Doing Bayesian Data Analysis

A Tutorial Introduction with R

Kruschke John

There is an explosion of interest in Bayesian statistics, primarily because recently created computational methods have finally made Bayesian analysis obtainable to a wide audience. Doing Bayesian Data Analysis, A Tutorial Introduction with R and BUGS provides an accessible approach to Bayesian data analysis, as material is explained clearly with concrete examples. The book begins with the basics, including essential concepts of probability and random sampling, and gradually progresses to advanced hierarchical modeling methods for realistic data. The text delivers comprehensive coverage of all scenarios addressed by non-Bayesian textbooks--t-tests, analysis of variance (ANOVA) and comparisons in ANOVA, multiple regression, and chi-square (contingency table analysis).

This book is intended for first year graduate students or advanced undergraduates. It provides a bridge between undergraduate training and modern Bayesian methods for data analysis, which is becoming the accepted research standard. Prerequisite is knowledge of algebra and basic calculus. Free software now includes programs in JAGS, which runs on Macintosh, Linux, and Windows.

Author website: http://www.indiana.edu/~kruschke/DoingBayesianDataAnalysis/

Audience
First-year Graduate Students and Advanced Undergraduate Students in Statistics, Psychology, Cognitive Science, Social Sciences, Clinical Sciences and Consumer Sciences in Business.

Buy this

Hardbound, 672 Pages

Published: October 2010

Imprint: Academic Press

ISBN: 978-0-12-381485-2

Reviews

I think it fills a gaping hole in what is currently available, and will serve to create its own market as researchers and their students transition towards the routine application of Bayesian statistical methods. -Prof. Michael lee, University of California, Irvine, and president of the Society for Mathematical Psychology
Kruschkes text covers a much broader range of traditional experimental designshas the potential to change the way
most cognitive scientists and experimental psychologists approach the planning and analysis of their experiments" -Prof. Geoffrey Iverson, University of California, Irvine, and past president of the Society for Mathematical Psychology
John Kruschke has written a book on Statistics. Its better than others for reasons stylistic. It also is better because itis Bayesian. To find out why, buy it -- its truly amazin!-James L. (Jay) McClelland, Lucie Stern Professor & Chair, Dept. Of Psychology, Standford University
"In a December article in The New Yorker, Jonah Lehrer pointed out that some phenomena in the psychology literature are not always repeatable. One reason for this failure to replicate results comes from the kinds of statistics often used in Psychology. We use a procedure called Null Hypothesis Testing that was developed over 100 years ago. More recently, statisticians and psychologists have been working to create a new form of statistical testing based on Bayesian statistics. These methods may help us to avoid publishing studies that are not likely to replicate. John Kruschke published a nice tutorial on how to use these methods." -2010s top ten advances in psychology on Psychology Todays blog
"The intended audience for this book is a first-year graduate student or advanced undergraduate in the social or biological sciences, but one whose mathematical background is sufficient for them to not be put off by occasional references to calculus Kruschke also provides a comprehensive solution manual for the exercises in each chapter. He says he has worked on his book for six years and it shows, not least because it has few typographical errors and is well-presented. In summary, this book has several features that could make it preferable to its competitorsit is impressive that Kruschke is able to quickly bring readers up to speed on techniques such as robust regression and repeated-measures regression that would be considered advanced in the conventional NHST curriculum. His extensions from linear regression to logistic, ordinal probit and Poisson regression are very clearly articulated and will outfit students with a very adaptable statistical toolbox This is the best introductory textbook on Bayesian MCMC techniques I have read, and the most suitable for psychology students. It fills a gap I described in my recent review of six other introductory Bayesian method texts (Smithson, 2010). I look forward to using it in my own teaching, and I recommend it to anyone wishing to introduce graduate or advanced undergraduate students to the emerging Bayesian revolution."--Journal of Mathematical Psychology
"In sum, this is a new kind of textbook to teach a kind of statistical analysis that will be new to its audience. It uses a tutorial approach and instills in its students the tools of the trade: coding, debugging, simulating, and plotting. Though some will surely look down on its folksy tone, its extended analogies and cautious commenting, these measures will probably do much more good than harm. The text has the potential to change the methodological toolbox of a new generation of social scientists, bringing them up to a level of computation, modeling, and analysis that they might not have thought to be within their grasp. Where past approaches to teaching statistics to those in psychology and economics have not lead to widespread insight, this tutorial approach might."--Journal of Economic Psychology
"I would describe this book as revolutionary, at least in the context of psychology. It is, to my knowledge, the first book of its kind in this field to provide a general introduction to exclusively Bayesian statistical methods. In addition, it does so almost entirely by way of Monte Carlo simulation methods. While reasonable minds may disagree, it is arguable that both the general Bayesian framework advocated here, and the heavy use of Monte Carlo simulations, are destined to be the future of all data-analysis, whether in psychology or elsewherethe ideas and methods presented here will eventually be seen as the foundations for new approaches to statistics that will become commonplace in the near future."--British Journal of Mathematical and Statistical Psychology
"There are quite a few books on Bayesian statistics, but what makes Doing Bayesian Data Analysis: A Tutorial With R and BUGS stand out for me is the authors focus of the book-writing for real people with real data. From the very first chapter, the engaging writing style will get readers excited about this topic, a comment one can rarely make about statistical books. Clearly a master teacher, the author, John Kruschke, uses plain language to explain complex ideas and concepts. A comprehensive website is associated with the book and provides program codes, examples, data, and solutions to the exercises. If the book is used to teach a statistics course, this set of materials will be necessary and helpful for students as they go through the materials in the book step by step."--PsycCritiques

