ANALYTIC COMPARISON of Pearl and Rubin CAUSAL FRAMEWORKS
Content Page Part I. General Considerations Chapter 1. What is the question? 16 Introduction 16 1. Randomization 17 1.1 An Example of Randomization 18 1.2 An example of observational study 18 1.3 The Difference between These Two Examples 18 2. Observational Study 19 2.1 Some Difficulties in Observational Study 19 2.1.1 Common Causes 19 2.1.2 Mutual Effect between Treatment and Response 20 2.1.3 Regression to Mean 20 2.2 Adjustment for confounders 21 2.2.1 Matching 21 2.2.2 Classification 21 3. Conditional Probability and Causal Relationship 22 3.1 The Necessity of Assignment Mechanismus Consideration 22 4. Some Works on Statistical Causal Inference Discussed in this Book 23 5. How to Approach Causality 24 6. Causal Languages 24 7. Outlines 25 8. Appendix 26 Chapter 2. Required Definitions and Clarifications 28 1. Introduction 28 2. Definitions 28 2.1 Statistics, Probability and Causality 30 3. Why Indeterministic Causality? 32 4. Some of My Observations 33 Chapter 3. Different Causal Views 34 6
1. Introduction 34 2. Different Views to Causal Effect 34 2.1 Subjective View 35 2.1.1 Causal Effect Measurement through Inter-Subjective View 35 2.2 Objective View 36 2.2.1 Propensity View 37 2.3 Conjugate View 37 3. Combination of Different Causal Views 38 4. Frameworks and Causal Effect Measurements 39 5. Causality Interpretation or Probability Interpretation 39 6. Appendix 40 6. 1 Causal Effect Measurement in Subjective View 40 6.1.1 Measurement by Observing Two Responses of One Unit 40 6.1.2 Measurement by Subjective Probability 41 6. 2 Causal Effect Measurement in Intersubjective View 42 7. My Observations 42 Part II. Rubin's Contributions to Causality and Related Works Chapter 4. Potential Response Framework 44 1. Introduction 44 2. Notations 45 3. Definition of Individual Causal Effect 47 3.1 Two Points about Individual Causal Effect 47 3.2 Treatment-Unit Additivity 48 3.3 Matching in the Potential Outcome Framework 48 3.4 Consistency Assumption 49 4. General Assumptions 49 4.1 Some Discussions on SUTVA 50 4.1.1 My Observation 50 4.2 Two Points 51 5. Assignment Mechanism (AM) 51 5.1 Formalizing Assignment Mechanism 52 7
6. A Particular Assumption 55 6.1 Unconfounded AM 55 7. Prediction from Bayesian View 56 8. Review of Essential Points in Potential Response Framework 58 8.1 The Science 59 8.2 Distinguish Between the Science and the Assignment Mechanism 59 8.3 Assumptions in potential response framework 60 9. Some of My Observations 61 Chapter 5. Some Works in Potential Response Framework 62 1. Introduction 62 2. Definition of Average Causal Effect 62 2.1 First Assumption for Identifiability of Average Causal Effect 62 2.1.1 Summary 64 2.2 Second Assumption for Identifiability of Average Causal Effect 64 2.3 Discussions on Average Causal Effect 65 2.4 Randomization 65 3. Propensity Score 66 3.1 A Comparison of Propensity Sore and Randomization 67 3.2 Some Discussions on Propensity Score 68 4. Some of My Observations 68 Chapter 6. Some Discussions on Potential Response Framework 70 1. Introduction 70 2. Piatonic Heaven 70 3. Potential Outcomes or Counterfactuals? 71 4. What is the Definition of Causal Effect? 72 5. The Dependence of Assignment Mechanism on Potential Outcomes 72 6. My Observations 73 Outlines of Part II 74 Part III. Dawid' S Contributions and the Necessary Preliminaries 8
Chapter 7. Extended Conditional Independency 76 1. Introduction 76 2. Conditional Independency 76 3. Some Properties and Axioms 77 4. Extension of Conditional Independency to Non-Stochastic Variables 78 5. Some Discussions on Conditional Independency and my Observations 79 Chapter 8. Exchangeability 81 1. Introduction 81 2. Exchangeability 81 3. Bayesian View 85 3.1 Two Rules Applied by Bayesians 86 4. A Bridge between Bayesian and Frequentist 87 5. One Point about Subjective Probability 88 6. Conditional Exchangeability 88 Chapter 9. Decision Theory Approach 89 1. Introduction 90 2. Decision-Theory Framework 90 3. Regimes Identification 91 4. The Language of AM 93 5. A Comparison of Rubin Causal Framework and Decision Theory Framework 93 5.1 In Notation 93 5.2 Estimating the Probability of Missing Potential Response 94 5.3 Dawid and Rubin's View 95 6. Some of My Observations in Part III 96 Outlines of Part III 98 Part IV. Pearl's Contributions and Some Computations 9
Chapter 10. Probabilistic Graphs 100 1. Introduction 100 2. Definition of Graph 100 2.1 Kinship Language 101 3. Three Types of Substructures in DAGs 102 4. DAG and Joint Distribution 103 5. Markov Equivalence 106 6. d-separation Criterion 107 7. Some of My Observations 109 Chapter 11. Causal Graph 110 1. Introduction 110 2. Graphs as an Illumination of AM 112 3. Two Different Mechanism 112 3.1 The Mechanism between Two Variables 113 3.2 Illumination of Treatment Assignment Mechanism 114 4. Representation of Assumptions 112 4.1 Missing Arrows as Representation of assumptions 115 4.2 Causal Markov Assumption 115 5. How to Draw Causal Graphs 116 5.1 My Definition of Causal Graph 116 5.2 Herman and Robin's Definition of Causal Graph 118 5.3 Pearl's Word to Draw a Causal Graph 119 6. A Comparison of Different Frameworks in Terms of Assumption 119 7. Whether Treatment must be Assignable? 119 8. Some of My Observations 120 Chapter 12. Causal Inference through Causal Graphs 121 1. Introduction 121 2. Missing Step in Pearl's Causal Approach 121 3. Pearl's Approach to Causal Discovery 122 4. Intervention 123 4.1 The Effect of Intervention in Markovian Model 123 4.2 Two Subsequences of Intervention 124 4.3 Causal Effect Definition 125 10
