Statistical Rethinking: A Bayesian Course with Examples in R and Stan

Citation

McElreath, R. (2020). Statistical Rethinking: A Bayesian Course with Examples in R and Stan (2nd ed.). Chapman & Hall/CRC.

Chapter Summary

The second edition has seventeen chapters, a preface that sets out the intended audience, the teaching strategy and how to use the book, and endnotes. Most chapters end with a summary and practice problems. Working code is built into the text, so the book is meant to be read at a keyboard.

Chapter 1: The Golem of Prague

The opening chapter introduces the golem metaphor from which the book takes its approach. Its sections on statistical golems, statistical rethinking and tools for golem engineering set up the case for building and checking models rather than applying procedures mechanically.

Chapters 2–3: Small Worlds and Large Worlds; Sampling the Imaginary

Chapter 2 builds Bayesian updating from the "garden of forking data", then walks through building a model, its components and making it run. Chapter 3 shows how to work with a posterior by sampling from a grid approximation, summarising the samples and simulating predictions.

Chapter 4: Geocentric Models

Linear regression is introduced under the heading of geocentric models. The chapter covers why normal distributions arise, a language for describing models, a Gaussian model of height, linear prediction, and curves built from lines.

Chapters 5–6: The Many Variables & The Spurious Waffles; The Haunted DAG & The Causal Terror

These two chapters carry the book's causal emphasis. Chapter 5 treats spurious association, masked relationships and categorical variables; Chapter 6 treats multicollinearity, post-treatment bias, collider bias and ways of confronting confounding. The publisher notes that the second edition integrates directed acyclic graphs (DAGs) into many examples and uses the dagitty R package.

Chapter 7: Ulysses' Compass

The chapter is about choosing among models without being fooled by fit. Its sections cover the problem with parameters, entropy and accuracy, regularisation, predicting predictive accuracy and model comparison.

Chapter 8: Conditional Manatees

Interactions are introduced by building an interaction, examining the symmetry of interactions and extending to continuous interactions.

Chapter 9: Markov Chain Monte Carlo

A story of an island kingdom introduces Metropolis algorithms, followed by Hamiltonian Monte Carlo, the ulam interface to it, and a section on checking and caring for a Markov chain.

Chapters 10–12: Big Entropy and the Generalized Linear Model; God Spiked the Integers; Monsters and Mixtures

Chapter 10 presents maximum entropy as the principle behind generalized linear models and for choosing priors. Chapter 11 covers binomial, Poisson, multinomial and categorical models for counts and categories. Chapter 12 extends this to over-dispersed counts, zero-inflated outcomes and ordered categorical outcomes and predictors.

Chapters 13–14: Models With Memory; Adventures in Covariance

Chapter 13 introduces multilevel models through a tadpole example, varying effects and the underfitting/overfitting trade-off, several types of cluster, divergent transitions with non-centered priors, and multilevel posterior predictions. Chapter 14 continues with varying slopes, instruments and causal designs, social relations as correlated varying effects, and continuous categories with the Gaussian process.

Chapters 15–16: Missing Data and Other Opportunities; Generalized Linear Madness

Chapter 15 treats measurement error, missing data, and categorical errors and discrete absences. Chapter 16 goes beyond generalized linear models to build scientific models directly into the analysis, through examples titled geometric people, hidden minds and observed behavior, ordinary differential nut cracking and population dynamics. The publisher describes this chapter as entirely new in the second edition.

Chapter 17: Horoscopes

A short closing chapter, with no subsections listed in the contents, followed by the endnotes.

Key Concepts

Chapters 1–4

• Statistical golem: the book's metaphor for statistical procedures, introduced in Chapter 1.
• Garden of forking data: the counting argument from which Bayesian updating is developed.
• Posterior sampling: working with a posterior distribution through samples rather than formulas.
• Geocentric model: the book's label for the linear regression models of Chapter 4.

Chapters 5–8

• Spurious association and masking: the ways a second predictor can create or hide a relationship.
• Collider and post-treatment bias: the causal traps named in Chapter 6.
• Entropy and out-of-sample accuracy: the information-theoretic basis for comparing models.
• Interaction: a predictor's effect depending on the value of another.

