Double Robustness in Estimation of Causal Treatment Effects

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2. Counterfactual Model Challenge, again: Based on observed data (Yi, Zi, Xi), i = 1, . . . , n, estimate ∆ = µ1 − µ0 = E(Y1) − E(Y0) Observed sample means: Averages among those observed to receive treatment or control n� n� (1 − Zi)Yi n1 = Y (1) = n −1 1 i=1 ZiYi, Y (0) = n −1 0 i=1 n� Zi = # subjects observed to receive treatment i=1 n0 = n� (1 − Zi) = # observed to receive control i=1 Double Robustness, EPID 369, Spring 2007 10
2. Counterfactual Model What is being estimated? If we estimate ∆ by Y (1) − Y (0) • Y (1) estimates E(Y |Z = 1) = population mean outcome among those observed to receive treatment • E(Y |Z = 1) = E{Y1Z + Y0(1 − Z)|Z = 1} = E(Y1|Z = 1). . . • . . . which is not equal to E(Y1) = mean outcome if entire population received treatment • Similarly, Y (0) estimates E(Y |Z = 0) = E(Y0|Z = 0) �= E(Y0) • Thus , what is being estimated in general is E(Y |Z = 1) − E(Y |Z = 0) �= ∆ = E(Y1) − E(Y0) Double Robustness, EPID 369, Spring 2007 11
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: 2. Counterfactual Model Conditional
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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