similarly, $X_{Z_1=0,Z_2=0}=X_{00}$, $X_{Z_1=0,Z_2=1}=X_{01}$, and $X_{Z_1=1,Z_2=1}=X_{11}$ for time-varying covariates under the various treatment regimes; and $S_{Z_1=0}=S_0$, $S_{Z_1=1}=S_1$ for survival states. Subjects who die at time $t$ have structurally missing data for outcomes and covariates i.e., $S_0=0$ implies that $Y_{00}=Y_{01}=NA$ and $X_{00}=X_{01}=NA$, while $S_1=0$ implies that $Y_{11}=NA$ and $X_{11}=NA$, where ‘NA’ indicates a structurally missing observation.

Marginal structural model \[chap3:msm\]
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To estimate the causal effect for confounding by indication and censoring by death problems, MSM makes the following assumptions. First, MSM assumes that $$\label{chap3:surv_mod}
    P(S_{z_1,\ldots,z_{t-1}}|z_1,\ldots,z_{t-1},y_{z_1},\ldots,y_{z_1,\ldots,z_{t-1}},x_{z_1},\ldots,x_{z_1,\ldots,z_{t-1}},w_1,\ldots,w_{t-1},v)>0.$$ and $$\label{chap3:treat_mod}
    P(Z_t|z_1,\ldots,z_{t-1},y_{z_1},\ldots,y_{z_1,\ldots,z_{t-1}},x_{z_1},\ldots,x_{z_1,\ldots,z_{t-1}},w_1,\ldots,w_{t-1},v)>0$$ for any $z_t$ i.e. the probability of survival under treatment profile $z_1,\ldots,z_{t-1}$ and the probability of treatment allocation for time $t$ is bounded away from 0. This is an extension of the standard positivity assumption to allow that at least some subjects will survive under a given treatment regimen. Second, MSM assumes that there is no interference between subjects i.e. the potential outcome of subject $i$, $Y_{i,Z_1,\ldots,Z_t}=Y_{i,z_1,\ldots,z_t}$, is independent of whatever treatment regimen subject $j$ is allocated to $i\neq j$. Third, MSM assumes no unmeasured confounding and sequential randomization condition $$Y_{Z_1,\ldots,Z_t}\bot Z_t|z_1,\ldots,z_{t-1},y_{z_1,\ldots,z_{t-1}},\ldots,y_{z_1},x_{z_1,\ldots,z_{t-1}},\ldots,x_{z_1},w_1,\ldots,w_{t-1},v.$$ Finally, MSM assumes that the model specifications for Equations \[chap3:surv\_mod\], \[chap3:treat\_mod\], and $$Y_{z_1,\ldots,z_t}|z_1,\ldots,z_{t},y_{z_1,\ldots,z_{t-1}},\ldots,y_{z_1},x_{z_1,\ldots,z_{t-1}},\ldots,x_{z_1},w_1,\ldots,w_{t-1},v$$ are correct.

With these assumptions in place, $E[Y_{z_1,\ldots,z_t}-Y_{z'_1,\ldots,z'_t}]$ 