## 3 Confounding Adjustment Methods  
Once the estimand is selected, the next task is to express the estimand in terms of the observed data, referred to as 'identification'. Identification often relies on the assumptions of causal consistency and exchangeability (or ignorability) with positivity [19; 20]. As described above, the causal consistency assumption links the potential outcomes to the observed outcomes. Exchangeability stipulates that the potential outcomes and treatment are independent marginally or conditionally. Conditional exchangeability is mathematically expressed as \(E[Y(a)\mid X]=E[Y(a)\mid A=a,\,X]\) for all values \(a\) of treatment. In other words, exchangeability is the assumption that there are no unobserved common causes of the exposure and outcome. For exchangeability to be mathematically well defined, we further assume that the probability of receiving a treatment is non-zero for every covariate combination relevant for exchangeability. This assumption is referred to as positivity and written as \(\text{Pr}(A=a\mid X)>0\)[21].  
In an RCT, marginal exchangeability is met by design and covariates are balanced in expectation. However, chance imbalances can still occur, particularly when a sample size is small. Accounting for chance imbalances by covariates strongly predictive of the outcome can provide more precise estimates of causal effects [22]. Here, the goal of covariate adjustment is to improve precision and power in estimating causal effects.  
If a study is not randomized, there can be systematic imbalance in covariates for different treatment arms (i.e., marginal exchangeability does not hold). Instead, one can assume there is a sufficient adjustment set for confounding (i.e., conditional exchangeability). Confounding adjustment can then be achieved via different ways, such as traditional outcome regression, g-computation, PS adjustment, and doubly robust approaches.  
Outcome regression methods were first developed to estimate conditional effects accounting for covariate imbalance between treatment arms [23; 24]. However, standard adjustment by parametric regression models is sensitive to model mis-specification [8]. G-computation has been proposed as a way to estimate the marginal causal effect using an outcome regression model [25]. G-computation allows for a treatment effect to be different for different levels of the covariates, and it is also relatively robust to model mis-specification if there is no unmeasured confounding [26]. Alternatively, propensity score (PS) methods that use PS in different ways to control confounding include matching [10; 27], stratification [28; 29], weighting, and conditional adjustment using PS as a covariate [30]. For example, the inverse probability weights (IPW) can be applied to subjects in each treatment arm to balance the covariate distributions, and the comparison is made between the weighted outcomes [17; 23; 31; 32; 33; 34]. Researchers [35] used a few large-scale cardiovascular observational studies to compare the performance of a conventional regression with PS methods. PS is the most widely applied causal inference tool for observational studies and it has theoretical advantages over conventional confounding adjustment using outcome regression. Another option is the doubly robust approach that combines PS methods and outcome regression. One of the doubly robust methods is the augmented IPW estimator, which can provide unbiased estimates if one of the models is mis-specified. We discuss each approach in more detail below.