# Compare effects of a treatment across groups

I am trying to conduct a retrospective study to understand the effect of the presence of heart disease (N and Y) on binary categorical outcomes of certain types of surgeries (let's call them A, B, C). I'd also like to be able to compare the magnitude of these effects across surgeries.

If I do an independent analysis within each surgery group (e.g. perform Propensity Score Matching on heart disease within patients receiving surgery A, find odds ratio of heart disease), my gut feeling is that the odds ratios are not comparable across these groups since there was no matching performed across groups.

Main Q: How should I go about yielding effect magnitudes that are comparable?

I've tried treating the above scenarios as effectively six different treatments (NA, YA, NB, YB, NC, YC).

I've looked into matching across multiple treatment groups from: A Tutorial on Propensity Score Estimation for Multiple Treatments Using Generalized Boosted Models (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3710547/). However, the primary metric estimated is the Average Treatment Effect (ATE) which is defined as $$E(Y [t′]) − E(Y [t″]) = \mu_{t'} - \mu_{t''}$$ where $$Y[t'']$$ is the outcome variable of interest under a different treatment $$t''$$ for patients who were originally treated under $$t`$$. So in essence, I'd like to compare $$\mu_{NA} - \mu_{YA}$$ vs $$\mu_{NB} - \mu_{YB}$$ vs $$\mu_{NC} - \mu_{YC}$$

My understanding is that this would mainly be applicable for continuous or ordinal outcome variables, as then it would make sense to compare outcome means of these two treatments. For binary categorical outcomes the interpretation would be comparing frequencies of the "Yes" categorical outcome.

This would be in contrast to comparing something like odds ratios between each group. Secondary Q: Should ATE be used for binary categorical variables, or should a different estimand/method be used?

Thank you for any help!