <span class="co"># number of items in each observation</span>
<span class="co">#> [1] 4 3 1 4 4 5</span>
<span class="kw">LIST</span>(<span class="kw">head</span>(Groceries, <span class="dv">3</span>)) <span class="co"># convert 'transactions' to a list, note the LIST in CAPS</span>
<span class="co">#> [[1]]</span>
<span class="co">#> [1] "citrus fruit" "semi-finished bread" "margarine" </span>
<span class="co">#> [4] "ready soups" </span>
<span class="co">#> </span>
<span class="co">#> [[2]]</span>
<span class="co">#> [1] "tropical fruit" "yogurt" "coffee" </span>
<span class="co">#> </span>
<span class="co">#> [[3]]</span>
<span class="co">#> [1] "whole milk"</span></code></pre></div>
<h2>How to see the most frequent items?</h2>
<p>The <code>eclat()</code> takes in a transactions object and gives the most frequent items in the data based the support you provide to the <code>supp</code> argument. The <code>maxlen</code> defines the maximum number of items in each itemset of frequent items.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">frequentItems <-<span class="st"> </span><span class="kw">eclat</span> (Groceries, <span class="dt">parameter =</span> <span class="kw">list</span>(<span class="dt">supp =</span> <span class="fl">0.07</span>, <span class="dt">maxlen =</span> <span class="dv">15</span>)) <span class="co"># calculates support for frequent items</span>
<span class="kw">inspect</span>(frequentItems)
<span class="co">#> items support </span>
<span class="co">#> 1 {other vegetables,whole milk} 0.07483477</span>
<span class="co">#> 2 {whole milk} 0.25551601</span>
<span class="co">#> 3 {other vegetables} 0.19349263</span>
<span class="co">#> 4 {rolls/buns} 0.18393493</span>
<span class="co">#> 5 {yogurt} 0.13950178</span>
<span class="co">#> 6 {soda} 0.17437722</span>
<span class="kw">itemFrequencyPlot</span>(Groceries, <span class="dt">topN=</span><span class="dv">10</span>, <span class="dt">type=</span><span class="st">"absolute"</span>, <span class="dt">main=</span><span class="st">"Item Frequency"</span>) <span class="co"># plot frequent items</span></code></pre></div>
<p><img src='screenshots/item_frequency_plot_arules.png' width='528' height='289' alt="Item Frequency Plot Arules" /></p>
<h1>How to get the product recommendation rules?</h1>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">rules <-<span class="st"> </span><span class="kw">apriori</span> (Groceries, <span class="dt">parameter =</span> <span class="kw">list</span>(<span class="dt">supp =</span> <span class="fl">0.001</span>, <span class="dt">conf =</span> <span class="fl">0.5</span>)) <span class="co"># Min Support as 0.001, confidence as 0.8.</span>
rules_conf <-<span class="st"> </span><span class="kw">sort</span> (rules, <span class="dt">by=</span><span class="st">"confidence"</span>, <span class="dt">decreasing=</span><span class="ot">TRUE</span>) <span class="co"># 'high-confidence' rules.</span>
<span class="kw">inspect</span>(<span class="kw">head</span>(rules_conf)) <span class="co"># show the support, lift and confidence for all rules</span>
<span class="co">#> lhs rhs support confidence lift </span>
<span class="co">#> 113 {rice,sugar} => {whole milk} 0.001220132 1 3.913649</span>
<span class="co">#> 258 {canned fish,hygiene articles} => {whole milk} 0.001118454 1 3.913649</span>
<span class="co">#> 1487 {root vegetables,butter,rice} => {whole milk} 0.001016777 1 3.913649</span>
<span class="co">#> 1646 {root vegetables,whipped/sour cream,flour} => {whole milk} 0.001728521 1 3.913649</span>
<span class="co">#> 1670 {butter,soft cheese,domestic eggs} => {whole milk} 0.001016777 1 3.913649</span>