1.) This Books Organization: Read Me First!
1.1 Real People Can Read This Book
1.2 Prerequisites
1.3 The Organization of This Book
1.3.1 What Are the Essential Chapters?
1.3.2 Wheres the Equivalent of Traditional Test X in This Book
1.4 Gimme Feedback (Be Polite)
1.5 Acknowledgments
Part 1.) The Basics: Parameters, Probability, Bayes Rule, and R
2.) Introduction: Models We Believe In
2.1 Models of Observations and Models of Beliefs
2.1.1 Prior and Posterior Beliefs
2.2 Three Goals for Inference from Data
2.2.1 Estimation of Parameter Values
2.2.2 Prediction of Data Values
2.2.3 Model Comparison
2.3 The R Programming Language
2.3.1 Getting and Installing R
2.3.2 Invoking R and Using the Command Line
2.3.3 A Simple Example of R in Action
2.3.4 Getting Help in R
2.3.5 Programming in R
2.4 Exercises
3.) What Is This Stuff Called Probability?
3.1 The Set of All Possible Events
3.1.1 Coin Flips: Why You Should Care
3.2 Probability: Outside or Inside the Head
3.2.1 Outside the Head: Long-Run Relative Frequency
3.2.2 Inside the Head: Subjective Belief
3.2.3 Probabilities Assign Numbers to Possibilities
3.3 Probability Distributions
3.3.1 Discrete Distributions: Probability Mass
3.3.2 Continuous Distributions: Rendezvous with Density
3.3.3 Mean and Variance of a Distribution
3.3.4 Variance as Uncertainty in Beliefs
3.3.5 Highest Density Interval (HDI)
3.4 Two-Way Distributions
3.4.1 Marginal Probability
3.4.2 Conditional Probability
3.4.3 Independence of Attributes
3.5 R Code
3.5.1 R Code for Figure 3.1
3.5.2 R Code for Figure 3.3
3.6 Exercises
4.) Bayes Rule
4.1 Bayes Rule
4.1.1 Derived from Definitions of Conditional Probability
4.1.2 Intuited from a Two-Way Discrete Table
4.1.3 The Denominator as an Integral over Continuous Values
4.2 Applied to Models and Data
4.2.1 Data Order Invariance
4.2.2 An Example with Coin Flipping
4.3 The Three Goals of Inference
4.3.1 Estimation of Parameter Values
4.3.2 Prediction of Data Values
4.3.3 Model Comparison
4.3.4 Why Bayesian Inference Can Be Difficult
4.3.5 Bayesian Reasoning in Everyday Life
4.4 R Code
4.4.1 R Code for Figure 4.1
4.5 Exercises
Part 2.) All the Fundamentals Applied to Inferring a Binomial Proportion
5.) Inferring a Binomial Proportion via Exact Mathematical Analysis
5.1 The Likelihood Function: Bernoulli Distribution
5.2 A Description of Beliefs: The Beta Distribution
5.2.1 Specifying a Beta Prior
5.2.2 The Posterior Beta
5.3 Three Inferential Goals
5.3.1 Estimating the Binomial Proportion
5.3.2 Predicting Data
5.3.3 Model Comparison
5.4 Summary: How to Do Bayesian Inference
5.5 R Code
5.5.1 R Code for Figure 5.2
5.6 Exercises
6.) Inferring a Binomial Proportion via Grid Approximation
6.1 Bayes Rule for Discrete Values of 0
6.2 Discretizing a Continuous Prior Density
6.2.1 Examples Using Discretized Priors
6.3 Estimation
6.4 Prediction of Subsequent Data
6.5 Model Comparison
6.6 Summary
6.7 R Code
6.7.1 R Code for Figure 6.2 and the Like
6.8 Exercises
7.) Inferring a Binomial Proportion via the Metropolis Algorithm
