AIPW (Augmented Inverse Probability Weighting)

Definition

$\hat{\tau}_{AIPW} = \frac{1}{n} \sum_{i=1}^{n} \left[ \hat{\mu}_1(X_i) - \hat{\mu}_0(X_i) + \frac{T_i(Y_i - \hat{\mu}_1(X_i))}{\hat{e}(X_i)} - \frac{(1-T_i)(Y_i - \hat{\mu}_0(X_i))}{1 - \hat{e}(X_i)} \right]$

• $\hat{\mu}_t(X)$: Outcome model ($E[Y|T=t, X]$)
• $\hat{e}(X)$: Propensity score model

Intuitive Understanding

A combination of IPW and outcome regression. It enjoys Double Robustness: it remains a consistent estimator as long as either one of the two models is correctly specified.

• IPW term: Propensity-based correction
• Augmentation term: Residual correction from the outcome model

Double Robustness

$\hat{e}(X)$ correct	$\hat{\mu}(X)$ correct	AIPW consistent?
O	O	O
O	X	O
X	O	O
X	X	X

Project Application

• AIPW ATE: $24 (95% CI: [-$56, $104])
• Positivity violation (PS AUC 0.989) causes weight explosion at extreme PS values
• More stable on the trimmed sample

Related Concepts

• IPW
• Doubly Robust Estimator
• Double Machine Learning

Key Papers

• Robins, Rotnitzky & Zhao (1994). Estimation of regression coefficients when some regressors are not always observed
• track2_report