## 6 Example  
This example uses the Study to Understand Prognoses and Preferences for Outcomes and Risks of Treatment (SUPPORT), which was a multi-center observational study on seriously ill and hospitalized patients to examine the effectiveness of Right Heart Catheterization (RHC) in the initial care of critically ill patients [70]. The dataset pertains to Day 1 of hospitalization. This example includes all 5735 subjects who were admitted to an ICU in the first 24 h after entering the study. RHC was coded as present if it was performed, and there were 2184 patients who received RHC. The outcome of interest is patients' chance of surviving by the end of first month. The original analysis by [PERSON] et al. [70] used a binary logistic model to obtain PS to match RHC patients with No-RHC patients. Their results provided evidence that RHC patients had a decreased survival time. Here, we apply the reviewed methods (i.e., regression with all covariates, g-computation, weighting, stratification, matching, and AIPW) and describe important steps for their implementation. In the Supplementary document, we provide code to replicate the analyses. All the covariates (-50) are included, as suggested by [PERSON] et al. [70]. The logistic regression is used to determine the PS, i.e., the probability of receiving RHC for each subject. The effectiveness of using PS to account for major covariates' imbalance is tested for differences between the two RHD groups (with and without RHC). The PS distributions for the two RHD groups are displayed in Figure 1. The graph shows apparent differences in the PS distributions. There are many subjects with small PS values (around 0) and large ones (close to 1), indicating some covariate combinations were highly predictive of receiving RHC. The patients whose PS and RHC status do not agree may be atypical but received large weights. The love plot in Figure 2 compares the standardized differences of selected covariates included in the PS model: (1) blue for the original data in the regular regression; (2) purple for ATT weighting; (3) red for ATE weighting; (4) black for matching; (5) green for overlap weighting. We can see that the mean differences for most covariates are quite large in the traditional regression but are reduced with the adjustment methods. For all the methods, the largest SMD in absolute value is less than the recommended limit of 0.1 [48]. These indicate the effect of those methods on reducing the differences in covariates. The SMD with overlap weighting is the smallest (very close to zero in green). PS matching and weighting seem to remove a greater portion of systematic differences between the treated and untreated subjects compared with many other approaches, such as stratification and traditional regression, which is in agreement with [PERSON]'s 2009 paper [71]. Among weighting approaches, overlap weighting seems to perform the best in terms of having the smallest mean difference, as expected. Figure 1: Histogram of the propensity score distributions for the two treatment groups. The blue and red curved lines are the probability density curves of the propensity score distributions for the two groups. The endpoint was surviving up to one month post-ICU admission. A binomial model with the logit link is used to obtain the risk difference and odds ratio for comparing the treatment effect of RHC versus No-RHC. We present the point estimates on the risk difference, standard errors, \(p\)-values, and corresponding confidence intervals for various methods. The results are displayed in Table 2, where the content is first arranged by the type of estimands, then by confounding adjust methods. In the Estimand column, we list the crude effect estimate, ATE, ATT, ATM, and ATO. For the crude effect, no covariates were included in the logistic model except for the treatment variable, and SAS's LOGISTIC procedure was used.