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CAUSAL INFERENCE IN STATISTICS
A Primer
Causality is central to the understanding and use of data. Without an understanding of cause-effect relationships, we cannot use data to answer questions as basic as "Does this treatment harm or help patients?" But though hundreds of introductory texts are available on statistical methods of data analysis, until now, no beginner-level book has been written about the exploding arsenal of methods that can tease causal information from data.
Causal Inference in Statistics fills that gap. Using simple examples and plain language, the book lays out how to define causal parameters; the assumptions necessary to estimate causal parameters in a variety of situations; how to express those assumptions mathematically; whether those assumptions have testable implications; how to predict the effects of interventions; and how to reason counterfactually. These are the foundational tools that any student of statistics needs to acquire in order to use statistical methods to answer causal questions of interest.
This book is accessible to anyone with an interest in interpreting data, from undergraduates, professors, researchers, or to the interested layperson. Examples are drawn from a wide variety of fields, including medicine, public policy, and law; a brief introduction to probability and statistics is provided for the uninitiated; and each chapter comes with study questions to reinforce the readers understanding.
Contents
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
