Introduction tocausalinference april02_2020
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The document provides an overview of causal inference techniques including: 1. Simpson's paradox and how propensity score matching and inverse probability of treatment weighting can resolve it. 2. The Rubin causal model and potential outcomes framework for defining average treatment effects. 3. Propensity score theory and how propensity scores can create balanced groups to estimate causal effects. 4. Inverse probability of treatment weighting to eliminate confounding by reweighting samples. 5. Heterogeneous treatment effect estimation using single and two model approaches, and transformed outcome modeling. 6. Applications of causal inference techniques like instrumental variables and counterfactual frameworks for ranking and churn analysis.
![Rubin Causal Model or Potential Outcome Framework
𝑌0
• Think of Potential Outcomes as the
outcomes we would see under each
possible treatment option
• Notation: Ya is the outcome that would
be observed if treatment was set to A=a
• Counterfactuals outcomes are ones that
would have been observed, had the
treatment been different.
• Average Causal Effect: E[Y1 – Y0]
𝑌1](https://image.slidesharecdn.com/introductiontocausalinferenceapril022020-200402150826/85/Introduction-tocausalinference-april02_2020-6-320.jpg)
![Average Causal Effect through Potential Outcome Framework
Population Of Interest
World 1
Everyone gets A=0
World 1
Everyone gets A=1
mean(Y) mean(Y)
Difference is Average Causal Effect: E[Y1-Y0]
• E[Y1 – Y0] ≠ E[Y|A=1] – E[Y|A=0]
• Condition Vs. Setting
• Comparing two different populations
• E[Y1/Y0] : Causal Relative Risk
• E[Y1-Y0|A=1 ]: Causal effect of treatment on
the treated
• Heterogeneity of Treatment Effects](https://image.slidesharecdn.com/introductiontocausalinferenceapril022020-200402150826/85/Introduction-tocausalinference-april02_2020-7-320.jpg)
![Untestable Causal Assumptions to link Observed and Potential Outcomes
• Stable Unit Treatment Value Assumption ( SUTVA )
• No Interference
• One Version Of Treatment
• Consistency Assumption: Y = Ya if A=a for all a
• Positivity Assumption: P(A=a | X=x) > 0
• Variability in treatment assignment for Causal Effect Identification
• Ignorability Assumption: (Y0, Y1 ) 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑜𝑓 𝐴 | X
• Ensures random assignment of treatment given co-variates
• In Simpson Paradox ignorability assumption is violated
• Linking Observed Data and Potential Outcomes
• E[Y|A=a, X=x]
• E[Y | A=a, X=x ] = E[Ya | A=a, X=x] by consistency assumption
• E[Ya | X=x ] by ignorability assumption](https://image.slidesharecdn.com/introductiontocausalinferenceapril022020-200402150826/85/Introduction-tocausalinference-april02_2020-9-320.jpg)
![IPTW
Resolving Simpson Paradox through IPTW
• E[𝑌1
] = ( ( 300 * (1/0.5) ) + ( 50*(1/0.2) ) ) / 11000 = 850/11,000 = 0.0772
• E[𝑌0
] = ( ( 300 * (1/0.5) ) + ( 200*(1/0.8) ) ) / 11000 = 850/11,000 =
0.0772
• E[𝑌1
] - E[𝑌0
] = 0
• Note:
• Propensity Scores for X=1 are 0.5, 0.5
• Propensity Scores for X=0 are 0.2, 0.8
X=1
Y=1 Y=0 Row Sums Success Rate
T=1 300 2700 3,000 0.100
T=0 300 2700 3,000 0.100
Col Sums 600 5,400 6,000
X=0
Y=1 Y=0 Row Sums Succes Rate
T=1 50 950 1,000 0.050
T=0 200 3800 4,000 0.050
Col Sums 250 4,750 5,000](https://image.slidesharecdn.com/introductiontocausalinferenceapril022020-200402150826/85/Introduction-tocausalinference-april02_2020-15-320.jpg)
Introduction tocausalinference april02_2020
- 1. A whirlwind tour of Causal Inference The Path From Cause To Effect By Viswanath Gangavaram Senior Data Scientist
- 2. Topics • Introduction to Causal Inference • Simpson Paradox: Let’s understand the damage done by Lurking Variables (Selection Bias at play) • Ceteris Paribus: Other Things Equal Comparison • Rubin Causal Model: Potential Outcomes, Observed Outcomes, Counterfactuals • Average Treatment Effects ( ATE ) • Five important theorems from the world of propensity score theory • IPTW: Slaying the Lurking Variable • Doubly Robust Estimation • Instrumental Variables: 2SLS, Compiler Average Causal Effect • Heterogeneous Treatment Effects • One Model Approach (OMA), Two Model Approach (TMA), • Transformed Outcome Approach • Applications of Causal Inference • Counterfactual Inference framework for Learning To Rank • Churn reason through Counterfactual Inference framework • Causal Explanations on Quasi-Experiments
