I'm comparing two populations of oncologic patients that were treated with two different regimens in two different hospitals, group A of patients from hospital A received one regimen exclusively (regimen A) and group B of patients from hospital B received another regimen exclusively (regimen B). Analysis is retrospective on prerecorded data, therefore there is huge bias here. My goal would be to obtain vague estimation is there an evident difference regarding survival of these patients depending on what regimen was used.

I decided to use Cox regression model and included regimen type, age and sex. A subset of population A was also treated with another factor X that improves survival, population B was not treated with X additionally. My question would be - does adjusting analysis for factor X (including X as independent variable for estimation of survival of population A+B) biases estimation of A/B effect on survival?

I guess it does because all patients from B are inappropriately weighted as 0 regarding factor X in a regression model. X improves survival in a subset of group A and I have a feeling that analysis is providing regimen B with unfair advantage when adjusting for this factor. Is this true? Or such analysis is unbiased and actually needed for objective comparison of these groups?