Equivalence causal frameworks: SEMs, Graphical models and Potential Outcomes
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The document discusses the concept of causality, defining it through interventionist causality and the importance of models for accurate causal estimates. It outlines three major frameworks for specifying causal models: causal graphical models, structural equation models, and the potential outcomes framework, all of which are equivalent in terms of their theorems. Additionally, it provides a historical perspective on the evolution of causality concepts and suggests best practices for choosing among these frameworks based on the nature of the study.
![ALL THREE ARE EQUIVALENT
• A theorem in one is a theorem in another (See Pearl [2009]).
• So what’s the problem?
• Different disciplines prefer one over another.
• Misconceptions abound about the frameworks.
• In general, no unified causal inference course in major
universities.](https://image.slidesharecdn.com/equivalencecausalframeworks-160707173812/85/Equivalence-causal-frameworks-SEMs-Graphical-models-and-Potential-Outcomes-9-320.jpg)
Equivalence causal frameworks: SEMs, Graphical models and Potential Outcomes
- 1. EQUIVALENCE OF SEM, POTENTIAL OUTCOMES AND CAUSAL GRAPHICAL MODELS AMIT SHARMA POSTDOCTORAL RESEARCHER MICROSOFT http://www.amitsharma.in @amt_shrma
- 2. WHAT IS CAUSALITY? • Debatable, from the times of Aristotle and Hume. • Practical definition: • Interventionist causality: X causes Y if changing X leads to a change in Y, keeping everything else constant.
- 3. CAUSALITY IS MEANINGLESS WITHOUT A MODEL • “Keeping everything else constant” requires knowing what everything else is. • Demand increases price is valid in most economies. So seems a universal causal law. • … except in a fully regulated economy. • Model: Explicit specification of “everything else” that can affect causal estimate.
- 4. WITHOUT A MODEL, EVEN EXPERIMENTS DO NOT TELL YOU ANYTHING ABOUT THE FUTURE • A/B experiments study the past. • Provide a counterfactual answer. • But we want to use the results for the future. • Model: The world stays the same between: • When the experiment was run, and • When its results will be applied.
- 5. HOW MIGHT WE SPECIFY A MODEL? • By qualitative knowledge about how the world works. Encouragement Effort Outcome
- 6. HOW MIGHT WE SPECIFY A MODEL? • By writing equations about how the world works. • E.g. F = ma • Encouragement (Z) • Effort (X) • Outcome (Y) 𝑦 ∶= 𝛽𝑥 + 𝜖 𝑦 𝑥 ∶= 𝛾𝑧 + 𝜖 𝑥
- 7. HOW MIGHT WE SPECIFY A MODEL? • By thinking about the different worlds that changing the causal variable creates (inspired by a randomized experiment). • Effort (X) • Outcome (Y) 𝑌𝑥1 𝑜𝑟 𝑌𝑥2 … . • Encouragement (Z) 𝑌𝑥1𝑧1 𝑜𝑟 𝑌𝑥2𝑧1 𝑜𝑟 𝑌𝑥1𝑧2 𝑜𝑟 𝑌𝑥2𝑧2 … .
- 8. THREE MAJOR FRAMEWORKS FOR SPECIFYING A CAUSAL MODEL • Causal Graphical model • Structural Equation Model 𝑦 ∶= 𝛽𝑥 + 𝜖 𝑦 𝑥 ∶= 𝛾𝑧 + 𝜖 𝑥 • Potential Outcomes Framework 𝑌𝑥1𝑧1 𝑜𝑟 𝑌𝑥2𝑧1 𝑜𝑟 𝑌𝑥1𝑧2 𝑜𝑟 𝑌𝑥2𝑧2 … . Encouragement Effort Outcome
- 9. ALL THREE ARE EQUIVALENT • A theorem in one is a theorem in another (See Pearl [2009]). • So what’s the problem? • Different disciplines prefer one over another. • Misconceptions abound about the frameworks. • In general, no unified causal inference course in major universities.
