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We have a longitudinal experiment, with interventions $\bar{A}=\{A_1,A_2,\ldots,A_K\}$ and outcomes $\bar{Y}=\{Y_1,Y_2,\ldots,Y_K\}$. Using some sequential conditional exchangeability assumptions, it should be possible to derive g-computation formula to estimate the causal effect of a treatment strategy $\bar{a}$.

Conditional exchangeability commonly assumed for a longitudinal trial are

$Y_{k'}^{\bar{a}}\ \bot A_k\ |\ {\bar{A}}_{k-1}={\bar{a}}_{k-1},{\bar{Y}}_k$ for $k=0,1,\ldots,k'$

I.e., we need to condition on treatments and observations occurring prior to $A_k$.

The resulting g-computation is

$$ p({\bar{Y}}^{\bar{a}})\ =\ \prod_{k=1}^{K}{p\left(Y_k\ \middle|\ \bar{Y}_{k-1},\bar{A}_{k-1}\right)} $$

How can this be derived?

I’m able to derive the g-computation, but only if I assume

$Y_{k\prime}^{\bar{a}}\bot A_k|{\bar{A}}_{k-1}={\bar{a}}_{k-1},{\bar{Y}}_{k'-1}$ for $k=0,1,\ldots,k'$

I.e., condition on treatments occurring prior to $A_k$ but observations prior to $Y_{k'}^{\bar{a}}$. Thus, condition also on observations that occur after the treatment $A_k$ but before the observation $Y_{k'}$. Details are written up at https://doi.org/10.6084/m9.figshare.20406687. It remains that I cannot derive the commonly used g-computation from the commonly made sequential conditional exchangeability assumptions, and I would appreciate help.

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