7.1 A Simple Case of the Metropolis Algorithm
7.1.1 A Politician Stumbles on the Metropolis Algorithm
7.1.2 A Random Walk
7.1.3 General Properties of a Random Walk
7.1.4 Why We Care
7.1.5 Why It Works
7.2 The Metropolis Algorithm More Generally
7.2.1 "Burn-in," Efficiency, and Convergence
7.2.2 Terminology: Markov Chain Monte Carlo
7.3 From the Sampled Posterior to the Three Goals
7.3.1 Estimation
7.3.2 Prediction
7.3.3 Model Comparison: Estimation of p(D)
7.4 MCMC in BUGS
7.4.1 Parameter Estimation with BUGS
7.4.2 BUGS for Prediction
7.4.3 BUGS for Model Comparison
7.5 Conclusion
7.6 R Code
7.6.1 R Code for a Home-Grown Metropolis Algorithm
7.7 Exercises
8.) Inferring Two Binomial Proportions via Gibbs Sampling
8.1 Prior, Likelihood, and Posterior for Two Proportions
8.2 The Posterior via Exact Formal Analysis
8.3 The Posterior via Grid Approximation
8.4 The Posterior via Markov Chain Monte Carlo
8.4.1 Metropolis Algorithm
8.4.2 Gibbs Sampling
8.5 Doing It with BUGS
8.5.1 Sampling the Prior in BUGS
8.6 How Different Are the Underlying Biases?
8.7 Summary
8.8 R Code
8.8.1 R Code for Grid Approximation (Figures 8. and 8.2)
8.8.2 R Code for Metropolis Sampler (Figure 8.3)
8.8.3 R Code for BUGS Sampler (Figure 8.6)
8.8.4 R Code for Plotting a Posterior Histogram
8.9 Exercises
9.) Bernoulli Likelihood with Hierarchical Prior
9.1 A Single Coin from a Single Mint
9.2 Multiple Coins from a Single Mint
9.2.1 Posterior via Grid Approximation
9.2.2 Posterior via Monte Carlo Sampling
9.2.3 Outliers and Shrinkage of Individual Estimates
9.2.4 Case Study: Therapeutic Touch
9.2.5 Number of Coins and Flips per Coin
9.3 Multiple Coins from Multiple Mints
9.3.1 Independent Mints
9.3.2 Dependent Mints
9.3.3 Individual Differences and Meta-Analysis
9.4 Summary
9.5 R Code
9.5.1 Code for Analysis of Therapeutic-Touch Experiment
9.5.2 Code for Analysis of Filtration-Condensation Experiment
9.6 Exercises
10.) Hierarchical Modeling and Model Comparison
10.1 Model Comparison as Hierarchical Modeling
10.2 Model Comparison in BUGS
10.2.1 A Simple Example
10.2.2 A Realistic Example with "Pseudopriors"
10.2.3 Some Practical Advice When Using Transdimensional MCMC with Pseudopriors
10.3 Model Comparison and Nested Models
10.4 Review of Hierarchical Framework for Model Comparison
10.4.1 Comparing Methods for MCMC Model Comparison
10.4.2 Summary and Caveats
10.5 Exercises
11.) Null Hypothesis Significance Testing
11.1 NHST for the Bias of a Coin
11.1.1 When the Experimenter Intends to Fix N
11.1.2 When the Experimenter Intends to Fix z
11.1.3 Soul Searching
11.1.4 Bayesian Analysis
11.2 Prior Knowledge about the Coin
11.2.1 NHST Analysis
11.2.2 Bayesian Analysis
11.3 Confidence Interval and Highest Density Interval
11.3.1 NHST Confidence Interval
11.3.2 Bayesian HDI
11.4 Multiple Comparisons
11.4.1 NHST Correction for Experimentwise Error
11.4.2 Just One Bayesian Posterior No Matter How You Look at It
11.4.3 How Bayesian Analysis Mitigates False Alarms
11.5 What a Sampling Distribution Is Good For
11.5.1 Planning an Experiment
