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Causal Inference in Statistics: A Primer

Judea Pearl, Madelyn Glymour, and Nicholas P. Jewell
John Wiley
Publication Date: 
Number of Pages: 
[Reviewed by
Robert W. Hayden
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If you have taught an introductory statistics course, you have probably uttered the phrase “correlation is not causation” many times. While most textbooks include that idea, fewer explain when it is relevant. Generally, a carefully designed experiment in which we control extraneous variables and randomly assign subjects to treatments allows us to reach causal conclusions, at least for the subjects at hand. The mantra is relevant primarily to situations where we take a survey of a randomly selected sample, or to observational studies where no randomization is applied. In these situations we cannot control any of the variables involved, and so it is hard to be sure of their effect.

Unfortunately, there are many research questions that are not amenable to a designed experiment. The variables may be intrinsically beyond our control, like the weather, or there may be ethical considerations when the subjects are human beings. We would like very much to be able to untangle causes in these situations, and statisticians have addressed that desire for decades. The earliest firm guidelines are known as “Hill’s Criteria.” These were formalized in part during a debate over the harmful effects (if any) of smoking. They are well worth discussing in a first course because most of the statistics our students will see in later life will not be from controlled experiments.

Hill’s Criteria are methodological principles described in words. In recent years, there have been attempts to devise mathematical techniques that help us assess causes in observational data. Judea Pearl, first author of the book at hand, has been prominent among those making this attempt. A mathematician who considers phrases like “causal chain” or “causal connections” will not be surprised to learn that Pearl’s work has involved the use of graph theory. A graph can be made of associational interconnections in a particular situation. We can focus on the directionality of the edges and the effects of cutting an existing edge or the implications of the existence of an edge we observe.

The work at hand offers a list of about sixty references for those wanting details, but the back cover describes this book as a “beginner-level book” that is “accessible… to the interested layperson.” Clearly the blurb writer has not read the book! The chapter template begins with a verbal explanation that tries to characterize the problem at hand. These are well-written and would be accessible to the intended audience. But nothing in these introductory explanations would allow anyone to understand or use the techniques. After these few words, the exposition becomes very mathematical. Reasonable prerequisites might be a year-long introductory statistics course, a mathematical statistics course (which would presuppose multi-variable calculus), and a basic familiarity with graph theory. So in fact the book is not likely to be accessible to a lay person or a beginner or even to most social scientists.

For those who do have the real prerequisites, however, this book serves the valuable purpose of gathering material scattered through the research literature into a single source.

After a few years in industry, Robert W. Hayden ( taught mathematics at colleges and universities for 32 years and statistics for 20 years. In 2005 he retired from full-time classroom work. He now teaches statistics online at and does summer workshops for high school teachers of Advanced Placement Statistics. He contributed the chapter on evaluating introductory statistics textbooks to the MAA's Teaching Statistics.

About the Authors ix

Preface xi

List of Figures xv

About the Companion Website xix

1 Preliminaries: Statistical and Causal Models 1

1.1 Why Study Causation 1

1.2 Simpson’s Paradox 1

1.3 Probability and Statistics 7

1.3.1 Variables 7

1.3.2 Events 8

1.3.3 Conditional Probability 8

1.3.4 Independence 10

1.3.5 Probability Distributions 11

1.3.6 The Law of Total Probability 11

1.3.7 Using Bayes’ Rule 13

1.3.8 Expected Values 16

1.3.9 Variance and Covariance 17

1.3.10 Regression 20

1.3.11 Multiple Regression 22

1.4 Graphs 24

1.5 Structural Causal Models 26

1.5.1 Modeling Causal Assumptions 26

1.5.2 Product Decomposition 29

2 Graphical Models and Their Applications 35

2.1 Connecting Models to Data 35

2.2 Chains and Forks 35

2.3 Colliders 40

2.4 d-separation 45

2.5 Model Testing and Causal Search 48

3 The Effects of Interventions 53

3.1 Interventions 53

3.2 The Adjustment Formula 55

3.2.1 To Adjust or not to Adjust? 58

3.2.2 Multiple Interventions and the Truncated Product Rule 60

3.3 The Backdoor Criterion 61

3.4 The Front-Door Criterion 66

3.5 Conditional Interventions and Covariate-Specific Effects 70

3.6 Inverse Probability Weighing 72

3.7 Mediation 75

3.8 Causal Inference in Linear Systems 78

3.8.1 Structural versus Regression Coefficients 80

3.8.2 The Causal Interpretation of Structural Coefficients 81

3.8.3 Identifying Structural Coefficients and Causal Effect 83

3.8.4 Mediation in Linear Systems 87

4 Counterfactuals and Their Applications 89

4.1 Counterfactuals 89

4.2 Defining and Computing Counterfactuals 91

4.2.1 The Structural Interpretation of Counterfactuals 91

4.2.2 The Fundamental Law of Counterfactuals 93

4.2.3 From Population Data to Individual Behavior – An Illustration 94

4.2.4 The Three Steps in Computing Counterfactuals 96

4.3 Nondeterministic Counterfactuals 98

4.3.1 Probabilities of Counterfactuals 98

4.3.2 The Graphical Representation of Counterfactuals 101

4.3.3 Counterfactuals in Experimental Settings 103

4.3.4 Counterfactuals in Linear Models 106

4.4 Practical Uses of Counterfactuals 107

4.4.1 Recruitment to a Program 107

4.4.2 Additive Interventions 109

4.4.3 Personal Decision Making 111

4.4.4 Sex Discrimination in Hiring 113

4.4.5 Mediation and Path-disabling Interventions 114

4.5 Mathematical Tool Kits for Attribution and Mediation 116

4.5.1 A Tool Kit for Attribution and Probabilities of Causation 116

4.5.2 A Tool Kit for Mediation 120

References 127

Index 133