Chapters 9–12

• Hamiltonian Monte Carlo: the sampling algorithm behind the book's model fitting.
• Maximum entropy: a rule for choosing likelihoods and priors given what is known.
• Generalized linear model: regression for outcomes that are counts or categories.
• Over-dispersion, zero inflation and ordered categories: complications of count and category data.

Chapters 13–17

• Multilevel (varying-effects) model: a model that lets parameters vary by cluster and shares information among clusters.
• Non-centered prior: a reparameterisation used to avoid divergent transitions.
• Instrumental variable and Gaussian process: tools for causal designs and continuous clustering.
• Measurement error and missing data: treated as parts of the model rather than data cleaning.

Critical Analysis

Strengths

• A single causal thread. The publisher states that the second edition emphasises the DAG approach to causal inference and integrates it into many examples, and the contents show it: Chapters 5 and 6 are organised around confounding, and instruments and causal designs return in Chapter 14.
• Computation as explanation. The publisher's features list says working code is integrated into the main text, assumptions are examined through how they appear in code, and optional sections give more detailed mathematics. The reader is pushed to carry out calculations that software usually hides.
• Range of the later chapters. Core material runs from basic regression to multilevel models, and goes on to measurement error, missing data, Gaussian processes for spatial and phylogenetic confounding, and scientific models built from scratch.
• Reviewer response. Reviews quoted on the publisher's page praise the clarity of the writing and the humour; Nathan Green in the Journal of the Royal Statistical Society (2021) recommended it to students old and new, and Abhirup Mallik in Technometrics (August 2021) described it as reframing the usual graduate methods so that readers appreciate their assumptions. The book received the 2023 De Groot Prize from the International Society for Bayesian Analysis.

Limitations

• Not a mathematical statistics text. A review in the Journal of Statistics Education, as summarised on Wikipedia, notes the lack of mathematical rigour that a doctoral student in statistics might need for developing Bayesian methodology.
• Tied to the author's software. The same review points to reliance on the author's rethinking R package. The package is provided on the author's website and on GitHub, so a reader who prefers another environment will have to translate the examples.
• Long. The book runs to about 600 pages, and because working code is part of the text, a full read-through takes sustained time at the keyboard.

Where It Fits in the Course

Within the research methods and statistics course, this is the one book that teaches Bayesian modelling and causal reasoning. Beth Morling's Research Methods in Psychology and the open textbook by Jhangiani and colleagues are introductions to research design, measurement and reporting, and Dancey and Reidy's Statistics without Maths for Psychology and Andy Field's Discovering Statistics Using IBM SPSS Statistics cover conventional testing and regression. Any of these makes a sensible foundation; McElreath builds models from Bayesian principles instead of working through a list of standard tests.

Maxwell, Delaney and Kelley's Designing Experiments and Analyzing Data is the closest companion in level: it treats the analysis of variance as comparison of statistical models, whereas McElreath moves to generalized and multilevel models fitted by simulation. Reading McElreath after a conventional course offers a second logic for analysing the same kinds of data.

How to Use This Book

The Journal of the Royal Statistical Society review describes the book as directed at working researchers and advanced doctoral students, while suggesting it deserves a wider audience. The publisher says the core material starts from the basics of regression, so introductory statistics, regression and some familiarity with R are the useful preparation.

A practical plan:

• Core path: Chapters 1 to 9 in order, running every code example. They set up the logic of Bayesian updating, regression, causal graphs, model comparison, interactions and MCMC, which everything after relies on.
• Next: Chapters 10 to 14 for generalized linear and multilevel models. Chapter 13 is the entry to multilevel models.
• Selective: Chapters 15 and 16 are best read when a project needs measurement error, missing data or a scientific model of its own; Chapter 16 is the most open-ended.
• Practice: Do the practice problems at the end of each chapter before moving on, and use the optional mathematical sections only when the code leaves a question unanswered.
• Alongside: Pair the book with one of the conventional methods texts in this course if a class also needs the standard hypothesis-testing toolkit.