11.5.2 Exploring Model Predictions (Posterior Predictive Check)
11.6 Exercises
12.) Bayesian Approaches to Testing a Point ("Null") Hypothesis
12.1 The Estimation (Single Prior) Approach
12.1.1 Is a Null Value of a Parameter among the Credible Values?
12.1.2 Is a Null Value of a Difference among the Credible Values?
12.1.3 Region of Practical Equivalence (ROPE)
12.2 The Model-Comparison (Two-Prior) Approach
12.2.1 Are the Biases of Two Coins Equal?
12.2.2 Are Different Groups Equal?
12.3 Estimation or Model Comparison?
12.3.1 What Is the Probability That the Null Value Is True?
12.3.2 Recommendations
12.4 R Code
12.4.1 R Code for Figure 12.5
12.5 Exercises
13.) Goals, Power, and Sample Size
13.1 The Will to Power
13.1.1 Goals and Obstacles
13.1.2 Power
13.1.3 Sample Size
13.1.4 Other Expressions of Goals
13.2 Sample Size for a Single Coin
13.2.1 When the Goal Is to Exclude a Null Value
13.2.2 When the Goal Is Precision
13.3 Sample Size for Multiple Mints
13.4 Power: Prospective, Retrospective, and Replication
13.4.1 Power Analysis Requires Verisimilitude of Simulated Data
13.5 The Importance of Planning
13.6 R Code
13.6.1 Sample Size for a Single Coin
13.6.2 Power and Sample Size for Multiple Mints
13.7 Exercises
Part 3.) Applied to the Generalized Linear Model
14.) Overview of the Generalized Linear Model
14.1 The Generalized Linear Model (GLM)
14.1.2 Scale Types: Metric, Ordinal, Nominal
14.1.3 Linear Function of a Single Metric Predictor
14.1.4 Additive Combination of Metric Predictors
14.1.5 Nonadditive Interaction of Metric Predictors
14.1.6 Nominal Predictors
14.1.7 Linking Combined Predictors to the Predicted
14.1.8 Probabilistic Prediction
14.1.9 Formal Expression of the GLM
14.1.10 Two or More Nominal Variables Predicting Frequency
14.2 Cases of the GLM
14.3 Exercises
15.) Metric Predicted Variable on a Single Group
15.1 Estimating the Mean and Precision of a Normal Likelihood
15.1.1 Solution by Mathematical Analysis
15.1.2 Approximation by MCMC in BUGS
15.1.3 Outliers and Robust Estimation: The t Distribution
15.1.4 When the Data Are Non-normal: Transformations
15.2 Repeated Measures and Individual Differences
15.2.1 Hierarchical Model
15.2.2 Implementation in BUGS
15.3 Summary
15.4 R Code
15.4.1 Estimating the Mean and Precision of a Normal Likelihood
15.4.2 Repeated Measures: Normal Across and Normal Within
15.5 Exercises
16.) Metric Predicted Variable with One Metric Predictor
16.1 Simple Linear Regression
16.1.1 The Hierarchical Model and BUGS Code
16.1.2 The Posterior: How Big Is the Slope?
16.1.3 Posterior Prediction
16.2 Outliers and Robust Regression
16.3 Simple Linear Regression with Repeated Measures
16.4 Summary
16.5 R Code
16.5.1 Data Generator for Height and Weight
16.5.2 BRugs: Robust Linear Regression
16.5.3 BRugs: Simple Linear Regression with Repeated Measures
16.6 Exercises
17.) Metric Predicted Variable with Multiple Metric Predictors
17.1 Multiple Linear Regression
17.1.1 The Perils of Correlated Predictors
17.1.2 The Model and BUGS Program