<span class="co">#> 1699 {citrus fruit,root vegetables,soft cheese} => {other vegetables} 0.001016777 1 5.168156</span>
rules_lift <-<span class="st"> </span><span class="kw">sort</span> (rules, <span class="dt">by=</span><span class="st">"lift"</span>, <span class="dt">decreasing=</span><span class="ot">TRUE</span>) <span class="co"># 'high-lift' rules.</span>
<span class="kw">inspect</span>(<span class="kw">head</span>(rules_lift)) <span class="co"># show the support, lift and confidence for all rules</span>
<span class="co">#> lhs rhs support confidence lift </span>
<span class="co">#> 53 {Instant food products,soda} => {hamburger meat} 0.001220 0.6315789 18.995</span>
<span class="co">#> 37 {soda,popcorn} => {salty snack} 0.001220 0.6315789 16.697</span>
<span class="co">#> 444 {flour,baking powder} => {sugar} 0.001016 0.5555556 16.408</span>
<span class="co">#> 327 {ham,processed cheese} => {white bread} 0.001931 0.6333333 15.045</span>
<span class="co">#> 55 {whole milk,Instant food products} => {hamburger meat} 0.001525 0.5000000 15.038</span>
<span class="co">#> 4807 {other vegetables,curd,yogurt,whipped/sour cream} => {cream cheese } 0.001016 0.5882353 14.834</span></code></pre></div>
<p>The rules with confidence of 1 (see <code>rules_conf</code> above) imply that, whenever the LHS item was purchased, the RHS item was also purchased 100% of the time.</p>
<p>A rule with a lift of 18 (see <code>rules_lift</code> above) imply that, the items in LHS and RHS are 18 times more likely to be purchased together compared to the purchases when they are assumed to be unrelated.</p>
<h2>How To Control The Number Of Rules in Output ?</h2>
<p>Adjust the <code>maxlen</code>, <code>supp</code> and <code>conf</code> arguments in the <code>apriori</code> function to control the number of rules generated. You will have to adjust this based on the sparesness of you data.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">rules <-<span class="st"> </span><span class="kw">apriori</span>(Groceries, <span class="dt">parameter =</span> <span class="kw">list</span> (<span class="dt">supp =</span> <span class="fl">0.001</span>, <span class="dt">conf =</span> <span class="fl">0.5</span>, <span class="dt">maxlen=</span><span class="dv">3</span>)) <span class="co"># maxlen = 3 limits the elements in a rule to 3</span></code></pre></div>
<ol style="list-style-type: decimal">
<li>To get <strong>‘strong‘</strong> rules, increase the value of <em>‘conf’</em> parameter.</li>
<li>To get <strong>‘longer‘</strong> rules, increase <em>‘maxlen’</em>.</li>
</ol>
<h2>How To Remove Redundant Rules ?</h2>
<p>Sometimes it is desirable to remove the rules that are subset of larger rules. To do so, use the below code to filter the redundant rules.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">subsetRules <-<span class="st"> </span><span class="kw">which</span>(<span class="kw">colSums</span>(<span class="kw">is.subset</span>(rules, rules)) ><span class="st"> </span><span class="dv">1</span>) <span class="co"># get subset rules in vector</span>
<span class="kw">length</span>(subsetRules) <span class="co">#> 3913</span>
rules <-<span class="st"> </span>rules[-subsetRules] <span class="co"># remove subset rules. </span></code></pre></div>
<h2>How to Find Rules Related To Given Item/s ?</h2>
<p>This can be achieved by modifying the <code>appearance</code> parameter in the <code>apriori()</code> function. For example,</p>
<h4>To find what factors influenced purchase of product X</h4>
