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I am trying to estimate the causal effects of a continuous treatment variable $T$ on a binary outcome $Y$. I have a set of nuisance variables $X$ that I know effect both $T$ and $Y$. I have read about a fair number of techniques but none seem to be set up to handle this situation. Matching, Propensity Scores, Double/Debiased, etc. they all seem to either be examples of having a non-continuous treatment or both a continuous treatment and outcome.I have a large data set but for the most part the $T$ is not randomly assigned. Recently I have been able to randomly assign a plus/minus 16% shift in whatever the baseline assigned $T$ is, I'm unsure though how I can use this fact to my advantage although intuitively the randomness seems like it should help.

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I am not aware of any causal inference methods that care whether the outcome is continuous or binary, this should not matter (e.g. $E[Y|A,X] = P(Y=1|A,X)$). Continuous treatments definitely require specialized methods. You could use this method: https://code.nimahejazi.org/txshift/ which estimates the causal effect of various shift interventions on the treatment (e.g. what would happen if everyone's treatment dose were increased by 20%).

Paper https://onlinelibrary.wiley.com/doi/10.1111/biom.13375

Also, $T$ need not be randomized. That is the main point of causal inference: to estimate causal effects using observational data.

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  • $\begingroup$ Thanks for the answer! If $T$ is partially randomized and I want to estimate counterfactual in the way that you are saying "(e.g. what would happen if everyone's treatment dose were increased by 20%)." Is it reasonable to just run a logistic regression and have it predict $Y$ at different dosages? $\endgroup$
    – Walnut
    Commented Aug 2, 2021 at 16:07
  • $\begingroup$ You want to use the specialized causal inference method/package I mentioned in my post. Behind the scenes, it does things like compute the estimated regression function at different doses, but it does it in a nonparametric way that allows for valid and efficient statistical inference. If you want to do some exploratory analysis for yourself, you can just change everyone's treatment value by 20% and get the regression values at the new doses. You can then take the empirical average of them for example, and plot these averages as a function of the increase dosage amount (e.g. 10%, 20%, 30%, 50%) $\endgroup$
    – user327671
    Commented Aug 4, 2021 at 3:22

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