I would recommend study of Paul Rosenbaum’s work on using an observational study to approximate an RCT. It would complement the recommendations of @f2harrell noted below.
Since you do not control the treatment allocation mechanism, you bear a burden of proof to demonstrate that no explanations other than the hypothesized treatment, are plausible. You will need to think very hard about the potential biases and confounding variables in order to create plausibly exchangeable groups for comparison with the data you have available.
As for actual regression models, I’d give strong consideration to logistic models, with log linear probability models also worth considering. See this thread:
Some related threads and posts
- Regression Modelling Strategies by Frank Harrell
- Ordinal Models for clinical trial design by Frank Harrell
- Links to recent papers on design of observational studies from statistical POV:
Rosenbaum, P. R. (2015). How to see more in observational studies: Some new quasi-experimental devices. Annual Review of Statistics and Its Application, 2, 21-48. (PDF)
Greenland, S. (2005). Multiple-bias modelling for analysis of observational data. Journal of the Royal Statistical Society: Series A (Statistics in Society), 168: 267-306. link
Branson, Z. (2021). Randomization Tests to Assess Covariate Balance When Designing and Analyzing Matched Datasets. Observational Studies 7(2), 1-36. doi:10.1353/obs.2021.0031. link