## 3 Confounding Adjustment Methods
### Propensity Score Methods
#### 3.3.3 PS Weighting  
PS weighting is another important tool in causal inference that can be implemented in different ways.  
1. Inverse Probability of Treatment Weighting (IPTW) By IPTW, subjects are weighted by the inverse probability of being assigned to the treatment: \(w_{i}=\frac{A_{i}}{e_{i}}+\frac{1-A_{i}}{1-e_{i}}\). However, we often need to estimate the weights in observational studies, so \(e_{i}\) would be estimated. Additionally, notice that weights can become quite large when \(e_{i}\) is near zero. For variance estimation, the distinction between the known weights and estimated weights is important [32]. The covariate imbalance between treatment groups is reduced by the weighting approach, and so an outcome can be compared directly between treated and untreated subjects in the weighted data. The IPTW can be used to estimate the ATE (Table 1).
2. ATT Weighting Alternate weighting \(w_{i}=A_{i}+\frac{(1-A_{i})e_{i}}{1-e_{i}}\) is used to obtain ATT, that is \(w_{i}=1\) when a subject is in the treatment group and \(w_{i}=\frac{e_{i}}{1-e_{i}}\) when a subject is in the control group (Table 1). A key requirement for both IPTW weighting and ATT weighting is the positivity assumption, meaning that PS should not be too close to 0 or 1. When this assumption fails, a small number of highly influential weights may lead to unstable weighting estimators. A few alternative methods have been proposed, including trimming and the overlap weighting [44].
3. Overlap Weighting (OW) The overlap weight is the probability that a subject is assigned to the opposite group, i.e., \(1-e_{i}\) for a subject in the treated group and \(e_{i}\) for a subject in the control group. The OW focuses on the causal effects on the population with the most overlap in covariates between two treatment groups [18]. Estimation under unconfounded or ignorable treatment assignment is often hampered by a lack of overlap in the covariates, which may result in imprecise estimates and lead to estimators that are sensitive to the choice of specification. The OW procedure involves the following a few steps: (1) estimating the \(e_{i}\) from a model;  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Population** & **Estimand** & **Weight** \\ \hline
**Combined** & ATE & \(w_{1}=\frac{1}{e_{i}(x)},w_{0}=\frac{1}{1-e_{i}(x)}\) \\ \hline
**Treated** & ATT & \(w_{1}=1,w_{0}=\frac{e_{i}(x)}{1-e_{i}(x)}\) \\ \hline
**Overlap** & ATO & \(w_{1}=1-e_{i}(x),w_{0}=e_{i}(x)\) \\ \hline
**Matchable Treated** & ATM & \(w_{1}=\frac{\min\{e_{i}(x),1-e_{i}(x)\}}{e_{i}(x)}\), \(w_{0}=\frac{\min\{e_{i}(x),1-e_{i}(x)\}}{1-e_{i}(x)}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of various types of population, corresponding estimands, and weights for both treated and control subjects.  
(2) calculating the weights based on the estimated PS: \(\hat{w}_{1}=1-\hat{e}_{i}(x)\) if in the treatment group and \(\hat{w}_{0}=\hat{e}_{i}(x)\) if in the control group; (3) normalizing the weights so that the sum of the weights equals 1 within each group; (4) estimating the average treatment effect for the overlap population by the difference of the OW-weighted average outcomes between the groups.  
Compared to the traditional IPTW weights and associated trimming methods, OW has several advantages: (1) there are no extreme weights; (2) it gives minimum variance of the weighted estimator of causal effects among all balancing weights (including IPTW); (3) the exact mean balance of covariates is achieved when PS is estimated via a logistic regression; and (4) there is no need to choose an artificial cutoff point, as required by trimming.