rest of the differences between Scenario 1 versus 2 and 3 were to ensure the resulting simulated population would have enough deaths and subjects in the various different treatment regimes for the assumptions used by MSM and our proposed method to be valid. The difference between Scenario 2 versus 3 lie in $\beta_{VZ}$, $\alpha_{Y_1}$, $\alpha_{Y_0}$, $\beta_{Y_0Z_{00}}$, $\beta_{Y_0Z_{01}}$, $\beta_{Y_1Z_{11}}$, $\alpha_{Y_0Z_{00}}$, $\alpha_{Y_0Z_{01}}$, $\alpha_{Y_1Z_{11}}$, $\beta_{Y_{00}Z_{000}}$, $\beta_{Y_{00}Z_{001}}$, $\beta_{Y_{01}Z_{011}}$, and $\beta_{Y_{11}Z_{111}}$ where the values for Scenario 2 is about 10 times smaller compared to Scenario 3.

To calculate the true parameters, we used the generated population data (size 1 million), and then took:

1.  $\Delta_{1,0}=\bar{Y}_1-\bar{Y}_0$;

2.  $\Delta_{01,00}=\bar{Y}_{01}-\bar{Y}_{00}$ given $S_0=1$;

3.  $\Delta_{11,00}=\bar{Y}_{11}-\bar{Y}_{00}$ given $S_0=S_1=1$;

4.  $\Delta_{11,01}=\bar{Y}_{11}-\bar{Y}_{01}$ given $S_0=S_1=1$;

5.  $\Delta_{001,000}=\bar{Y}_{001}-\bar{Y}_{000}$ given $S_{00}=1$;

6.  $\Delta_{011,000}=\bar{Y}_{011}-\bar{Y}_{000}$ given $S_{00}=S_{01}=1$;

7.  $\Delta_{111,000}=\bar{Y}_{111}-\bar{Y}_{000}$ given $S_{00}=S_{11}=1$;

8.  $\Delta_{011,001}=\bar{Y}_{011}-\bar{Y}_{001}$ given $S_{00}=S_{01}=1$;

9.  $\Delta_{111,001}=\bar{Y}_{111}-\bar{Y}_{001}$ given $S_{00}=S_{11}=1$; and

10. $\Delta_{111,011}=\bar{Y}_{111}-\bar{Y}_{011}$ given $S_{01}=S_{11}=1$.

We measured performance using the empirical bias, root mean squared error (RMSE), 95% coverage, and the average 95% Confidence Interval (CI) length (AIL). 1000 simulations were used to estimate these quantities. Under each simulation, a simple random sample of 4,000 or 8,000 subjects was drawn from the target population data. All methods were then implemented on the sampled data to obtain the effect 