I have a dataset where the main question of interest concerns the association between the use of a particular medication prior to surgery and the occurrence of a particular surgical adverse event - both are binary.

Unsurprisingly there are some potential confounders that could be associated with both the medication use and the outcome (e.g. age) that I would like to account for. These have been selected by the clinicians, but even though there are not very many (10 covariates with 400+ observations), the event rate is quite low (< 5%) and if I just fit a standard logistic regression model that includes all of the covariates, I get complete separation.

If this was a ‘normal’ modelling / variable selection problem I would use some regularisation method, but I’m not sure if that is the best approach when I am specifically interested in the adjusted effect of one particular covariate, and it is the set of adjustment covariates that I want to ‘select’ (in whatever sense).

Some of the covariates (e.g. type of surgery) are likely to be associated with the outcome but not with the medication use. Is it necessary to account for these in the regression model?

Any advice would be much appreciated!