## 7 Discussions
If there are not many strata, residual confounding within strata may cause bias. To mitigate this bias, we can increase the number of strata. The more strata used, the closer the stratification will be to the matching method. However, stratification may perform poorly when the data size is small by giving inconsistent results across different strata, as shown in our example. To choose the proper number of strata, we should consider both the need for confounding control and the need of having enough events. Previous research has shown that five strata may reduce confounding bias by up to 90% [35]. PS-based matching is simple to use and often performs well in most cases. It provides nice covariate balance among the matched subjects. However, it occasionally tends to provide imprecise estimates as a result of the fact that some data without matches are dropped. Matching can result in large variance in estimates if a great deal of data is taken out. There are a number of matching techniques in the literature but little guidance to how to select among them in practice; the primary advice seems to select the one with the best balance [23]. Multivariate matching with the Mahalanobis distance or coarsened exact matching [74] are competitive, if not preferable, to PS matching in their ability to reduce imbalance and estimation bias. PS weighting keeps all the data and often provides good balance among groups in most case [35]. When a covariate imbalance is large, the PS will be close to either 0 or 1, meaning that some subjects can have very large weights. It produces unbiased estimates but can have large variances if many subjects have large weights. Trimming can be employed in the case of many large weights. [PERSON] and [PERSON] (2018) [75] provided asymptotic theories for trimming, pointing out that trimming may introduce extra arbitrariness to the process while stabilizing PS weighting. Treating PS as a covariate in a regression model is very convenient to achieve and is efficient in balancing covariates. A disadvantage is that it requires that the regression is correctly specified [71]. Researchers have also demonstrated that confounding adjustment using PS removes less of the systematic differences if compared with other approaches, such as the PS matching and weighting [71]. Doubly robust estimators, like AIPW and TMLE, offer clear advantages over g-computation and PS methods [76; 77]. First, doubly robust estimators are unbiased as long as either the treatment model or outcome model is correctly specified. Second, doubly robust estimators are more efficient than IPW when the outcome model is not grossly mis-specified. Third, the variance estimator based on the influence curve has a closed-form. Lastly, some doubly robust methods allow for valid use of machine learning to estimate the PS and outcome models, unlike g-computation and PS methods [78; 79]. Overlap weighting does not involve issues related to large weight problems, unlike standard inverse probability weights, as overlap weights are bounded between 0 and 1. Overlap weighting can obtain the exact mean balance of any covariates and minimize the asymptotic variance, as shown in the example. The variance estimators of the OW estimator of the marginal treatment effect have a closed-form, whereas the bootstrap or simulation used to estimate the variances with non-linear estimators do not [37]. As an alternative to PS methods, covariate balancing can be achieved by multivariate reweighting methods such as entropy balancing [80]. It can exactly match the first, second, and possibly higher moments of specified covariates. These balance improvements can potentially reduce model dependence for the subsequent estimation of treatment effects. We have carried out a comprehensive review of common confounding adjusting approaches, including outcome regression models, g-computation, PS methods, and doubly robust methods. Estimands and target population should be considered in determining which methods produce the most valid results.