- 3. Simpson Paradox Y=1 Y=0 Row Sums Success Rate T=1 350 3,650 4,000 0.088 T=0 500 6,500 7,000 0.071 Col Sums 850 10,150 11,000 𝐸 𝑌1 | 𝑇 = 1 𝐸 𝑌0|𝑇 = 0 ( 𝐸 𝑌1 | 𝑇 = 1 − 𝐸 𝑌0 | 𝑇 = 0 ) = 0.017 X=1 Y=1 Y=0 Row Sums Success Rate T=1 300 2,700 3,000 0.100 T=0 300 2,700 3,000 0.100 Col Sums 600 5,400 6,000 X=0 Y=1 Y=0 Row Sums Succes Rate T=1 50 950 1,000 0.050 T=0 200 3,800 4,000 0.050 Col Sums 250 4,750 5,000
- 4. • Standardization: Stratification followed by Normalization 𝐸 𝑌1 | 𝑇 = 1 = (6000/11000)*0.1 + ( 5000/1000)*0.05 = 0.077 𝐸 𝑌0 |𝑇 = 0 = (6000/11000)*0.1 + ( 5000/1000)*0.05 = 0.077 In future slides, we will resolve Simpson Paradox with Inverse Propensity Treatment Weighting(IPTW) a Causal Inference Technique Standardization Pitfalls: • Stratification • Some strata might not have representation from other treatment groups
- 5. Ceteris Paribus: Other Things Equal Comparison “The notion of Ideal Experiment disciplines our approach to causal inference” • Comparisons made under ceteris paribus conditions have a causal interpretation • The Causal Inference craft uses data to get to other things equal in-spite of obstacles-called selection bias or omitted variables found on the path running from raw numbers to causal knowledge • Random Assignment with Law Large Of Numbers, ensures Ceteris Paribus in Randomized Control Trails • Random assignment isn’t the same as holding everything else fixed, but it has the same effect. • The notion of Ideal Experiment disciplines our approach to causal inference • Nothing but ensures Ceteris Paribus before marking Causal Statement
- 6. Rubin Causal Model or Potential Outcome Framework 𝑌0 • Think of Potential Outcomes as the outcomes we would see under each possible treatment option • Notation: Ya is the outcome that would be observed if treatment was set to A=a • Counterfactuals outcomes are ones that would have been observed, had the treatment been different. • Average Causal Effect: E[Y1 – Y0] 𝑌1
- 7. Average Causal Effect through Potential Outcome Framework Population Of Interest World 1 Everyone gets A=0 World 1 Everyone gets A=1 mean(Y) mean(Y) Difference is Average Causal Effect: E[Y1-Y0] • E[Y1 – Y0] ≠ E[Y|A=1] – E[Y|A=0] • Condition Vs. Setting • Comparing two different populations • E[Y1/Y0] : Causal Relative Risk • E[Y1-Y0|A=1 ]: Causal effect of treatment on the treated • Heterogeneity of Treatment Effects
- 8. • Fundamental Problem Of Causal Inference • How do we use observed data to link Observed Outcomes to Potential Outcomes ? • What assumptions are necessary to estimate causal effect from observed data ?
- 9. Untestable Causal Assumptions to link Observed and Potential Outcomes • Stable Unit Treatment Value Assumption ( SUTVA ) • No Interference • One Version Of Treatment • Consistency Assumption: Y = Ya if A=a for all a • Positivity Assumption: P(A=a | X=x) > 0 • Variability in treatment assignment for Causal Effect Identification • Ignorability Assumption: (Y0, Y1 ) 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑜𝑓 𝐴 | X • Ensures random assignment of treatment given co-variates • In Simpson Paradox ignorability assumption is violated • Linking Observed Data and Potential Outcomes • E[Y|A=a, X=x] • E[Y | A=a, X=x ] = E[Ya | A=a, X=x] by consistency assumption • E[Ya | X=x ] by ignorability assumption
- 10. 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑖𝑛 𝑔𝑟𝑜𝑢𝑝 𝑀𝑒𝑎𝑛𝑠 = 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐶𝑎𝑢𝑠𝑎𝑙 𝐸𝑓𝑓𝑒𝑐𝑡 + 𝑆𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝐵𝑖𝑎𝑠 𝐸 𝑌𝑖|𝑇𝑖 = 1 − 𝐸 𝑌𝑖|𝑇𝑖 = 0 = 𝐸 𝑌1𝑖|𝑇𝑖 = 1 − 𝐸 𝑌0𝑖|𝑇𝑖 = 0 = 𝐸 𝑌0𝑖 + 𝑘|𝑇𝑖 = 1 − 𝐸 𝑌0𝑖|𝑇𝑖 = 0 = 𝑘 + 𝐸 𝑌0𝑖|𝑇 = 1 − 𝐸 𝑌0𝑖|𝑇𝑖 = 0 = k + Selection Bias Random Assignment Eliminates Selection Bias: 𝐸 𝑌𝑖|𝑇𝑖 = 1 − 𝐸 𝑌𝑖|𝑇𝑖 = 0 = 𝐸 𝑌1𝑖|𝑇𝑖 = 1 − 𝐸 𝑌0𝑖|𝑇𝑖 = 0 = 𝐸 𝑌0𝑖 + 𝑘|𝑇𝑖 = 1 − 𝐸 𝑌0𝑖|𝑇𝑖 = 0 = 𝑘 + 𝐸 𝑌0𝑖|𝑇𝑖 = 1 − 𝐸 𝑌0𝑖|𝑇𝑖 = 0 = k 𝐸 𝑌0𝑖|𝑇𝑖 = 1 = 𝐸 𝑌0𝑖|𝑇𝑖 = 0 Constant Causal Effect: Y1i = Y0i + k
