Double Robustness in Estimation of Causal Treatment Effects
Double Robustness in Estimation of Causal Treatment Effects
Double Robustness in Estimation of Causal Treatment Effects
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E<br />
5. Doubly Robust Estimator<br />
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{Z − e(X)}<br />
{Y1 − m1(X, α1)}<br />
e(X)<br />
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{Z − e(X)}<br />
�<br />
= E E<br />
{Y1 − m1(X, α1)} �<br />
e(X)<br />
� Y1,<br />
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X<br />
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{Z − e(X)} �<br />
= E {Y1 − m1(X, α1)} E<br />
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e(X) � Y1,<br />
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X<br />
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= E {Y1 − m1(X, α1)} {E(Z|Y1,<br />
�<br />
X) − e(X)}<br />
e(X)<br />
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�<br />
{E(Z|X) − e(X)}<br />
= E {Y1 − m1(X, α1)}<br />
e(X)<br />
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{e(X) − e(X)}<br />
= E {Y1 − m1(X, α1)} = 0<br />
e(X)<br />
(1) uses no unmeasured confounders<br />
<strong>Double</strong> <strong>Robustness</strong>, EPID 369, Spr<strong>in</strong>g 2007 30<br />
(1)
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