Double Robustness in Estimation of Causal Treatment Effects

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5. Doubly Robust Estimator Recap: ∆ = E(Y1) − E(Y0) • Estimator for ∆ based on regression modeling requires correct postulated regression model • Estimator for ∆ based on inverse propensity score weighting requires correct postulated propensity model Modified estimator: Combine both approaches in a fortuitous way Double Robustness, EPID 369, Spring 2007 24
5. Doubly Robust Estimator Modified estimator: n� � �∆DR = n −1 −n −1 i=1 i=1 = �µ1,DR − �µ0,DR ZiYi e(Xi, � β) − {Zi − e(Xi, � β)} e(Xi, � β) m1(Xi, �α1) n� � (1 − Zi)Yi 1 − e(Xi, � β) + {Zi − e(Xi, � β)} 1 − e(Xi, � β) m0(Xi, � �α0) • e(X, β) is a postulated model for the true propensity score e(X) = E(Z|X) (fitted by logistic regression ) • m0(X, α0) and m1(X, α1) are postulated models for the true regressions E(Y |Z = 0, X) and E(Y |Z = 1, X) (fitted by least squares ) Double Robustness, EPID 369, Spring 2007 25 �
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 and 8: 2. Counterfactual Model Problem: Do
Page 9 and 10: 2. Counterfactual Model Conditional
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: 4. Adjustment by Inverse Weighting
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:

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