- 10. A HISTORICAL TOUR OF CAUSALITY • 1850s: John Snow uses a natural experiment to detect causal connection between water and cholera. • 1910s: Buoyed by triumphs in physics, Bertrand Russell argues that causality is irrevelant. • 1920s: Sewall and Philip Wright develop path diagrams and simultaneous equation modelling (SEM) for determining supply or demand from price and quantity. • 1920s: Neyman uses potential outcomes to analyze experiments. • 1930s: Ronald Fisher popularizes the randomized experiment.
- 11. A HISTORICAL TOUR OF CAUSALITY • 1960s: Blalock and Duncan solve path diagrams using regression equations. • 1960-now?: Age of regression. • Path diagrams lose their original causal interpretation. • SEMs, Path diagrams and regression become entangled. • 1970s: Rubin builds on potential outcomes framework. • Becomes popular with social scientists. • 1980s: Pearl builds on SEM framework. • Starting to become popular with computer scientists.
- 12. EQUIVALENCE OF GRAPHICAL MODELS AND SEM Encouragement (Z) Effort (X) Outcome (Y) P(X, Y, Z ) = P(Y|X) P(X|Z) P(Z) 𝑦 ∶= 𝑓 𝑥, 𝜖 𝑦 𝑥 ∶= 𝑓(𝑧, 𝜖 𝑥)
- 13. EQUIVALENCE OF GRAPHICAL MODELS AND SEM Encouragement (Z) Effort (X) Outcome (Y) P(Y|do(X)) = P(Y|X) 𝑦 ∶= 𝑓 𝑥, 𝜖 𝑦 Effect = 𝑑𝑦 𝑑𝑥
- 14. EQUIVALENCE OF POTENTIAL OUTCOMES AND SEM Encouragement (Z) Effort (X) Outcome (Y) 𝑦 ∶= 𝑓 𝑥, 𝜖 𝑦 𝑥 ∶= 𝑓(𝑧, 𝜖 𝑥) 𝑌𝑥1, 𝑌𝑥2, … 𝑋𝑧1, 𝑋𝑧2, … 𝑌𝑥1𝑧1, 𝑌𝑥1𝑧2, …
- 15. EQUIVALENCE OF POTENTIAL OUTCOMES AND SEM Encouragement (Z) Effort (X) Outcome (Y) Effect = 𝑌𝑥2 − 𝑌𝑥1 𝑦 ∶= 𝑓 𝑥, 𝜖 𝑦 Effect = 𝑑𝑦 𝑑𝑥
- 17. IV BY GRAPHICAL MODEL Encouragement (Z) Effort (X) Outcome (Y) Unobserved Confounders (U) 𝑃 𝑦 𝑑𝑜 𝑥 = 𝑢 𝑃 𝑦 𝑥, 𝑢 𝑃(𝑢) Average Causal Effect = 𝑃 𝑦2 𝑑𝑜 𝑥2 − 𝑃 𝑦2 𝑑𝑜 𝑥1 = 𝑢 𝑃 𝑦2 𝑥2, 𝑢 − 𝑃(𝑦2|𝑥1, 𝑢) 𝑃(𝑢)
- 18. IV BY STRUCTURAL EQUATIONS 𝑦 ∶= 𝛽𝑥 + 𝜖 𝑦 𝑥 ∶= 𝛾𝑧 + 𝜖 𝑥 Local average Causal effect = 𝐶𝑜𝑣(𝑌, 𝑋) 𝐶𝑜𝑣(𝑋,𝑍)
- 19. IV IN POTENTIAL OUTCOMES • Assumptions: • 𝑌𝑥1𝑧1 = 𝑌𝑥1𝑧2 Local average Causal Effect = 𝐸 𝑌 𝑋2 – 𝐸 𝑌 𝑋1 𝐸 𝑋 𝑍2 −𝐸 𝑋 𝑍1
- 20. BEST PRACTICES • A randomized experiment, or a problem with few variables: • Use Potential outcomes framework: Simple and practical. • An observational study with many confounders, or any problem with many variables: • Use graphical models to encode causal assumptions. • If functional forms unknown, use graphical criteria or do-calculus to estimate causal effect. • Else, a domain where functional forms are known or can be approximated • Use structural equation models to solve for causal effects, based on the causal graph.