Compare ORs between strata with log(OR) difference or use interactions?

I ran a nice model to predict mortality in a sub-cohort of patients with obesity who underwent surgery and presented that in a meeting. Upon writing the paper, the coauthors wanted to know if these predictors will predict mortality similarly in patients without obesity who undergo the same surgery. Thus, we used propensity score to find age- and sex-matched cohort to compare these predictors.

I wonder if the right way to compare the OR (e.g. female vs males) in the obesity vs no-obesity cohorts is to do OR1-OR2, or log(OR1)-log(OR2) and then exponentiate? In my results, females had a significant higher risk of mortality if they were obese, vs no effect if they are non-obese. In such a case, does it matter to compare ORs whether significant or not?

Some suggested that I should measure ORs for all the cohort (Obese+noneObese) and then add an interaction effect of obesity to each predictor in the model. The results give me no significant interaction across the predictors but only a few significant main effects (e.g. now female sex is significant, for all patients, with no significant interaction). I don’t know how to interpret this now and I’m not sure if this serves the reason why we started the comparison in the first place, but this certainly added a great AIC penalty to my model. I wonder if comparing ORs has merit in this case or if a better method is established to answer the study’s question better than what I did?

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