5. 6. Intervention and Observation Computation of the Intervention Effect 6.1 Conclusion 125 126 127 7. Identifiability 128 7.1 Identifiability through Other Notations 128 8. Finding Confounders and Coping with Immeasurable Confounders 129 8.1 The Conditions of the Back-door Criteria by Regime Indicator Notation 130 8.2 Back-Door Criteria by Regime Indicator Notation 131 8.3 Front-door criteria 131 8.4 The Intuition behind Front-Door Criteria 132 8.5 Some Explanations for the Computation of the Quantity in Front-Door Criteria 133 8.6 Application of Front-Door Criteria 9. Computations of Some Causal Quantities 133 9.1 Computation of the Quantity in Introduction of Chapter 3, Pearl (2009a) 135 10. The Difference between immeasurable and Unobservable Confounders 136 ll.summery 136 12. Appendix 136 13. Some of My Works 137 Chapter 13. Structural Equation Modeling (SEM) 138 1. Introduction 138 2. SEM Surrounded by Controversies 138 3. Assumptions and Representations 139 4. The Error Term in Regression and Exogenous Variable U in SEM 141 4.1 Discussion 141 5. Causal Assumptions in Nonparametric Models 142 6. Intervention in Non-Parametric Models 145 7. Estimating the Effect of Intervention in Linear Model 146 8. Estimating the Effect of Intervention in Non-Linear Model 147 9. My Work 148 Chapter 14. Predictions and Actions 149 1. Introduction 149 2. Computation of Prediction and Action through DAGs 149 2.1 Indeterministic Model 150 2.2 Deterministic Model 150
3. Demand and Price 151 3.1 Indeterministic Causal Model 152 3.2 Deterministic model 153 3.3 Counterfactuals 154 4. Discussion 155 4.1 Chain Graph 155 4.1.1 Some Properties of Chain Graphs 156 4.2 Dawid's Approach to the Example "Demand and Price" 156 4.3 My Observation 158 Outlines of Part IV 159 Part V. Comparison Chapter 15. Assumptions in Causal Inference 161 1. Introduction 161 2. The Review of Applied Notations 161 3. Causality versus Exchangeability 163 4. The Number of Assumptions to Discover Causal Relationships 164 Chapter 16. Common Lessons from Pearl and Rubin on Average Causal Effect 166 1. Introduction 166 1.1 Formalizing a Causal Quantity for Average Causal Effect 167 2. Notations 169 3. Necessary Conditions for Causal Inference 171 4. Causal Quantity 172 4.1 Other Causal Quantities 173 4.2 Nonassignable Treatment 174 5. The Causal Quantity in Rubin and Pearl's Notation 174 5.1 A Comparison of Rubin and Pearl's Causal Notations 5.2 Representations of AM through Both Notations 176 5.3 Some Points about Dawid's Work in Causality 177 12
6. Identification of the Causal Element in Observational Study 177 6.1 Graphical method 178 6.2 Propensity Score 178 7. Causal Inference toward Individual Causal Effect 178 7.1 Individual causal effect 179 Chapter 17. Causal Effect in Different Levels 180 1. Introduction 180 2. Definition and Measurement of Individual Causal Effect 181 2.1 Rubin's Approach to Unit Level Causal Effect 181 2.2 Pearl's Approach to Unit Level Causal Effect 181 2.3 Discussion 182 3. Relationship to the Potential Response Framework 183 3.1 In Individual level 183 3.2 In Population Level 183 3.3 My Observations 184 4. Causal Effect Measurement 185 4.1 Average causal effect 185 4.2 Considering more Knowledge rather than Identification of AM 186 4.3 One Point about Dawid's Framework 188 5. The more Covariates the More Accurate Individual Causal Effect 188 Chapter 18. Three Features of a Causal Framework 189 1. Introduction 189 2. Probability or Causality 189 3. Considering AM 190 3.1 Intervention 191 4. Formalizing AM on the Whole Knowledge 191 4.1 Potential Responses are not Observations 192 4.2 Formalizing AM upon the Potential Responses Brings about Difficulties 193 5. Confounder 193 Chapter 19. A Summary of Comparison 195 13
Introduction 195 1. Association and Causation 195 2. Causal Effect Definition Deterministically 196 3. Inference based on Assumption 197 4. Propensity Score and Graphical Methods 197 5. Which Framework is more proper in Statistics? 198 6. Can We Answer any Question in Potential Outcome Framework? 198 7. Can We Interpret Coefficients in Structural Equations as Causal Parameters? 199 8. The Main Difference between Pearl and Rubin 199 9. Assessing Pearl and Rubin's Frameworks in Teaching and Reasoning 200 9.1 Teaching Causal Questions 200 9.2 Articulation a Causal Question 201 9.3 Representation of the Assumptions Transparently 201 9.4 Articulation an Example in the Two Frameworks 202 REFERENCE 206 Abbreviation 211 14