- 11. Propensity Score & Propensity Score Matching Short comings • Discards some data Propensity Score Propensity Score as a balancing score Propensity Score Matching
- 12. Distribution of Propensity score before & after matching Propensity Score Matching
- 13. Five important theorems from the world of propensity score theory The propensity score is a balancing score Any score that is finer than the propensity score is a balancing score; moreover, x is the finest balancing score and the propensity score is the coarsest If treatment assignment is strongly ignorable given x, then it is strongly ignorable given any balancing score At any value of a balancing score, the difference between the treatment and control means is an unbiased estimate of the average treatment effect at that value of the balancing score if treatment assignment is strongly ignorable. Consequently, with strongly ignorable treatment assignment, pair matching on balancing score, subclassification on a balancing score and covariance adjustment on a balancing score can all produce unbiased estimates of treatment effects. Using sample estimates of balancing scores can produce sample balance on x
- 14. Inverse Propensity Treatment Weighting • Rather than match, we could use all of the data, but down-weight some and up-weight others • This is accomplished by weighting by the inverse of the probability of treatment received • For treated subjects, weight by the inverse of P(A=1 | X ) • For control subjects, weight by the inverse of P(A=0 | X) This is known as Inverse Probability Of Treatment Weighting • There is confounding in the original population • IPTW creates a pseudo-population where treatment assignment no-longer depends on X • No Confounding in the pseudo-population
- 15. IPTW Resolving Simpson Paradox through IPTW • E[𝑌1 ] = ( ( 300 * (1/0.5) ) + ( 50*(1/0.2) ) ) / 11000 = 850/11,000 = 0.0772 • E[𝑌0 ] = ( ( 300 * (1/0.5) ) + ( 200*(1/0.8) ) ) / 11000 = 850/11,000 = 0.0772 • E[𝑌1 ] - E[𝑌0 ] = 0 • Note: • Propensity Scores for X=1 are 0.5, 0.5 • Propensity Scores for X=0 are 0.2, 0.8 X=1 Y=1 Y=0 Row Sums Success Rate T=1 300 2700 3,000 0.100 T=0 300 2700 3,000 0.100 Col Sums 600 5,400 6,000 X=0 Y=1 Y=0 Row Sums Succes Rate T=1 50 950 1,000 0.050 T=0 200 3800 4,000 0.050 Col Sums 250 4,750 5,000
- 16. Some Terminology • Marginal Treatment Probability • Conditional Treatment Probability ( Propensity Score ) • The unit-level causal effect: • Observed outcome: • Conditional Average Treatment Effect( CATE ) • Population Treatment Effect ( ATE )
- 17. Heterogenous Treatment Effects: SMA & TMA Single Model Approach: We can use conventional ML techniques with the observed outcome 𝑌𝑖 𝑜𝑏𝑠 as the outcome and both the treatment Wi and Xi as the features. Two Model Approach: Separate Trees for the Observed Outcome by Treatment Group
- 18. Heterogenous Treatment Effects: Transformed Outcome Approach
- 19. • Unbiased Learning-to-Rank with Biased Feedback: Optimizing Propensity-Weighted ERM Objective • Position bias is search rankings strongly influences how many clicks a result receives, so that directly using click data as a training signal in Learning-To-Rank methods yields sub-optimal results • To overcome this bias problem, this paper presents a Counterfactual Inference Framework that provides the theoretical basis for unbiased LTR via Empirical Risk Minimization despite biased data • Propensity-Weighted Ranking SVM for discriminative learning from implicit feedback data, where click models take the role of the propensity estimators • Other Quasi-Experiment Designs • Use IPTW Use Cases of Causal Inference
- 21. References 1. Coursera’s Causality Crash Course 2. https://eng.uber.com/causal-inference-at-uber/ 3. The Book Of Why 4. Mastering Metrics 5. Mostly Harmless Econometrics