Inverse probability weighting

${\displaystyle {\hat {\mu }}_{a,n}^{IPWE}={\frac {1}{n}}\sum _{i=1}^{n}Y_{i}{\frac {\mathbf {1} _{A_{i}=a}}{{\hat {p}}_{n}(A_{i}=a|X_{i})}}}$

1. ${\displaystyle \mu _{a}=\mathbb {E} \{Y1_{A=a}/p(A|X)\}}$ where ${\displaystyle p(a|x)=P(A=a,X=x)/P(X=x)}$
2. construct ${\displaystyle {\hat {p}}_{n}(a|x)}$ or ${\displaystyle p(a|x)}$ using any propensity model (often a logistic regression model)
3. ${\displaystyle {\hat {\mu }}_{a,n}^{IPWE}=n^{-1}\Sigma _{i=1}^{n}Y_{i}1_{A_{i}=a}/{\hat {p}}_{n}(A_{i}|X_{i})}$

With the mean of each treatment group computed, a statistical t-test or ANOVA test can be used to judge difference between group means and determine statistical significance of treatment effect.

1. Consistency: ${\displaystyle Y=Y^{*}(A)}$
2. No unmeasured confounders: ${\displaystyle \{Y^{*}(0),Y^{*}(1)\}\perp A|X}$
   • Treatment assignment is based solely on covariate data and independent of potential outcomes.
3. Positivity: ${\displaystyle P(A=a|X=x)>0}$ for all ${\displaystyle a}$ and ${\displaystyle x}$

The Inverse Probability Weighted Estimator (IPWE) can be unstable if estimated propensities are small. If the probability of either treatment assignment is small, then the logistic regression model can become unstable around the tails causing the IPWE to also be less stable.

${\displaystyle {\hat {\mu }}_{a,n}^{AIPWE}={\frac {1}{n}}\sum _{i=1}^{n}{\Biggl (}{\frac {Y_{i}1_{A_{i}=a}}{{\hat {p}}_{n}(A_{i}|X_{i})}}-{\frac {1_{A_{i}=a}-{\hat {p}}_{n}(A_{i}|X_{i})}{{\hat {p}}_{n}(A_{i}|X_{i})}}{\hat {Q}}_{n}(X_{i},a){\Biggr )}}$

1. Construct regression estimator ${\displaystyle {\hat {Q}}_{n}(x,a)}$ to predict outcome ${\displaystyle Y}$ based on covariates ${\displaystyle X}$ and treatment ${\displaystyle A}$
2. Construct propensity estimate ${\displaystyle {\hat {p}}_{n}(A_{i}|X_{i})}$
3. Combine in AIPWE to obtain ${\displaystyle {\hat {\mu }}_{a,n}^{AIPWE}}$