## 7 Discussions  
In randomized control trials, it is reasonable to assume there are no systematic differences in covariates between treatment groups. Therefore, the causal effect of a treatment can be directly estimated by comparing the observed outcome in the active treatment group versus that in the control group. To evaluate the treatment effect from observational data, additional effort must be made to remove the impact of confounding variables, which are related to both the treatment and outcome. For observational data, making causal inference needs to elect an appropriate estimand to start. The choice of estimands relies on the research objectives. For example, if we want \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Estimand**} & \multirow{2}{*}{**Methods**} & **Risk Differ-** & \multirow{2}{*}{**Standard Error**} & \multirow{2}{*}{_p-Value_} & **95\% CI of Risk Difference** \\ \cline{3-5}  & & **Differ-** & & & **Error** \\ \hline \multirow{2}{*}{**Crude**} & Regression without covariates & \multirow{2}{*}{0.074} & \multirow{2}{*}{0.013} & \multirow{2}{*}{\textless{0.0001}} & \multirow{2}{*}{[0.048, 0.099]} \\ \hline \multirow{2}{*}{**CATE**} & Traditional regression with all covariates & \multirow{2}{*}{0.072} & \multirow{2}{*}{0.021} & \multirow{2}{*}{\textless{0.0001}} & \multirow{2}{*}{[0.031, 0.114]} \\ \hline \multirow{4}{*}{**ATE**} & G-computation & \multirow{2}{*}{0.059} & \multirow{2}{*}{0.014} & \multirow{2}{*}{\textless{0.0001}} & \multirow{2}{*}{[0.032, 0.087]} \\ \cline{2-2} \cline{5-5}  & PS stratification (5 strata) & & & & \\ \cline{2-2} \cline{5-5}  & PS weighting: IPTW & \multirow{2}{*}{0.053} & \multirow{2}{*}{0.010} & \multirow{2}{*}{\textless{0.0001}} & \multirow{2}{*}{[0.035, 0.070]} \\ \cline{2-2} \cline{5-5}  & Doubly robust & \multirow{2}{*}{0.059} & \multirow{2}{*}{0.014} & \multirow{2}{*}{\textless{0.0001}} & \multirow{2}{*}{[0.032, 0.087]} \\ \hline \multirow{4}{*}{**ATT**} & G-computation & \multirow{2}{*}{0.056} & \multirow{2}{*}{0.014} & \multirow{2}{*}{\textless{0.0001}} & \multirow{2}{*}{[0.028, 0.085]} \\ \cline{2-2} \cline{5-5}  & PS stratification (5 strata) & & & & \\ \cline{2-2} \cline{5-5}  & PS weighting & \multirow{2}{*}{0.053} & \multirow{2}{*}{0.014} & \multirow{2}{*}{\textless{0.0001}} & \multirow{2}{*}{[0.025, 0.081]} \\ \cline{2-2} \cline{5-5}  & Doubly robust & \multirow{2}{*}{0.062} & \multirow{2}{*}{0.016} & \multirow{2}{*}{\textless{0.0001}} & \multirow{2}{*}{[0.029, 0.094]} \\ \hline \multirow{2}{*}{**ATM**} & PS matching & \multirow{2}{*}{0.049} & \multirow{2}{*}{0.017} & \multirow{2}{*}{0.0035} & \multirow{2}{*}{[0.016, 0.08]} \\ \hline \multirow{4}{*}{**ATO**} & Overlap weighting with doubly robust & \multirow{2}{*}{0.059} & \multirow{2}{*}{0.012} & \multirow{2}{*}{\textless{0.0001}} & \multirow{2}{*}{[0.031, 0.083]} \\ \cline{2-2} \cline{5-5}  & Overlap weighting with doubly robust & \multirow{2}{*}{0.059} & \multirow{2}{*}{0.012} & \multirow{2}{*}{\textless{0.0001}} & \multirow{2}{*}{[0.033, 0.085]} \\ \cline{1-1} \cline{5-5}  & & & & & \\ \hline \hline \end{tabular}  
* Note: The results from using SAS are presented whenever there is a suitable procedure available to use in SAS. For the doubly robust estimator for ATT and overlapping weighting, the PSWeight package in R was used.  
\end{table} 
Table 2: Analysis results for the binary outcome of passing the first month: The point estimates on risk difference in having 30-day mortality between RHC and No-RHC groups, associated standard errors, _p_-values, and confidence intervals for various methods. All methods except for the crude model include all available covariates. to make an inference about the effect on outcomes for all subjects, then the ATE should be the preferred estimand. If the population of interest is among subjects who selected treatment, then the ATT should be used. If one is interested in comparing causal effects estimated from an observational study to that from a randomized trial, then ATO might be more appropriate.