<p>To find out what customers had purchased before buying ‘Whole Milk’. This will help you understand the patterns that led to the purchase of ‘whole milk’.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">rules <-<span class="st"> </span><span class="kw">apriori</span> (<span class="dt">data=</span>Groceries, <span class="dt">parameter=</span><span class="kw">list</span> (<span class="dt">supp=</span><span class="fl">0.001</span>,<span class="dt">conf =</span> <span class="fl">0.08</span>), <span class="dt">appearance =</span> <span class="kw">list</span> (<span class="dt">default=</span><span class="st">"lhs"</span>,<span class="dt">rhs=</span><span class="st">"whole milk"</span>), <span class="dt">control =</span> <span class="kw">list</span> (<span class="dt">verbose=</span>F)) <span class="co"># get rules that lead to buying 'whole milk'</span>
rules_conf <-<span class="st"> </span><span class="kw">sort</span> (rules, <span class="dt">by=</span><span class="st">"confidence"</span>, <span class="dt">decreasing=</span><span class="ot">TRUE</span>) <span class="co"># 'high-confidence' rules.</span>
<span class="kw">inspect</span>(<span class="kw">head</span>(rules_conf))
<span class="co">#> lhs rhs support confidence lift </span>
<span class="co">#> 196 {rice,sugar} => {whole milk} 0.001220132 1 3.913649</span>
<span class="co">#> 323 {canned fish,hygiene articles} => {whole milk} 0.001118454 1 3.913649</span>
<span class="co">#> 1643 {root vegetables,butter,rice} => {whole milk} 0.001016777 1 3.913649</span>
<span class="co">#> 1705 {root vegetables,whipped/sour cream,flour} => {whole milk} 0.001728521 1 3.913649</span>
<span class="co">#> 1716 {butter,soft cheese,domestic eggs} => {whole milk} 0.001016777 1 3.913649</span>
<span class="co">#> 1985 {pip fruit,butter,hygiene articles} => {whole milk} 0.001016777 1 3.913649</span></code></pre></div>
<h4>To find out what products were purchased after/along with product X</h4>
<p>The is a case to find out <em>the Customers who bought ‘Whole Milk’ also bought . .</em> In the equation, ‘whole milk’ is in LHS (left hand side).</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">rules <-<span class="st"> </span><span class="kw">apriori</span> (<span class="dt">data=</span>Groceries, <span class="dt">parameter=</span><span class="kw">list</span> (<span class="dt">supp=</span><span class="fl">0.001</span>,<span class="dt">conf =</span> <span class="fl">0.15</span>,<span class="dt">minlen=</span><span class="dv">2</span>), <span class="dt">appearance =</span> <span class="kw">list</span>(<span class="dt">default=</span><span class="st">"rhs"</span>,<span class="dt">lhs=</span><span class="st">"whole milk"</span>), <span class="dt">control =</span> <span class="kw">list</span> (<span class="dt">verbose=</span>F)) <span class="co"># those who bought 'milk' also bought..</span>
rules_conf <-<span class="st"> </span><span class="kw">sort</span> (rules, <span class="dt">by=</span><span class="st">"confidence"</span>, <span class="dt">decreasing=</span><span class="ot">TRUE</span>) <span class="co"># 'high-confidence' rules.</span>
<span class="kw">inspect</span>(<span class="kw">head</span>(rules_conf))
<span class="co">#> lhs rhs support confidence lift </span>
<span class="co">#> 6 {whole milk} => {other vegetables} 0.07483477 0.2928770 1.5136341</span>
<span class="co">#> 5 {whole milk} => {rolls/buns} 0.05663447 0.2216474 1.2050318</span>
<span class="co">#> 4 {whole milk} => {yogurt} 0.05602440 0.2192598 1.5717351</span>
<span class="co">#> 2 {whole milk} => {root vegetables} 0.04890696 0.1914047 1.7560310</span>
<span class="co">#> 1 {whole milk} => {tropical fruit} 0.04229792 0.1655392 1.5775950</span>
<span class="co">#> 3 {whole milk} => {soda} 0.04006101 0.1567847 0.8991124</span></code></pre></div>
