ADjusting for variables in regression model

Dear all,

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?
You can make a within group judgement about gender but cannot make a between group statement.

In short, gender is not controlled for between all levels of treatment and thus is a nuisance variable in your design.

So if there is an effect due to a variable that is not controlled for and that variable is a significant source of variance, then its lack of inclusion will likely result in a biased estimate.
So factor X (additional treatment in a subset of group A patients) is definitely a nuisance variable because because it cannot be controlled for between two levels of treatment (A and B) but it also improves survival.
I didn't understand completely. Do you suggest it should be included or excluded? Estimate is biased in both cases whether it is included or excluded?
there is an interaction between a treatment level and factor x, BUT you cannot say that the effect between treatment a and b is not due to factor x.

You know gender increases survival rate in one treatment but you cannot know if the reason that the survival rates between the two treatments is because of gender.

So you can make a within group claim but cannot make a between group.
I recommend on including it in the model and then discussing it in the limitations section of your manuscript. The interaction is a valuable contribution and should be noted.