Double Robustness in Estimation of Causal Treatment Effects

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2. Counterfactual Model Unconditional (marginal) expectation: Conceptually, the “average ” across all possible values a random variable can take on in the population Statistical independence: For two random variables Y and Z • Y and Z are independent if the probabilities with which Y takes on its values are the same regardless of the value Z takes on • Notation – Y � Z Double Robustness, EPID 369, Spring 2007 8
2. Counterfactual Model Conditional expectation: For two random variables Y and Z, E(Y |Z = z) is the “average ” across all values of Y for which the corresponding value of Z is equal to z • E(Y |Z) is a function of Z taking on values E(Y |Z = z) as Z takes on values z • E{ E(Y |Z) } = E(Y ) (“average the averages ” across all possible values of Z) • E{ Y f(Z)|Z } = f(Z)E(Y |Z) – f(Z) is constant for any value of Z so factors out of the average • If Y � Z, then E(Y |Z = z) = E(Y ) for any z Double Robustness, EPID 369, Spring 2007 9
Page 1 and 2: Double Robustness in Estimation of
Page 3 and 4: 1. Introduction Simplest situation:
Page 5 and 6: Define: 2. Counterfactual Model Z =
Page 7: 2. Counterfactual Model Problem: Do
Page 11 and 12: 2. Counterfactual Model What is bei
Page 13 and 14: 2. Counterfactual Model In contrast
Page 15 and 16: 3. Adjustment by Regression Modelin
Page 21 and 22: 4. Adjustment by Inverse Weighting
Page 23 and 24: 4. Adjustment by Inverse Weighting
Page 25 and 26: 5. Doubly Robust Estimator Modified
Page 27 and 28: 5. Doubly Robust Estimator �µ1,D
Page 29 and 30: 5. Doubly Robust Estimator Scenario
Page 31 and 32: 5. Doubly Robust Estimator Result:
Page 33 and 34: E 5. Doubly Robust Estimator �
Page 35 and 36: Summary: If OR 5. Doubly Robust Est
Page 37 and 38: 6. Some Interesting Issues Issue 1:

Double Robustness in Estimation of Causal Treatment Effects

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