from the parameter MSM estimates which is $E[Y_{Z_1,\ldots,Z_t}-Y_{Z'_1,\ldots,Z'_t}]$.

Proposed method {#chap3:prop_meth}
---------------

We make the same four assumptions used by MSM (See Section \[chap3:msm\]). We impose a further fifth assumption of strict monotonicity in that $$\label{monotonicity}
    \text{If}\,Z_1\leq Z'_1,Z_2\leq Z'_2,\ldots,Z_{t-1}\leq Z'_{t-1},\,Z_i\neq Z'_i\,\text{for any }i,\,\text{then}\,S_{Z_1,\ldots,Z_{t-1}}\geq S_{Z'_1,\ldots,Z'_{t-1}}.$$ As a consequence, we have for example, when $t=2$, if $S_0=0$ then $S_1=0$ and if $S_1=1$ then $S_0=1$. This means that we rule out the possibility of a subject who does not receive a negative wealth shock and dies but would survive if having received a neagtive wealth shock. Conversely, if a subject survives after having received a negative wealth shock, we rule out the possibility that this same subject would die if he or she did not receive a negative wealth shock.

Our proposed method then estimates $E[Y_{Z_1,\ldots,Z_t}-Y_{Z'_1,\ldots,Z'_t}|S_{Z_1,\ldots,Z_{t-1}}=S_{Z'_1,\ldots,Z'_{t-1}}=1]$ by imputing the survival status of each subject at the current time $t$ and then combine the imputed counterfactual survival status together with the observed survival status to determine which principal stratum a subject belongs to. We then use a slightly modified PENCOMP to impute the counterfactual outcomes among the potentially surviving subjects to account for the bias due to confounding by indication. This approach is doubly 