<p>One drawback with this is, you will get only 1 item on the RHS, irrespective of the support, confidence or minlen parameters.</p>
<h2>Caveat with using Lift</h2>
<p>The directionality of the rule is lost when <em>lift</em> is used. That is, the lift of any rule, <em>A => B</em> and the rule <em>B => A</em> will be the same. See the calculation below:</p>
<h4><em>A -> B</em></h4>
<ul>
<li><p>Support: <span class="math inline"><em>P</em>(<em>A</em>∩<em>B</em>)</span></p></li>
<li><p>Confidence: <span class="math inline">$\frac{P\left( A \cap B \right)}{P\left( A \right)}$</span></p></li>
<li><p>Expected Confidence: <span class="math inline"><em>P</em>(<em>B</em>)</span></p></li>
<li><p>Lift: <span class="math inline">$\frac{Confidence}{Expected\ Confidence}$</span> = <span class="math inline">$\frac{P\left( A \cap B \right)}{P\left( A \right).P\left( B \right)}$</span></p></li>
</ul>
<h4><em>B -> A</em></h4>
<ul>
<li><p>Support: <span class="math inline"><em>P</em>(<em>A</em>∩<em>B</em>)</span></p></li>
<li><p>Confidence: <span class="math inline">$\frac{P\left( A \cap B \right)}{P\left( B \right)}$</span></p></li>
<li><p>Expected Confidence: <span class="math inline"><em>P</em>(<em>B</em>)</span></p></li>
<li><p>Lift: <span class="math inline">$\frac{Confidence}{Expected\ Confidence}$</span> = <span class="math inline">$\frac{P\left( A \cap B \right)}{P\left( A \right).P\left( B \right)}$</span></p></li>
</ul>
<h4>Important Note</h4>
<p>For both rules <em>A -> B</em> and <em>B -> A</em>, the value of <em>lift</em> and support turns out to be the same. This means we cannot use lift to make recommendation for a particular <em>directional</em> ‘rule’. It can merely be used to club frequently bought items into groups.</p>
<h2>Caveat with using Confidence</h2>
<p>The <em>confidence</em> of a rule can be a misleading measure while making product recommendations in real world problems, especially while making <em>add-ons</em> product recommendations. Lets consider the following data with 4 transactions, involving IPhones and Headsets:</p>
<ol style="list-style-type: decimal">
<li>Iphone, Headset</li>
<li>Iphone, Headset</li>
<li>Iphone</li>
<li>Iphone</li>
</ol>
<p>We can create 2 rules for these transactions as shown below:</p>
<ol style="list-style-type: decimal">
<li><em>Iphone -> Headset</em></li>
<li><em>Headset -> IPhone</em></li>
</ol>
<p>In real world, it would be realistic to recommend <em>headphones</em> to a person who just bought an <em>iPhone</em> and not the other way around. Imagine being recommended an iPhone when you just finished purchasing a pair of headphones. Not nice!.</p>
<p>While selecting rules from the <code>apriori</code> output, you might guess that higher the confidence a rule has, better is the rule. But for cases like this, the headset -> iPhone rule will have a higher confidence (2 times) over iPhone -> headset. Can you see why? The calculation below show how.</p>
<h4>Confidence Calculation:</h4>
<p><strong>iPhone -> Headset</strong>: <span class="math inline">$\frac{P(iPhone\ \cap\ Headset)}{P(iPhone)}$</span> = 0.5 / 1 = <strong>0.5</strong></p>
<p><strong>Headset -> iPhone</strong>: <span class="math inline">$\frac{P(iPhone\ \cap\ Headset)}{P(Headset)}$</span> = 0.5 / 0.5 = <strong>1.0</strong></p>
<p>As, you can see, the <em>headset -> iPhone</em> recommendation has a higher confidence, which is misleading and unrealistic. So, confidence should not be the <em>only measure</em> you should use to make product recommendations.</p>
<p>So, you probably need to check more criteria such as the price of products, product types etc before recommending items, especially in cross selling cases.</p>
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