# Unbiased estimation of treatment regime contrast with time-varying treatment and outcome

I have some troubles finding a strategy to identify the causal effect I am looking for from my observed data. I am assuming the following DAG:

where $$Z$$ is the result of a coin toss (randomization), $$A$$ is the exposure of interest and $$Y$$ is the outcome of interest. I am assuming that $$A$$ is binary (0 or 1) and $$Y$$ is continuous. The estimand of interest is:

$$E\left(Y_6^{\bar{A} = \bar{1}} - Y_6^{\bar{A} = \bar{0}}\right)$$

where $$Y_6^{\bar{A}=\bar{1}}$$ denotes the value of $$Y$$ that would be observed at $$t = 6$$ if $$A$$ was set to $$1$$ at every considered point in time. $$Y_6^{\bar{A}=\bar{0}}$$ follows the same definition, but with $$A$$ being set to $$0$$ for all points in time. All variables are observed completely without any missing values or measurement error.

I do think it should be possible to estimate this effect in an unbiased manner given the data, but I am not sure how to do it. I have thought about using a mixed model, but that would not really give me the marginal estimate i am interested in.

Would a structural marginal model work? If yes, how would I apply one to this data? If anyone could point me in the right direction, that would be great.

• Yes, a marginal structural model would work for the causal structure provided. More generally, any of the g-methods would work for estimation of the parameter of interest. For a general introduction to g-methods see ncbi.nlm.nih.gov/pmc/articles/PMC6074945 Commented Nov 10, 2023 at 14:15
• Thanks, I will definitly look into it. Commented Nov 10, 2023 at 14:20