## 2 Causal Framework and Estimands  
Assume that there are \(N\) subjects and, for each subject, there is a binary treatment indicator \(A_{i}\) (\(i=1,\ 2,\ \dots,\ N\)), where \(A_{i}=0\) for the control and \(A_{i}=1\) for the active treatment. The observed endpoint variable is \(Y_{i}\), and \(Y_{i}(0)\) denotes the potential outcome under the control (\(A_{i}=0\)) for subject \(i\), whereas \(Y_{i}(1)\) is for the potential outcome under the active treatment (\(A_{i}=1\)). There are \(p\) covariates \(X_{i}=(X_{11},X_{2},\dots,\ X_{ip})\) for subject \(i\), such as the baseline characteristics, demographic features, risk factors, etc. The covariates can be binary, categorical, or continuous. The individual treatment effect is the difference between the two potential outcomes for a subject, \(Y_{i}(1)-Y_{i}(0)\). The causal consistency assumption relates an observed outcome to the potential outcome [11]. \[Y_{i}=A_{i}Y_{i}(1)+(1-A_{i})Y_{i}(0),\]  
which states that the observed response \(Y_{i}\) is equal to the potential outcome with a treatment level that matches the actual treatment level. It is not possible to observe both \(Y_{i}(1)\) and \(Y_{i}(0)\) for a single individual. Therefore, the individual treatment effect is not identifiable, and, instead, we focus on the causal effects averaged over subjects. A commonly used population-level treatment effect is the average treatment effect (ATE), which is defined as the expected individual difference in potential outcomes as below:  
\[\text{ATE}=E[Y(1)-Y(0)].\]  
The ATE estimand gives the average effect of treatment in the population and is most relevant when we want to compute an average effect estimate that summarizes the effect for all members in the target population. The treatment effect may be heterogeneous if it affects individuals differently. In this case, we can divide the population into subsets (e.g., male versus female) and contrast the average effects by subsets. This type of ATE is called the conditional average treatment effect (CATE) and is conditioned on covariates in the subset, as follows:  
\[\text{CATE}=E[Y(1)-Y(0)|X=x].\]  
In some scenarios, average treatment effects for subpopulations related to treatment are of interest. Two common subpopulations are the subjects in the active treatment group and the subjects in the control group, which are the basis for the average treatment effect for the treated (ATT) and the average treatment effect for the control (ATC), as follows:  
\[\text{ATT}=E[Y(1)-Y(0)|A=1],\]\[\text{ATC}=E[Y(1)-Y(0)|A=0].\]  
ATT is the estimand that is most relevant when we want to evaluate the effect of a treatment among those who are treated, whereas ATC is most relevant when we want to consider the cost incurred by subjects who are not given treatment. The same approaches can be used to estimate the ATT and ATC. For this reason, we only discuss the ATT hereafter. Moreover, if a matching method is used to pair subjects from the active treatment group to those in the control group, we have a matched subpopulation that may be different from the target (such as the whole or treated) population. The matching approach creates matched sets of treated and untreated subjects who have similar values of PS. The response variable is then compared between the treated and untreated subjects in the matched set. The corresponding estimand is called the average treatment for the matched set (ATM), as follows:  
\[\text{ATM}=E[Y(1)-Y(0)|M=1],\]  
where \(M=1\) for matched observations. Recently, another subpopulation is considered to be of interest, which is the set with the most overlap in the observed covariates between the control and treated [16; 17; 18]. This subpopulation contains subjects that are eligible to be recruited and assigned to either treatment arm with a similar probability, and thus may more closely mimic a population enrolled in a RCT [17]. The estimand for the overlap population is called ATO (Web appendix of [PERSON] et al.