Estimate Causal Effect of Countinuous Treatment on Binary Outcome

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.

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%).
Also, $$T$$ need not be randomized. That is the main point of causal inference: to estimate causal effects using observational data.
• 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? Commented Aug 2, 2021 at 16:07