17.1.3 The Posterior: How Big Are the Slopes?
17.1.4 Posterior Prediction
17.2 Hyperpriors and Shrinkage of Regression Coefficients
17.2.1 Informative Priors, Sparse Data, and Correlated Predictors
17.3 Multiplicative Interaction of Metric Predictors
17.3.1 The Hierarchical Model and BUGS Code
17.3.2 Interpreting the Posterior
17.4 Which Predictors Should Be Included?
17.5 R Code
17.5.1 Multiple Linear Regression
17.5.2 Multiple Linear Regression with Hyperprior on Coefficients
17.6 Exercises
18.) Metric Predicted Variable with One Nominal Predictor
18.1 Bayesian Oneway ANOVA
18.1.1 The Hierarchical Prior
18.1.2 Doing It with R and BUGS
18.1.3 A Worked Example
18.2 Multiple Comparisons
18.3 Two-Group Bayesian ANOVA and the NHST t Test
18.4 R Code
18.4.1 Bayesian Oneway ANOVA
18.5 Exercises
19.) Metric Predicted Variable with Multiple Nominal Predictors
19.1 Bayesian Multifactor ANOVA
19.1.2 The Hierarchical Prior
19.1.3 An Example in R and BUGS
19.1.4 Interpreting the Posterior
19.1.5 Noncrossover Interactions, Rescaling, and Homogeneous Variances
19.2 Repeated Measures, a.k.a. Within-Subject Designs
19.2.1 Why Use a Within-Subject Design? And Why Not?
19.3 R Code
19.3.1 Bayesian Two-Factor ANOVA
19.4 Exercises
20.) Dichotomous Predicted Variable
20.1 Logistic Regression
20.1.1 The Model
20.1.2 Doing It in R and BUGS
20.1.3 Interpreting the Posterior
20.1.4 Perils of Correlated Predictors
20.1.5 When There Are Few 1s in the Data
20.1.6 Hyperprior Across Regression Coefficient
20.2 Interaction of Predictors in Logistic Regression
20.3 Logistic ANOVA
20.3.1 Within-Subject Designs
20.4 Summary
20.5 R Code
20.5.1 Logistic Regression Code
20.5.2 Logistic ANOVA Code
20.6 Exercises
21.) Ordinal Predicted Variable
21.1 Ordinal Probit Regression
21.1.1 What the Data Look Like
21.1.2 The Mapping from Metric x to Ordinal y
21.1.3 The Parameters and Their Priors
21.1.4 Standardizing for MCMC Efficiency
21.1.5 Posterior Prediction
21.2 Some Examples
21.2.1 Why Are Some Thresholds Outside the Data?
21.3 Interaction
21.4 Relation to Linear and Logistic Regression
21.5 R Code
21.6 Exercises
22.) Contingency Table Analysis
22.1 Poisson Exponential ANOVA
22.1.1 What the Data Look Like
22.1.2 The Exponential Link Function
22.1.3 The Poisson Likelihood
22.1.4 The Parameters and the Hierarchical Prior
22.2 Examples
22.2.1 Credible Intervals on Cell Probabilities
22.3 Log Linear Models for Contingency Tables
22.4 R Code for the Poisson Exponential Model
22.5 Exercises
23.) Tools in the Trunk
23.1 Reporting a Bayesian Analysis
23.1.1 Essential Points
23.1.2 Optional Points
23.1.3 Helpful Points
23.2 MCMC Burn-in and Thinning
23.3 Functions for Approximating Highest Density Intervals
23.3.1 R Code for Computing HDI of a Grid Approximation
23.3.2 R Code for Computing HDI of an MCMC Sample
23.3.3 R Code for Computing HDI of a Function
23.4 Reparameterization of Probability Distributions
23.4.1 Examples
23.4.2 Reparameterization of Two Parameters
REFERENCES
INDEX

Doing Bayesian Data Analysis

Reviews

Contents